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DISS. ETH NO. 20845
FUSION OF DIGITAL ELEVATION MODELS
A dissertation submitted to
ETH ZURICH
for the degree ofDoctor of Sciences
presented by
Charis Papasaika-HanuschM.Sc. in Mathematics, Vision and Learning, Ecole Normale Supérieure de Cachan, France
Dipl.-Ing. Rural and Surveying Engineering, National Technical University of Athens, Greece
born13th of August, 1978
citizen of Greece
accepted on the recommendation of
Prof. Dr. Konrad Schindler, examinerZurich, Switzerland
Dr. Emmanuel Baltsavias, co-examinerZurich, Switzerland
Prof. Dr. Ross Purves, co-examinerZurich, Switzerland
2012
IGP Mitteilungen Nr.109Fusion of Digital Elevation ModelsCharis Papasaika-Hanusch
Copyright c©2012, Charis Papasaika-Hanusch
Published by:Institute of Geodesy and PhotogrammetryETH ZURICHCH-8093 Zurich
All rights reserved
ISBN 978-3-906467-96-2
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AbstractDigital Elevations Models are one of the most important types of geodata. They are needed in alarge number of applications, ranging from virtual globes and visualization to engineering andenvironmental planning. DEMs of larger areas are usually generated either by photogrammet-ric processing of aerial and satellite images, InSAR interferometry, or laser scanning (mainlyfrom airborne platforms). Each sensing technology has its own strengths and weaknesses, andeven within one technology the variations in DEM quality are large. DEMs are available atdifferent scales from tailor-made local models to national and even global coverage. We areprimarily interested in large-scale national and global products, whose resolution, accuracy,error characteristics, and homogeneity vary a lot. In most cases, the DEM producers provideusers with information only on production technology, date of acquisition, and resolution, butonly with coarse accuracy measures that fail to capture the local variations in data quality –sometimes only a single global number.
In an ideal world, one would of course obtain the raw measurements and sensor modelsfrom all sensors, and merge them by fitting a single DEM to the entire set of heterogeneousobservations, along the way computing quality measures for every single height value. Un-fortunately, this is usually not feasible in practice. Thus, one resorts to the next best solution,namely to fuse DEMs from different providers into a higher-quality product, and estimate itsquality in the process.
DEM fusion – and its necessary prerequisite, fine-grained quality characterization of theinputs – has several benefits: improved accuracy, homogeneity and completeness, as well asfine-grained quality information for the final product. We deal only with 2-D surfaces in regulargrid format, which constitute the vast majority of large-scale DEMs (although our frameworkcould in principle be extended to TINs).
In this work we make two contributions: First, we develop a computationally efficient andflexible mathematical method for robust fusion of 2.5-D surface models. The formulation isgeneric and can be applied with any two input DEMs, independent of the sensor technologyand processing with which they were created, making it useful for practical applications; ittakes into account both prior information about plausible terrain shapes (in the form of a dic-tionary), and the local accuracy of the inputs, controlled by interpretable weights; and it posesthe complete fusion as a clean, convex mathematical optimisation problem that can be solvedto global optimality, and in which the influence of the input DEMs is controlled by an inter-pretable set of local fusion weights. Comparison of the proposed method is done with the datafitting and weighted average methods
Second, we propose a data-driven method, which allows one to derive local measures ofDEM quality (and thus also fusion weights) for each point or segment of a DEM, if no suchinformation is available. To this end we use as input geomorphological characteristics of the ter-rain (slope, aspect, roughness) which are derived directly from the DEMs, as well as optionallysemantic information such as land-cover maps. Using existing high-quality ground-truth DEMsas reference, we learn regression functions relating the available geomorphological character-istics to the DEM quality, which then allow one to estimate the local quality of a new DEM.
The proposed method is evaluated in detail with different datasets, and shows improvementin DEM quality, consistently over all the combinations of inputs.
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ZusammenfassungDigitale Höhenmodelle bilden die Grundlage für eine Vielzahl von Anwendungen, beginnendbei virtuellen Globen, über verschiedenste Visualisierungen bis hin zu Planungsaufgaben inverschiedensten Bereichen. Großflächige digitale Höhenmodelle können effizient mittels pho-togrammetrischen Prozessen aus Luft- und Satellitenbilddaten, mittels SAR (Synthetic Aper-ture Radar) Interferometrie oder mittels Lasersystemen generiert werden. Jede dieser Tech-nologien hat Stärken und Schwächen, selbst innerhalb einer Systemgruppe ist die Qualitätder gewonnen Daten mitunter höchst unterschiedlich. Digitale Höhenmodelle liegen in ver-schiedenen Maßstäben und verschiedensten Ausdehnungen, von der lokal projektbezogen übergroßmaßstäbig nationale bis hin zur globalen Anwendung, vor.
Die hier präsentierte Arbeit fokussiert vorwiegend auf großmaßstäbige nationale und glob-ale Produkte, wobei Auflösung, Genauigkeit, Fehlercharakteristik und Homogenität stark un-terschiedlich sind. In den meisten Fällen stellen die Datenlieferanten den Kunden lediglich In-formationen über die verwendete Technologie, das Erhebungsdatum und die Bodenauflösungzur Verfügung. Problematisch ist meist die Genauigkeitsinformation. Hier steht oft nur einegeneralisierte Genauigkeitsangabe zur Verfügung, welche die lokalen Variationen der Daten-qualität nicht ausreichend beschreiben kann. In manchen Fällen besteht diese Information auslediglich einer einzigen Zahl.
Im optimalen Fall, würden die Rohdaten und Sensormodelle aller Sensoren, welche Höhen-modellinformationen über ein Gebiet erfasst haben, aufgezeichnet. Diese Daten würden an-schließend in einem Arbeitsschritt dahingehend verbunden, dass ein digitales Höhenmodellin diese Datenwolke eingepasst wird. Zusätzlich ist es möglich für jede einzelne Höhenkoteein Qualitätsmaß zu ermitteln. Dieses Vorgehen ist jedoch aus verschiedensten Gründen nichtmöglich, da zum Beispiel die Höhenmodelle von verschiedenen Anbietern erstellt werden unddiese den Zugriff auf die entsprechenden Rohdaten verweigern. Die nächstbeste Lösung diesesProblems besteht in der Fusion von Höhenmodellen verschiedener Anbieter um ein qualitativhöherwertiges Produkt, inklusive einer umfassenden und aussagekräftigen Qualitätsbeschrei-bung zu erhalten.
Die Fusion von Höhenmodellen mit der notwendigen Analyse der Qualitätscharakteristikader Eingabedaten bringt viele Vorteile, wie die Qualitätsverbesserung, Homogenisierung undVervollständigung, wie auch die Bereitstellung detaillierter Qualitätsinformationen, mit sich.In dieser Arbeit wird ausschließlich mit 2.5D Oberflächen, welche eine homogene Rasterstruk-tur, wie sie die überwiegende Mehrheit von großmaßstäbigen Höhenmodellen aufweisen, gear-beitet. Der hier verfolgte Ansatz kann jedoch auch auf Dreiecksvermaschungen übertragenwerden.
In dieser Arbeit wurde eine recheneffiziente, flexible mathematische Methode zur robustenFusion von 2.5D Oberflächendaten entwickelt. Die Formulierung ist allgemeingültig und er-laubt es zwei Höhenmodelle ohne Berücksichtigung der verwendeten Erfassungstechnologieund Vorverarbeitung zu verschmelzen. Es werden sowohl Informationen über mögliche Ober-flächenstrukturen mittels einer Bibliothek, als auch lokale Qualitätsinformationen der zugrundeliegenden Höhenmodelle verwendet. Mittels physikalisch interpretierbarer Gewichtungen wirdder konvexe mathematische globale Optimierungsprozess gesteuert.
Im Weiteren wird eine datenbasierte Methode zur Bestimmung von lokalen Qualitätskri-
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terien aufgezeigt, welche notwendig ist um entsprechende Gewichtungen für die Fusion zubestimmen. Dies ist im Fall von fehlenden oder unzureichenden Qualitätsinformationen für dieeinzelnen Ausgangshöhenmodelle notwendig. Hierfür werden geomorphologische Beschrei-bungen wie Neigung, Ausrichtung und Rauigkeit verwendet, welche aus dem Höhenmodelldirekt bestimmt werden. Falls semantische Informationen, wie Oberflächenbedeckung zur Ver-fügung stehen, werden auch diese in den Prozess integriert.
Anhand existierender qualitativ hochwertiger Referenzdaten ist es möglich mittels Regres-sion die geomorphologischen Informationen mit der Qualität des Höhenmodells in Verbindungzu bringen. Dies ermöglicht es die lokale Qualität des neuen Höhenmodells zu bestimmen.
Die hier aufgezeigten Methoden werden mittels verschiedener Datensätze verifiziert undzeigen das Potential zur Verbesserung der Qualität von Höhenmodellen im Vergleich zu denzugrunde liegenden Daten.
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RésuméLes Modèles Numériques d’Élévation sont un des plus importants types de géodonnées. Ils sontnécessaires dans un grand nombre d’applications, comme les globes virtuels, la visualisation etla planification de l’environnement. MNEs de grandes zones sont habituellement générés soitpar le traitement photogrammétrique des images satellitaires, interférométrie ou télédétectionpar laser . Chaque technologie a ses propres avantages et faiblesses, et même au sein d’uneseule technologie, les variations de la qualité DEM sont grandes. Les MNEs sont disponiblesà différentes échelles, de modèles locaux et même nationaux à une couverture mondiale. Nousnous intéressons principalement dans des produits à grande échelle, nationales et mondiales,dont la résolution, la précision, les caractéristiques de l’erreur et l’homogénéité varient beau-coup. Dans la plupart des cas, les producteurs des MNEs fournissent les utilisateurs seulementavec des informations sur la technologie de production, la date d’acquisition et la résolution,et avec des mesures de précision grossière qui ne parviennent pas à saisir les variations localesdans la qualité des données.
Dans un monde idéal, on pourrait prendre les points brutes des MNEs initiales et les fusion-ner en un seul MNE et calculer des mesures de qualité pour chaque point. Malheureusement,cet option n’est pas réalisable dans la pratique. Ainsi, on recourt à la meilleure solution, qui estde fusionner des différents MNEs dans un produit de meilleure qualité, et d’estimer son qualité.
La fusion des MNEs – et sa condition sine qua non, la caractérisation de qualité des MNEsinitiales – présente plusieurs avantages: une meilleure précision, homogénéité des données,ainsi que des informations sur la qualité du produit final. Nous traitons uniquement des surfaces2.5-D sous forme de grille régulière, qui constituent la grande majorité des MNEs (même sinotre cadre pourrait en principe, être étendu aussi pour des TIN).
Dans ce travail, on a fait deux contributions: on a développé une méthode mathématiquesouple et efficace pour la fusion robuste des MNEs 2.5-D. La formulation est générique et peutêtre appliquée avec n’importe quelles MNEs d’entrée, ce qui est utile pour des applicationspratiques; elle prend en compte à la fois de l’information préalable sur les formes du terrainplausibles (sous la forme d’un dictionnaire), et la précision locale des MNEs, contrôlée par despoids interprétables; et elle pose la fusion complète comme un problème mathématique propre,d’optimisation convexe qui peut être résolu avec l’optimalité globale, et dans lequel l’influencede les MNEs d’entrée est contrôlée par l’ensemble des poids de fusion locales.
On propose une méthode axée sur les données, ce qui permet d’obtenir des mesures dequalité locales pour les MNEs (et donc aussi des poids pour la fusion) pour chaque point ousegment de le MNE, si aucune information n’est disponible. Nous utilisons comme entrée descaractéristiques géomorphologiques du terrain (pente, exposition, rugosité) qui sont directe-ment dérivées par les MNEs, ainsi que des informations sémantiques tels que des cartes decouverture du sol. En utilisant pour référence des MNEs de haute qualité nous apprenons desfonctions de régression qui lient les caractéristiques géomorphologiques à la qualité des MNEs,qui permettent ensuite d’estimer la qualité locale d’un nouveau MNE.
La méthode proposée est évaluée en détail en utilisant différents ensembles de données, etelle présente toujours une amélioration de la qualité des MNEs.
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Acknowledgements
This dissertation is the cumulative result of six years of Doctoral Studies at the Institute ofGeodesy and Photogrammetry (IGP), Department of Civil, Environmental and Geomatic Engi-neering at ETH Zurich.
I gratefully acknowledge my first supervisor Professor em. Armin Grün who accepted meas a PhD student at the Institute at end of summer 2006. I cordially acknowledge my lastsupervisor Professor Dr. Konrad Schindler who accepted to supervise me till the end of myPhD thesis. Special thanks are reserved for my main supervisor Dr Emmanuel Baltsavias. Iwant to thank Manos for all the knowledge, the discussions and the guidance he provided mein all my steps in the field.
Professor Ross Purves (Department of Geography, University of Zurich) is warmly ac-knowledged for reviewing the thesis.
I thank Dr. Devrim Akça, Dr. Daniela Poli and Gilles Bourbon for all their help anddiscussions during the early stages of my research and Dr. Effrosyni Kokiopoulou, and Prof.Daniel Kressner for giving me the setup to built the theoretical part of my research.
There are no words to express my gratitude to my colleague, to my FRIEND Dr. NusretDemir, whose support during the full course of my research has been really valuable.
I also want to thank Prof. Grün and Prof. Schindler for creating such a group, with peoplehighly competent and nice. I want to thank all the members of the group both the ones whoalready left before me and the ones who are still there.
I warmly thank all the past and present administrative and technical staff of the group whohelped me during these years: Monique Berger Lande, Zsusanne Sebestyén, Liliane Stein-brückner and the GREAT Beat Rüedin.
EU FP6, the Swiss Space Office, Dassault Aviation, and the Commission for Technologyand Innovation, are acknowledged for their financial support. Swisstopo, Swissphoto, Dassaultaviation, WSL and and Sarmap SA are acknowledged for providing data for my research.
I want to thank my friends, for the funny and bitter moments we shared together.I will always feel gratitude for my charismatic teachers who introduced me to the magic
world of mathematics and research: Prof. Petros Maragos, Panagiotis Ginos, Prof. ChristinaFili, Prof. Andreas Georgopoulos, Prof. Jean-Michel Morel, and Prof. Dimitris Argialas.
My BIGGEST thanks goes to my parents Efthalia and Konstantinos, my brother Panagiotisand my sister Niki. Thank you for always being there (HERE), for inspiring me, for helpingme, for LOVING me so much!
The last thoughts are dedicated to my new FAMILY, my lovely husband Thomas and ourlovely children Thalia and Reinhardt.
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Table of Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiZusammenfassung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivRésumé . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixAbbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xixList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxii
1 Introduction 11.1 Motivation for Combining DEMs . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Aims . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Structure of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Digital Elevation Models 72.1 DEMs Types and Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.1 Categorization According to the Coverage . . . . . . . . . . . . . . . . 92.1.2 Categorization According to the Data Structure . . . . . . . . . . . . . 142.1.3 Categorization According to the Technology . . . . . . . . . . . . . . 15
2.2 Error Types in DEMs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2.1 Systematic Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2.2 Blunders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2.3 Random Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3 DEM Quality Assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.3.1 Global Accuracy Parameters . . . . . . . . . . . . . . . . . . . . . . . 232.3.2 Local Accuracy Parameters . . . . . . . . . . . . . . . . . . . . . . . 25
3 Fusion Processing Chain 293.1 Overview of DEM Fusion Processing . . . . . . . . . . . . . . . . . . . . . . 293.2 Pre-processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2.1 DEM Co-registration . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.2.2 DEM Resampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.3 Core Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.3.1 Quality Assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.3.2 DEM Fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.4 Post-processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
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3.4.1 Quality Assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.5 DEM Fusion Literature Survey . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.5.2 Fusion of InSAR DEMs . . . . . . . . . . . . . . . . . . . . . . . . . 343.5.3 Fusion of DEMs Produced with Different Technology . . . . . . . . . 353.5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4 Quality Characterisation of DEMs and Derivation of Weights 414.1 Quality Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.1.1 Geomorphological Parameters . . . . . . . . . . . . . . . . . . . . . . 444.1.2 Edginess . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.1.3 Further Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.2 Derivation of Quality Values for DEM Points based on Single Criteria . . . . . 524.3 Quality Characterisation through Combination of Criteria . . . . . . . . . . . . 57
4.3.1 Combination of Two Criteria . . . . . . . . . . . . . . . . . . . . . . . 574.3.2 Combination of Slope Aspect, and Roughness . . . . . . . . . . . . . . 60
4.4 Derivation of Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5 DEM Fusion 635.1 Mathematical Models of Fusion . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.1.1 Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.1.2 Data Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.1.3 Weighted Average . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655.1.4 Sparse Representations . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.2 Quality Assessment of the Fused DEM . . . . . . . . . . . . . . . . . . . . . . 71
6 Results and Discussion 736.1 Test Area and Test Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
6.1.1 InSAR DEMs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 756.1.2 Optical DEMs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 756.1.3 Reference DEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
6.2 Test Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 776.2.1 Co-registration Results . . . . . . . . . . . . . . . . . . . . . . . . . . 776.2.2 DEM Fusion Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 776.2.3 Example 1: ALOS – IKONOS . . . . . . . . . . . . . . . . . . . . . . 796.2.4 Example 2: IKONOS – PRISM . . . . . . . . . . . . . . . . . . . . . 796.2.5 Example 3: ALOS – SPOT . . . . . . . . . . . . . . . . . . . . . . . . 826.2.6 Example 4: ALOS – ERS . . . . . . . . . . . . . . . . . . . . . . . . 826.2.7 Example 6: ERS – SPOT . . . . . . . . . . . . . . . . . . . . . . . . . 896.2.8 Example 7: ERS – ASTER . . . . . . . . . . . . . . . . . . . . . . . . 1036.2.9 Example 8: SPOT – SRTM band-C . . . . . . . . . . . . . . . . . . . 1036.2.10 Example 9: SRTM band-C – ASTER . . . . . . . . . . . . . . . . . . 1036.2.11 Example 10: SRTM band-X – SRTM band-C . . . . . . . . . . . . . . 110
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6.3 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
7 Conclusions and Outlook 1237.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1237.2 Open Issues and Further Improvements . . . . . . . . . . . . . . . . . . . . . 124
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Abbreviations
ALOS Advanced Land Observing SatelliteALS Airborne Laser ScanningASPRS American Society for Photogrammetry and Remote SensingASTER Advanced Spaceborne Thermal Emission and Reflection RadiometerDEM Digital Elevation ModelDLR Deutsches Zentrum fur Luft- und RaumfahrtDOM High precision digital surface model by the Swisstopo (Digitales Oberflächenmodell)DSM Digital Surface ModelDTED Digital Terrain Elevation DataDTM Digital Terrain ModelDTM-AV Digital terrain model of the cadastral surveying by the Swisstopo (Digitales Terrainmodell
der Amtlichen Vermessung)EROS Earth Resources Observation and ScienceERS Earth Remote Sensing SatelliteESA European Space AgencyGIS Geographical Information SystemGRASS Geographic Resources Analysis Support SystemGNSS Global Navigation Satellite SystemGPS Global Positioning SystemHRS High-Resolution StereoscopicICP Iterative Closest PointIGARSS International Geoscience And Remote Sensing SymposiumIMU Inertial Measurement UnitINS Inertial Navigation SystemInSAR Interferometric Synthetic Aperture RadarIRS Indian Remote SensingISPRS International Society for Photogrammetry and Remote SensingJDL Joint Directors of LaboratoriesLASER Light Amplification by Stimulated Emission of RadiationLiDAR Light Detection and RangingMAD0 Median Absolute Deviation from the meanMAD1 Median Absolute Deviation from the medianNASA National Aeronautics and Space AdministrationMETI Ministry of Economy, Trade and Industry of Japan
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NED National Elevation DatasetNGA National Geospatial-Intelligence AgencyPALSAR Phased Array type L-band Synthetic Aperture RadarPCA Principal Component AnalysisPOS Position and Orientation SystemRADAR RAdio Detection and RangingRAR Real Aperture RadarRMSE Root Mean Squared ErrorRST Regularized Spline with TensionSAR Synthetic Aperture RadarSPOT Système Pour l’Observation de la TerreSRTM Shuttle Radar Topography MissionTIN Triangulated Irregular NetworkUSGS United States Geological SurveyUTM Universal Transverse MercatorWGS World Geodetic System
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List of Figures
2.1 Shaded and coloured surface image produced from ETOPO1 data (NationalGeophysical Data Center, 2012). . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2 Various data structure DEMs. . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.1 Processing flow chart for DEM fusion. . . . . . . . . . . . . . . . . . . . . . . 31
4.1 SPOT DEM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.2 Thun satellite images. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.3 3X3 Height matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.4 Slope map the SPOT DEM. The bar unit is in degrees. . . . . . . . . . . . . . 464.5 Aspect map of the SPOT DEM. . . . . . . . . . . . . . . . . . . . . . . . . . . 474.6 Standard deviation of DEM points. Principle of orthogonal regression plane
fitting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.7 Roughness maps of the SPOT DEM. For (a)-(b) the bars units are in meters, (c)
unitless. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.8 Edge detection on the SPOT orthoimage. . . . . . . . . . . . . . . . . . . . . . 514.9 Absolute height residual map of the SPOT DEM in meters. . . . . . . . . . . . 524.10 (a) Residuals between the LiDAR and the SPOT DEM versus slope (mean ev-
ery degree). (b) The slope histogram with bins of one degree. . . . . . . . . . . 544.11 (a) Residuals between the LiDAR and the SPOT DEM versus aspect (mean
every degree). (b) The aspect histogram with bins of one degree. . . . . . . . . 554.12 (a) Residuals between the LiDAR and the SPOT DEM versus roughness (mean
every 0.01). (b) The roughness histogram, with bins of 0.01. . . . . . . . . . . 564.13 (a) Loess surface fitting. Residuals between the LiDAR and the SPOT DEM
(z-axis) versus slope (x-axis) and aspect (y-axis). (b) Contour plot. . . . . . . . 584.14 (a) Loess surface fitting. Residuals between the LiDAR and the SPOT DEM
(z-axis) versus slope (x-axis) and roughness (y-axis). (b) Contour plot. . . . . . 594.15 (a) Loess surface fitting. Residuals between the LiDAR and the SPOT DEM
(z-axis) versus roughness (x-axis) and aspect (y-axis). (b) Contour plot. . . . . 604.16 Colour coded residuals between the LiDAR and the SPOT DEM versus slope
(x-axis), aspect (y-axis) and roughness (z-axis). . . . . . . . . . . . . . . . . . 61
5.1 Reconstruction of a 9 × 9 patch. (a) from three non-zero atoms (b)-(d), where[a0, a1, a2] are the sparse non-zero coefficient elements. . . . . . . . . . . . . . 69
5.2 Fusion parameters optimization. . . . . . . . . . . . . . . . . . . . . . . . . . 70
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6.1 Overlap area of Thun. ALOS DEM. . . . . . . . . . . . . . . . . . . . . . . . 746.2 The coloured absolute Z residuals (m) between the LiDAR DEM and the trans-
formed ALOS, IKONOS and PRISM DEMs after co-registration. . . . . . . . 906.3 The coloured absolute Z residuals (m) between the LiDAR DEM and the trans-
formed ERS, and SPOT DEMs after co-registration. . . . . . . . . . . . . . . . 916.4 The coloured absolute Z residuals (m) between the LiDAR DEM and the trans-
formed ASTER, SRTM-C and SRTM-X DEMs after co-registration. . . . . . . 926.5 Profiles of fused DEM, and reference LiDAR DEM along with that of (a) ALOS
DEM and (b) IKONOS DEM. . . . . . . . . . . . . . . . . . . . . . . . . . . 936.6 ALOS–IKONOS fusion example. (a)-(c) Input and output DEMs, (d) Residuals
between the ALOS and the LiDAR DEMs, (e) Residuals between the IKONOSand the LiDAR DEMs, (f) Residuals between the fusion and the LiDAR DEMs.The residuals bar unit is in meters. . . . . . . . . . . . . . . . . . . . . . . . . 94
6.7 Profiles of fused DEM, and reference LiDAR DEM along with that of (a)IKONOS DEM and (b) PRISM DEM. . . . . . . . . . . . . . . . . . . . . . . 95
6.8 IKONOS–PRISM fusion example. (a)-(c) Input and output DEMs, (d) Resid-uals between the IKONOS and the LiDAR DEMs, (e) Residuals between thePRISM and the LiDAR DEMs, (f) Residuals between the fusion and the LiDARDEMs. The residuals bar unit is in meters. . . . . . . . . . . . . . . . . . . . . 96
6.9 Profiles of fused DEM, and reference LiDAR DEM along with that of (a) ALOSDEM and (b) SPOT DEM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6.10 ALOS–SPOT fusion example. (a)-(c) Input and output DEMs, (d) Residualsbetween the ALOS and the LiDAR DEMs, (e) Residuals between the SPOTand the LiDAR DEMs, (f) Residuals between the fusion and the LiDAR DEMs.The residuals bar unit is in meters. . . . . . . . . . . . . . . . . . . . . . . . . 98
6.11 Profiles of fused DEM, and reference LiDAR DEM along with that of (a) ALOSDEM and (b) ERS DEM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6.12 ALOS–ERS fusion example. (a)-(c) Input and output DEMs, (d) Residualsbetween the ALOS and the LiDAR DEMs, (e) Residuals between the ERS andthe LiDAR DEMs, (f) Residuals between the fusion and the LiDAR DEMs.The residuals bar unit is in meters. . . . . . . . . . . . . . . . . . . . . . . . . 100
6.13 Profiles of fused DEM, and reference LiDAR DEM along with that of (a) ERSDEM and (b) SPOT DEM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.14 ERS–SPOT fusion example. (a)-(c) Input and output DEMs, (d) Residualsbetween the ERS and the LiDAR DEMs, (e) Residuals between the SPOT andthe LiDAR DEMs, (f) Residuals between the fusion and the LiDAR DEMs.The residuals bar unit is in meters. . . . . . . . . . . . . . . . . . . . . . . . . 102
6.15 Profiles of fused DEM, and reference LiDAR DEM along with that of (a) ERSDEM and (b) ASTER DEM. . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.16 ERS–ASTER fusion example. (a)-(c) Input and output DEMs, (d) Residualsbetween the ERS and the LiDAR DEMs, (e) Residuals between the ASTERand the LiDAR DEMs, (f) Residuals between the fusion and the LiDAR DEMs.The residuals bar unit is in meters. . . . . . . . . . . . . . . . . . . . . . . . . 113
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6.17 Profiles of fused DEM, and reference LiDAR DEM along with that of (a) SPOTDEM and (b) SRTM-C DEM. . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.18 SRTM-C–SPOT fusion example. (a)-(c) Input and output DEMs, (d) Residualsbetween the SPOT and the LiDAR DEMs, (e) Residuals between the SRTM-Cand the LiDAR DEMs, (f) Residuals between the fusion and the LiDAR DEMs.The residuals bar unit is in meters. . . . . . . . . . . . . . . . . . . . . . . . . 117
6.19 Profiles of fused DEM, and reference LiDAR DEM along with that of (a)SRTM-C DEM and (b) ASTER DEM. . . . . . . . . . . . . . . . . . . . . . . 118
6.20 SRTM-C–ASTER fusion example. (a)-(c) Input and output DEMs, (d) Resid-uals between the SRTM-C and the LiDAR DEMs, (e) Residuals between theASTER and the LiDAR DEMs, (f) Residuals between the fusion and the Li-DAR DEMs. The residuals bar unit is in meters. . . . . . . . . . . . . . . . . . 119
6.21 SRTM-X–SRTM-C fusion example. (a)-(c) Input and output DEMs, (d) Resid-uals between the SRTM-X and the LiDAR DEMs, (e) Residuals between theSRTM-C and the LiDAR DEMs, (f) Residuals between the fusion and the Li-DAR DEMs. The residuals bar unit is in meters. . . . . . . . . . . . . . . . . . 120
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List of Tables
2.1 Definitions of DEM, DSM and DTM. . . . . . . . . . . . . . . . . . . . . . . 82.2 Terms that are recognised as a DEM. . . . . . . . . . . . . . . . . . . . . . . . 82.3 DEM uses and applications (Sulebak, 2000). . . . . . . . . . . . . . . . . . . . 82.4 Characteristics of important wide-area DEMs. . . . . . . . . . . . . . . . . . . 20
4.1 The parameters of the sum of sines model for fitting the slope to the Z residuals. 554.2 The parameters of the Gaussian model for fitting the aspect to the Z residuals. . 564.3 The coefficients of the polynomial model for the fitting of roughness. . . . . . . 57
6.1 Characteristics of the used DEMs. The accuracy values of the DEMs are esti-mated for the test area using the LiDAR DEM as reference. . . . . . . . . . . . 77
6.2 Co-registration results of the input DEMs to with the reference LiDAR DEM.σ0 is the σ a-priori, σ0σ0σ0 is the σ a-posteriori and Tx, Ty, and Tz are the threetranslations. All units are in meters. . . . . . . . . . . . . . . . . . . . . . . . 79
6.3 Statistical results of the fusion of the ALOS and IKONOS DEMs for the com-plete area. All units are in meters. . . . . . . . . . . . . . . . . . . . . . . . . 80
6.4 Fused DEM RMSE (m) in relation to slope. Fusion of the ALOS and IKONOSDEMs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.5 Statistical results of the fusion of the IKONOS and PRISM DEMs for the com-plete area. All units are in meters. . . . . . . . . . . . . . . . . . . . . . . . . 83
6.6 Fused DEM RMSE (m) in relation to slope. Fusion of the PRISM and IKONOSDEMs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
6.7 Statistical results of the fusion of the ALOS and SPOT DEMs for the completearea. All units are in meters. . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.8 Fused DEM RMSE (m) in relation to slope. Fusion of the ALOS and SPOTDEMs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.9 Statistical results of the fusion of the ALOS and ERS DEMs for the completearea. All units are in meters. . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6.10 Fused DEM RMSE (m) in relation to slope. Fusion of the ERS and ALOS DEMs. 886.11 Statistical results of the fusion of the ERS and SPOT DEMs for the complete
area. All units are in meters. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1046.12 Fused DEM RMSE (m) in relation to slope. Fusion of the ERS and SPOT DEMs.1056.13 Statistical results of the fusion of the ERS and ASTER DEMs for the complete
area. All units are in meters. . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
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6.14 Fused DEM RMSE (m) in relation to slope. Fusion of the ERS and ASTERDEMs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.15 Statistical results of the fusion of the SPOT and SRTM-C DEMs for the com-plete area. All units are in meters. . . . . . . . . . . . . . . . . . . . . . . . . 108
6.16 Fused DEM RMSE (m) in relation to slope. Fusion of the SRTM-C and SPOTDEMs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.17 Statistical results of the fusion of the SRTM-C and ASTER DEMs for the com-plete area. All units are in meters. . . . . . . . . . . . . . . . . . . . . . . . . 111
6.18 Fused DEM RMSE (m) in relation to slope. Fusion of the SRTM-C and ASTERDEMs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.19 Statistical results of the fusion of the SRTM-X and SRTM-C DEMs for thecomplete area. All units are in meters. . . . . . . . . . . . . . . . . . . . . . . 114
6.20 Fused DEM RMSE (m) in relation to slope. Fusion of the SRTM-C and SRTM-X DEMs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
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Chapter 1
Introduction
The introduction chapter provides an insight to the conducted research and its importance. Itfamiliarises the reader with the term of “fusion” and explains the general idea behind this work.
1.1 Motivation for Combining DEMs
Accuracy is always the main concern. Although nowadays it is easy to access DEM data, ac-curate data are requiring a lot of effort and cost. The amount of DEMs produced in the worldincreases every year and this rate will only increase since the proliferation of inexpensive sen-sor devices and new advanced techniques makes possible the collection of elevation data fromdifferent and diversified sensors operating with different technologies, different resolution, anddifferent reliability. The whole globe has been mapped so far many times at different levels ofaccuracy, resolution and using different sensors. A growing demand exists for procedures thatcan exploit existing multi-source DEMs. Numerous mapping applications may benefit fromthis. Research work is therefore still being carried out in order to develop efficient methodolo-gies aiming at a combination of different DEMs to generate a value-added product that is morecomplete, accurate and reliable. However, a definitive solution has not been reached yet.
Although the concept of data fusion is easy to understand, it is very difficult to providea precise definition of data fusion. The exact interpretation of data fusion varies from onescientist to another. Furthermore, several words exist such as merging, combination, synergy,integration that express the same concept. Several definitions can be found in the literature. Acommon theme in all data fusion techniques is that they combine data from multiple sensors,and related information from associated databases, to achieve improved accuracies and morespecific inferences than what could be achieved by the use of one sensor alone (Waltz & Llinas,1990; Hall, 1992; Klein, 1999).
In the mid-1980s, the JDL formed the Data Fusion Subpanel which later became known asthe Data Fusion Group. The initial JDL Data Fusion Lexicon defined data fusion as: “A processdealing with the association, correlation, and combination of data and information from singleand multiple sources to achieve refined position and identity estimates, and complete and timelyassessments of situations and threats, and their significance. The process is characterized bycontinuous refinements of its estimates and assessments, and the evaluation of the need for
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additional sources, or modification of the process itself, to achieve improved results” (White,1991).
The notion of “fusion” is not only used in connection with computers. Humans and animalshave evolved the capability to integrate sensory input from multiple sources to improve theirability to survive. A combination of sight, touch, smell, and taste is naturally performed byanimals and humans to achieve more accurate assessment of the surrounding environment andidentification of threats, thereby improving their chances of survival. Additionally, humansperform planning of future actions based on what may be characterized as advanced data fusion.The operation of data fusion by itself is not new in any domain of application. Techniques tocombine or fuse data are drawn from a diverse set of more traditional disciplines including:digital signal processing, statistical estimation, control theory, artificial intelligence and classicnumerical methods (Hall & Llinas, 1997; Kessler, 1992; Wright, 1980).
Pohl & Van Genderen (1998) defined image fusion as “the combination of two or moredifferent images to form a new image by using a certain algorithm”, which is a definition re-stricted to images. Mangolini (1994) defines data fusion as a “set of methods, tools and meansusing data coming from various sources of different nature, in order to increase the quality (ina broad sense) of the requested information”. (Hall & Llinas, 1997) involve in their definitionthe aspect of quality: “data fusion techniques combine data from multiple sensors, and relatedinformation from associated databases, to achieve improved accuracy and more specific in-ferences that could be achieved by the use of a single sensor alone”. Based upon the worksof (Buchroithner, 1998; Wald, 1998), the following definition was adopted in January 1998:“data fusion is a formal framework in which are expressed means and tools for the alliance ofdata originating from different sources. It aims at obtaining information of greater quality; theexact definition of “greater quality” will depend upon the application”. This definition puts anemphasis on quality. Here, quality has not a very specific meaning. It is a generic word denot-ing that the resulting information is more satisfactory for the “customer” when performing thefusion process than without it (Wald, 1999).
In principle, fusion of multi-sensor data provides significant advantages over single sourcedata (Waltz & Llinas, 1990). It usually results in an improvement of the results. There aremany advantages in using data fusion. In addition to the statistical advantage gained by com-bining same-source data (e.g., obtaining an improved estimate of a physical phenomena viaredundant observations), the use of multiple types of sensors may increase the accuracy andreproducibility with which a quantity can be measured (Hall & Llinas, 1997).
In this dissertation, it was investigated how the fusion of existing elevation data could leadto the generation of a higher quality DEM. The term quality means a DEM with higher ac-curacy, less blunders, higher density and completeness. This has been done by developing amethodology which fuses registered DEMs with different characteristics and produces a com-posite DEM with quality equal to or better than that of the individual contributors.
1.2 AimsThe main objectives of the present work are:
Quality characterisation of input DEMs. When considering the fusion of multiple DEMs, it
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is important to simultaneously account for the errors that may be present in each DEM,as well as the impact these may have on the merged DEM (Buckley & Mitchell, 2004).In the research community a lot of effort was put in DEM quality assessment. Sincemany years a lot of work has been done to obtain quality measures for existing DEMs.There is always however the problem of accounting for missing information and derivingappropriate quality characterization. Quality information can exist at a global level (a solequality number i.e. root mean square error) but seldom at a local level (e.g. for everyDEM point). There is a strong need for careful accuracy assessment of each availableDEM. In this context, it was decided to concentrate on the assessment of quality of DEMswithout any a priori information. The main objective is to acquire quality measures foreach individual point of every given DEM.
DEM Fusion. This thesis aims mainly at showing the potential for combined usage of sev-eral DEMs, from different epochs (multi-temporal), derived with different sensors andmethods, at different resolutions (multi-resolution) to produce a composite DEM withcharacteristics equal to or better than that of the individual DEMs. This is not an easyproblem to solve. Actual implementation of effective DEM fusion is by far not straight-forward. Fusion of elevation data may actually produce worse results than those thatcould be obtained by the best input DEM. This is caused by the attempt to combine ac-curate (i.e., good data) with inaccurate or biased data, especially if the uncertainties orvariances of the data are unknown (Hall & Llinas, 1997). The simple agglomerationof points has the potential to introduce blunders, systematic and random errors into themerged surface, brought about by a host of possible disparities between the input data(Kweon & Kanade, 1992; Schiewe, 2000).
The sub-objectives of image fusion are:
1. Blunder reduction
2. Reduction of systematic and random errors
3. Extension of DEM area coverage
4. Production of a denser DEM
5. Production of a higher accuracy DEM
6. DEM updating and change detection
The erroneous areas may be replaced with the corresponding data from the best inputDEM or alternatively each component DEM can contribute to the overall DEM, accord-ing to its quality.
Quality characterisation of the output DEM. The last aim of this thesis is to analyse theaccuracy of a DEM produced after the fusion of the input DEMs.
The methodology was built by combining state-of-the-art methods extended or adapted tothe needs of DEM fusion. The input DEMs for fusion must be geometrically co-registeredand cropped to a common spatial coverage. They must be in grid format with the same gridsize and the data type may be floating-point, or integers. Multiple and extensive experiments
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were carried out to validate the proposed methodology. All the tests were done using two inputDEMs with different characteristics.
Before we enter a more detailed discussion of these topics, it seems useful to list the publi-cations in which this work was published in:
1. Papasaika, H., Kokiopoulou, E., Baltsavias, E., & Schindler, K. (2011a). Accuracy esti-mation and fusion of digital elevation models. In Proc. 32nd Asian Conference on RemoteSensing Taipeh, Taiwan
2. Papasaika, H., Kokiopoulou, E., Baltsavias, E., Schindler, K., & Kressner, D. (2011b).Fusion of digital elevation models using sparse representations. In Proceedings of the2011 ISPRS conference on Photogrammetric image analysis, PIA’11 (pp. 171–184).Berlin, Heidelberg: Springer-Verlag
3. Papasaika, H., Schütz, S., Baltsavias, E., & Schindler, K. (2011c). Verbesserung gross-flächiger DHMs mittels fusion. Geomatik Schweiz, 9, 448–452
4. Schindler, K., Papasaika, H., Schutz, S., & Baltsavias, E. (2011). Improving wide-areaDEMs through data fusion - chances and limits (invited paper). In Proc. 53rd Photogram-metric Week Stuttgart, Germany
5. Papasaika, H. & Baltsavias, E. (2010). Quality evaluation of DEMs. In P. Tate, N.; Fisher(Ed.), Accuracy 2010 - Proceedings of the Ninth International Symposium on SpatialAccuracy Assessment in Natural Resources and Environmental Sciences
6. ASTER GDEM Validation Team (2009). ASTER Global DEM Validation. Summaryreport, METI/ERSDAC, NASA/LPDAAC, USGS/EROS, In cooperation with NGA andOther Collaborators
7. Papasaika, H. & Baltsavias, E. (2009a). Effects of geomorphological characteristics onthe DEM accuracy. In Geomorphometry 2009 Zurich, Switzerland
8. Papasaika, H., Poli, D., & Baltsavias, E. (2009). Fusion of digital elevation models fromvarious data sources. In Advanced Geographic Information Systems Web Services, 2009.GEOWS ’09 (pp. 117–122)
9. Papasaika, H. & Baltsavias, E. (2009b). Fusion of LiDAR and photogrammetric gener-ated digital elevation models. In ISPRS Hannover Workshop on High-Resolution EarthImaging for Geospatial Information
10. Papasaika, H., Poli, D., & Baltsavias, E. (2008). A framework for the fusion of digitalelevation models. In The international archives of the photogrammetry, remote sensingand spatial information sciences, volume 37 (pp. 811–818)
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1.3 Structure of the ThesisIn chapter 2, a general overview about DEMs is given and some basic concepts used through-out the text are presented. The DEMs are classified according to the coverage, data structureand production technology and different types of DEMs are shortly described and their maincharacteristics, strengths and weak points are presented. Another important aspect related toDEMs is the definition of parameters which can be used to assess their accuracy. Thus, thesecond part of the second chapter concentrates on quality parameters and draws attention to theimportance of using different approaches to quantify the quality of a DEM, both globally andlocally. Finally, common error types of DEMs are described.
In chapter 3, a complete stratified approach to DEM fusion is proposed. An overview of thecomplete proposed processing chain and the steps needed for fusion are described and DEMfusion methods proposed by others are presented here.
Chapter 4 deals with issues concerning the derivation of the weights that are used duringthe DEM fusion. In this context, some important quality criteria of DEMs are also described.A flexible derivation method is proposed which can deal with DEMs with unknown a priorilocal accuracy and some problems of this method are also discussed and possible solutions aredescribed.
DEM fusion is introduced in Chapter 5. First, some theoretical considerations are presented.Then, then focus is brought on our proposed DEM fusion approach. This approach is based onsparse representations. The mathematical framework is described and all critical aspects of themethodology are discussed in detail.
In Chapter 6 the developed methodology is applied to fuse DEMs of different spatial reso-lution, different technologies and acquired in different epochs. Thereupon, the flexibility andthe potential of the approach is demonstrated. Several experiments are performed in order tocompare the proposed method to other existing methods. Finally, results and applications arepresented, and discussed.
The conclusions of our work are presented in Chapter 7 together with the open issues andthe possible further improvements to the current methodology.
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6
Chapter 2
Digital Elevation Models
A Digital Elevation Model (DEM) is one of the simplest and most commonly used digitalrepresentations of the topography. The earliest definition of a Digital Terrain Model as “astatistical representation of the continuous surface of the ground by a large number of selectedpoints with known xyz coordinates in an arbitrary coordinate field” dates back to the 1950s(Miller & Laflamme, 1958) and it was introduced by engineers at Massachussetts Institute ofTechnology (MIT). The definition was based on the nature of the work at that point in time i.e.engineering. However, in the following decades the use of the terms Digital Elevation Model(DEM) and Digital Surface Model (DSM) has been confusing (Table 2.1). Several terms canbe recognized as a DEM (Table2.2). Despite of the different terms that have been used, all ofthem represent the same phenomenon, i.e. the topography of the earth surface. According toYue et al. (2007) “DEMs can be described as a function of geographic location representingterrain elevations and providing information about the topographic attributes of terrain”.
Elevation has been recognized as one of the most essential geodata in GIS for two rea-sons(Atkinson, 2002):
1. It represents the third spatial dimension. It provides the Z-value (third dimension) tocomplement theX-value and Y -value that are used to represent location in two-dimensional(2D) space.
2. It is also the most needed and probably the most fundamental geodata since elevationsurfaces are required in multiple GIS applications (Table 2.3).
Throughout this thesis, a DEM is considered identical to DSM.
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Table 2.1: Definitions of DEM, DSM and DTM.
Term DefinitionDEM Generic term covering digital topographic (and bathymetric) data in all its var-
ious forms as well as the method(s) for interpreting implicitly the elevationsbetween observations. Normally implies elevations of bare earth without veg-etation and buildings, but may include other manmade features, such as roadembankments. Elevations of hydrological features (e.g. lakes and rivers) nor-mally imply a free water surface (Maune et al., 2001).
DSM Model depicting elevations of the top of surfaces, such as buildings and veg-etation (Maune et al., 2001).
DTM DTM is a synonym of bare-earth DEM (Maune et al., 2001).
Table 2.2: Terms that are recognised as a DEM.
Term DefinitionDEM Digital Elevation ModelDTM Digital Terrain ModelDSM Digital Surface ModelDHM Digital Height ModelDGM Digital Ground ModelDTED Digital Terrain Elevation Data
Table 2.3: DEM uses and applications (Sulebak, 2000).
1. Scientific applications: climate impact studies, water and wildlife management,geological and hydrological modelling, geographic information technology, geo-morphology and landscape analysis, mapping purposes and educational programs.
2. Commercial applications: telecommunication, air traffic routing and navigation,planning and construction, geological exploration, hydrological and meteorologicalservices, geocoding of remote sensing and market of multimedia applications andcomputer games.
3. Industrial applications: telecommunication industry, avionics industry, telematicsindustry, mining and oil industry, tourism industry.
4. Operational applications: reconnaissance for mineral and water resources, air-craft guidance systems - flight simulations, forest planning and management, plan-ning of breakwater constructions, mass movement and hazard prediction, floodingrisk assessment, disaster management (prevention, relief, assessment).
5. Military applications: site planning, battlefield management, missile guidance,planning of communication networks.
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2.1 DEMs Types and Characteristics
Several types of DEMs exist. DEMs are classified below according to their coverage, datastructure and production technology.
2.1.1 Categorization According to the Coverage
DEMs are produced at a number of spatial scales.
Global DEMs
Global DEMs provide topographic data including topography and bathymetry for the entireglobe. The highest resolution DEM data set for the entire world (excluding bathymetry) is theASTER Global DEM (G-DEM), which has a nominal ground resolution of 30m. This sectiondescribes six very important and well-known, existing and planned data products that includeelevation data for all or most of the Earth’s surface.
SRTM. The SRTM DEM is one of the most used datasets in the Geomatics community andby the general public (e.g. SRTM DEMs are used in Google Earth). SRTM acquired DEMsin February 2000 by single-pass SAR interferometry in the C- and X-bands (Farr et al., 2007).The SRTM data were acquired by the NGA and the NASA using two radar systems that flew onboard the Space Shuttle Endeavour during an 11-day mission in February, 2000. The C-bandradar data were processed at the Jet Propulsion Laboratory and are being distributed throughthe United States Geological Survey’s EROS Data Center. Data from the X-band radar areused to create slightly higher resolution DEMs but without the global coverage of the C-bandradar. The SRTM X-band radar data were processed and distributed by the German AerospaceCenter, DLR. The X-band data has large gaps due to the limited strip width.
The resulting SRTM DEMs from the X-band information have a resolution of 1 arc sec-ond (nominally 30 m, SRTM-1) and the resulting SRTM DEMs from the C-band informationhave a resolution of 3 arc second (90 m, SRTM-3). The X-band data with a wavelength ofapproximately 3 cm barely penetrates vegetation, while it is expected that the C-band data witha wavelength of about 5.6 cm will penetrate somewhat more into the canopy. SRTM DEM usesWGS84 datum and EGM96 geoid model. The collected elevation data underwent extensiveprocessing and noise filtering before they were released to the public.
Despite the lower resolution of SRTM-3, it has been extensively used since it is the bestfree digital topographic dataset in many countries and regions. The SRTM-3 has horizontaland vertical accuracies of about 20 m (circular error at 90% confidence) and 16 m (linear errorat 90% confidence), respectively (Smith & Sandwell, 2003; Slater et al., 2006; Rodríguez et al.,2006; Farr et al., 2007). Reinartz et al. (2005) conclude that SRTM data accuracy decreasesdrastically in forest areas, since it represents neither the tree canopy nor the ground.
The data currently being distributed by NASA/USGS (C-band) contain “no-data” holeswhere water or heavy shadow prevented the quantification of elevation. These small holes arecalled SRTM voids. Andy Jarvis and Edward Guevara of the CIAT Agroecosystems Resilience
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project, Hannes Isaak Reuter (JRC-IES-LMNH) and Andy Nelson (JRC-IES-GEM) have fur-ther processed the original DEMs to fill in these no-data voids (Jarvis et al., 2008). This in-volved the production of vector contours and points, and the re-interpolation of these derivedcontours back into a raster DEM. These interpolated DEM values are then used to fill in theoriginal no-data holes within the SRTM C-band data. These processes were implemented usingArc/Info and an AML script. These files are available for download in both Arc-Info ASCIIformat, and as GeoTiff, for easy use in most GIS and Remote Sensing software applications. Inaddition, a binary Data Mask file is available for download, allowing users to identify the areasthat have been interpolated within each DEM.
ASTER Global DEM. The ASTER Global DEM (G-DEM) is a new global DEM set with aresolution of 1 arc second (30m) for the whole world, which was released in December 2008(ASTER G-DEM Version 1). ASTER G-DEM was developed in partnership of the Ministry ofEconomy, Trade and Industry of Japan and NASA and it is based on satellite imagery from theASTER sensor (1.4 million scenes in total).
It was created from data acquired between 1999 and 2009 with stereo matching of imagedata in the visible and near-infrared range. The methodology used to produce the ASTER G-DEM involved automated processing of the entire 1.5-million-scene ASTER archive, includingstereo-correlation to produce 1,264,118 individual scene-based ASTER DEMs, cloud maskingto remove cloudy pixels, stacking all cloud-screened DEMs, removing residual bad values andoutliers, averaging selected data to create final pixel values, and then correcting residual anoma-lies before tessellating the data into 1◦-by-1◦tiles. It took approximately one year to completeproduction of the beta version of the ASTER G-DEM using a fully automated approach. Itcovers the landmasses between 83◦N and 83◦S at 30 m grid spacing, with some small holes. Itis referenced to the WGS84/EGM96 geoid.
The officially announced vertical accuracy of the G-DEM Version 1 is 7 m, less than halfthat of the SRTM-3. Although the G-DEM was expected to be a better source of global topo-graphic information for various scientific applications, it did not fulfil the expectations. The“real” accuracy (95% confidence) is 20 m. Empirical valuations have shown that ASTER hassomewhat inhomogeneous quality – while in most tiles the specifications (see Table 2.4) aremet, there are regions with a significant amount of blunders as well as systematic artifacts(Reuter et al., 2009). In the first semester of 2009 a big study has been made for the valida-tion of the ASTER G-DEM (ASTER GDEM Validation Team, 2009). This study was made byseveral organizations and institutes, including ourselves. It presents highlights and the most rel-evant results from initial studies conducted to validate and otherwise characterize the ASTERG-DEM. Results and findings presented in this report were instrumental in the decision byMETI and NASA to release the ASTER G-DEM.
On the 17th October, 2011 the ASTER G-DEM Version 2 was released. The improvedversion of the map added 260,000 additional stereo-pair images to improve coverage. It featuresimproved spatial resolution, increased horizontal and vertical accuracy, more realistic coverageover water bodies and the ability to identify lakes as small as 0.6 miles (1 km) in diameter.
10
Reference3D. Reference3D was developed in partnership by Spot Image and the French sur-vey and mapping agency IGN. It is produced by stereo matching of optical images from theFrench SPOT-5 satellite acquired since 2002. The DEM product currently covers large parts ofEurope, Africa and Central America, and the coastal regions of China and Australia, as wellas some smaller regions. However, data is available for almost the entire landmass of the earthand can be expected to be processed in the future.
Reference3D is a geocoded database containing three layers of information:
A DTED level 2 HRS DEM. These files contain a uniform grid of surface elevation values ofan area of interest and are obtained through automatic correlation of SPOT HRS stere-opairs. The horizontal grid spacing is 1 second of arc (approximately 30 m). The absoluteelevation accuracy is 10 m and the absolute planimetric accuracy is 10 m for a slope lessthan 20◦. These accuracies have been confirmed empirically (Bouillon et al., 2006).
GPS-compatible HRS Orthoimage. These images are orthorectified from the DEM and havea high degree of geometric quality and location accuracy. The absolute planimetric accu-racy is 10 m and the sampling step is 1/6 second of arc (approximately 5 m).
Quality and traceability metadata. These are quality indicators for data sources and providereferences and footprints for the DEMs and the orthoimages. They describe the process-ing done for producing DEMs with mask layers, and they give accuracy estimates.
GTOPO30. GTOPO30 is a global DEM, developed by USGS with a horizontal grid spacingof 30 arc seconds (approximately 1km). It does not contain any bathymetric data. It was derivedfrom several raster and vector sources of topographic information. GTOPO30, completed inlate 1996, was developed over a three year period through a collaborative effort led by staff atthe USGS Center for EROS. GTOPO30 tiles are available for download from USGS’ EROSData Center. GTOPO60, a resampled and untiled version of GTOPO30, is available throughthe USGS’ Seamless Data Distribution Service.
ETOPO1. It is an 1 arc-minute (approximately 1.9 km) global terrain model of Earth’s sur-face that integrates land topography and ocean bathymetry and it is freely available. It was builtfrom numerous global and regional data sets, and is available in “Ice Surface” (top of Antarcticand Greenland ice sheets) and “Bedrock” (base of the ice sheets) versions. Historic ETOPO2v2and ETOPO5 global relief grids are deprecated but still available (Amante & Eakins, 2009).Source data, and thus data quality, vary from region to region.
TanDEM-X DEM. Together with the almost identical TerraSAR-X (in orbit since June 2007),the new TanDEM-X (TerraSAR-X add-on for Digital Elevation Measurement) circles the Earthas a unique satellite formation since June 2010 (Krieger et al., 2007), when the German radarsatellite TanDEM-X was launched from Baikonur in Kazakhstan. The mission is currently inits operational phase, and is expected, over the next 2.5 years, to cover all land masses of theearth at 12 m grid spacing at an accuracy of about 10 m, but with relative accuracies < 4 m.
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Figure 2.1: Shaded and coloured surface image produced from ETOPO1 data (National Geo-physical Data Center, 2012).
Regional DEMs
NEXTMap. The company Intermap offers a DEM of the USA and Western Europe, acquiredwith airborne SAR interferometry. The DEM has a grid spacing of 5 m, and a nominal accuracyof 2 m for the planimetry and 1 m for the height, which is reached in open terrain, but notnecessarily in vegetation areas (Dowman et al., 2003).
National DEMs
A national DEM is the primary elevation data product of a country. They are designed toprovide national elevation data in a seamless form with a consistent datum, elevation unit, andprojection. We refer to the main national DEMs of three example countries, the United States,Canada and Switzerland.
United States.
National Elevation Dataset (NED). It is the primary elevation data product of the USGS.Highest-resolution, best quality elevation data available across the United States intoa seamless raster format. Resolution: Complete coverage at 1 arc-seconds and 1/3 arc-seconds. Partial coverage at 1/9 arc-seconds. The NED is updated on a nominal twomonth cycle to integrate newly available, improved elevation source data. All NED dataare public domain.
SRTM30. Shuttle Radar Topography Mission data of the United States. Resolution: 1 arc-seconds (approx. 30 m) and it is derived from the C band.
Canada.
Canadian Digital Elevation Data (CDED). Scanned digital elevation model data in DEM for-mat derived from elevation data from the National Topographic Database data or variousscaled positional data acquired from the provinces and territories. Resolution: 30 m(1:50,000) and 90 m (1:250,000).
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Digital Elevation Model, DMTI. Extracted from the National Topographic Database 1:50,000.Resolution: 30 m and 90 m.
Canada3D. A digital elevation model of the Canada’s landmass that covers all of Canada.Resolution: 30 arc-seconds (approximately 1 km) and 300 arc-seconds (approximately10 km). It has been derived from the cells of the Canadian Digital Elevation Data (CDED)at the 1:250, 000 scale.
Switzerland.
DHM25 The digital height model DHM25 is a DTM. It is essentially based on the SwissNational Map 1:25,000. The resolution is 25 m. Comparisons with photogrammetricallydetermined control points show that the average accuracy reaches 1.5 m for the SwissPlateau and the Jura Mountains, 2 m for the pre-Alps and Canton Ticino, and 3 m for theAlps (Swisstopo, 2004).
swissALTI3D It is an extremely precise digital elevation model which describes the surfaceof Switzerland without vegetation and objects. It is updated in an cycle of 6 years.swissALTI3D is the result of merging multiple data sets. The data is delivered as a gridwith a grid width of 2 m, 5 m or 10 m. The quality information provided corresponds tothe accuracies of all three dimensions:
• Laser points (less than 2000 m above sea level): ±0.5 m 1σ
• DHM25 (more than 2000 m above sea level): 3− 8 m average error
• Manual updates (individual points, breaklines and areas): 25 cm - 1 m average error
DOM & DTM-AV DOM is a DSM and DTM-AV is a DTM. These models cover wholeSwitzerland to altitudes up to 2000 m. The accuracy of the DOM in open terrain andfor the DTM-AV everywhere is ±0.5 m (1σ) and for the DOM in terrain with vegetation±1.5 m (1σ). The two models are derived from classification of the original laser pointcloud. The resolution is 2 m and it is produced from a cloud of raw points with a densityof about 1 point per 2 m2. Both models are available in point cloud and in regular gridformat.
Local/Regional DEMs
High-resolution DEMs are available for small regions within every country. Regional and localDEMs are finer (higher resolution) than the national or global DEMs and they are characterizedby higher accuracy levels. High-resolution DEMs (1 m or less) are based on airborne LiDAR,photogrammetry and terrestrial geodesy and they are typically used for physical disasters mon-itoring, for archaeological sites mapping and land administrative studies. While such data arebecoming increasingly available, their potential is far from being fully utilised because of theirhigh cost.
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2.1.2 Categorization According to the Data Structure
A DEM is a representation of the earth’s relief, thus in order to create a DEM the raw data(original observations) must be ordered and structured. Today, the majority of digital elevationdata are organized into two data structures: regular grid (raster DEM) or TIN.
Raster DEMs
Traditionally, this is the simplest and the most frequently used data structure. In this case, aDEM is a set of elevation values representing the elevation at points in a regular grid on theEarth’s surface (McDonnell & Kemp, 1995). Grids usually have a matrix structure. This ele-vation matrix implicitly encodes the topological relations between the data points. The points(X, Y, Z coordinates) are normally arranged in a series of rows and columns or in a X − YCartesian system and in most cases the data are stored in a uniform grid width. Since a gridcan be stored as a two-dimensional array of elevations the handling of elevation matrices issimple, and grid-based surface modelling algorithms tend to be relatively straightforward andfast. The major drawbacks are a) the inability to accommodate relief changes in rough terrain,b) uniform resolution although DEMs vary non-uniformly and c) there is no explicit modellingof characteristic points and lines. Raster DEMs cannot handle abrupt changes in elevation eas-ily and they will often skip important details of the land surface (Carter, 1988). An exampleterrain using a regular grid is shown in Fig. 2.2(a). Raster DEMs are constructed via interpo-lation at the desired resolution. Numerous interpolation methods, ranging from sophisticatedbut computationally expensive methods to simpler and efficient methods, for grid DEMs havebeen developed, see (Mitas & Mitášová, 1999) for a review.
TIN DEMs
Another possibility of storing a model is a TIN. TINs are based on triangular elements (facets),with vertices at the sample original points (Peucker et al., 1978; Moore & Hutchinson, 1991).These facets consist of planes joining the three adjacent points in the network and are usuallyconstructed using Delauney triangulation (Weibel & Heller, 1991). Points are stored as a set ofX , Y , and Z values together with pointers to their neighbours in the net (Moore & Hutchinson,1991). TIN models are used to provide better control over surface slope, aspect, surface areas,volumetric and cut-fill analysis and generating contours.
Fewer points than for a raster DEM are needed for a DEM of the same accuracy, and surfacefeatures can be incorporated into the model because the density of the triangles can be varied tomatch the roughness of the surface (Moore & Hutchinson, 1991). TINs can be organized intoa hierarchical model so that they can represent a surface in various levels of detail. However,topological relations have to be stored explicitly (Kumler, 1994). Thus, TINs become morecomplex and also more difficult to handle. The resolution adapts to the terrain and the initialconstruction is time consuming. TIN data can be stored in several methods, namely, triangle-by-triangle, points and the neighbours, and side-based. An example of a triangulated irregularnetwork can be seen in Fig. 2.2(b).
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(a) Raster DEM. (b) TIN DEM.
Figure 2.2: Various data structure DEMs.
Other DEMs
The above described models represent the most common types that are used for surface mod-elling, but they are not the only ones.
Quadtree DEMs. The quadtree data structure is a grid-based structure and has variable reso-lution. The details of the principle and concept of this structure can be referred to Samet (1984).A major drawback of this data structure is the difficulty to modify the data, as in such a casea recalculation of the quadtree is needed. In terms of storage space, quadtree needs significantless space compared to the raster DEM data structure.
Contours. Contour lines are the traditional method for representing a 3D surface. They areisolines of elevation. They are excellent for human interpretation, but inferior to DEMs, andTINs for computer display and analysis. Contours are stored as 2D lines with an attribute orlabel containing the appropriate elevation value. The contour interval is chosen to representas much surface detail as possible without making the map overly crowded. Flat areas haveusually smaller contour intervals while hilly areas are represented with larger contour intervals.
2.1.3 Categorization According to the TechnologyThe base data for DEM estimation are terrain points. Historically, these points were acquiredmanually through ground-based surveying or stereo measurement of contour lines or terrainpoints. These methods have now been superseded by less labour-intensive alternatives. Nowa-days, there are three main technologies for acquiring 3D points over large areas, photogram-metry, radar and lidar.
The choice of the technology is critical for the quality of the resulting DEM. All data ac-quisition methods have advantages and disadvantages. There may be other criteria apart formDEM quality requirements, which will guide the selection of a particular technology for a givenapplication (e.g. purpose, efficiency, availability of source materials, cost). The cost for gener-ating DEMs can become significant for increased resolution and accuracy. In order to choose
15
the most effective technique, a trade-off between accuracy and production cost always has tobe taken into consideration.
Photogrammetric DEMs
As a tool for topographic mapping, photogrammetry has a long history. Traditionally elevationdata was extracted from stereo aerial photography in the form of contours. DEM generationwas computer-supported with the advent of analytical stereoplotters and then further processautomation accompanied the introduction of digital aerial imagery. The generation of a DEMfrom digital aerial or satellite imagery is today a fully automatic batch process, with the result-ing elevation model often being employed to support orthoimage generation.
Optical DEMs are derived applying photogrammetric techniques on two or more opticalimages (aerial or satellite imagery) by measuring conjugate points in these images.
Automatic image matching is the direct successor of manual stereo-photogrammetric mea-surements. The technology is nowadays mature. Since the advent of digital cameras, imageblocks are routinely recorded with large overlaps and thus high redundancy, which allows oneto generate DEMs with an accuracy comparable to airborne LiDAR (Leberl et al., 2010). Themain disadvantages of the technology is that it can only deliver DSMs, and that, being a pas-sive technique, it cannot cope well with untextured regions. When applied to optical satelliteimagery, automatic matching can cover very large areas, making it suitable for wide-area DEMgeneration.
Although digital photogrammetric methods can produce very satisfactory results (West-away et al., 2000), there are a number of concerns with digital DEM production.
1. The parallax estimation procedure fails in the presence of large textureless surfaces andareas of poor definition (Baltsavias, 1999), i.e. wetlands, clouds, shadows, sand coasts,dunes, desserts, swamps, snow and ice fields, or glaciers.
2. Strong differences between images (radiometric and geometric, partly caused by multi-temporal differences, for example due to occlusion or differences in shadows), can resultin errors in the derived heights.
3. Stereo optical DEMs become much more rugged because of the spikes induced by con-jugate point mismatches in the stereo-pair (Honikel, 1998).
LiDAR DEMs
LiDAR is an active remote sensing technique that provides an attractive alternative to pho-togrammetry. Airborne laser scanning has over the last decade become the dominant technol-ogy for smaller DEMs. The technology offers high density and accuracy with little processingoverhead. A further advantage is that in vegetation areas the laser pulse partially penetratesthe canopy, which allows one to acquire both a DSM and a DTM from the same sensor data.However, the technology is difficult to scale up to wide areas, because the costs for data captureincreases considerably. At this point, LiDAR DEMs are not available for most of the world:complete coverage is only available for a few small countries like for example the Netherlands;
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for several other countries, especially in Europe, significant parts are covered (e.g. Switzerland,UK, Germany).
ALS utilises a narrow laser beam for a high frequent range determination to illuminatedobjects (Wehr & Lohr, 1999; Pfeifer & Briese, 2007). The ground surface, vegetation canopy,or other obstacles reflect the pulses, and the instrument’s receiver detects some of the backscat-ter. The power received by the sensor will depend upon target characteristics, including thephysical properties of the target (Andersen et al., 2006). LiDAR systems used for topographicmapping applications usually operate in the near infrared range of the electromagnetic spectrum(800− 1550nm) and use pulsed lasers.
The ALS measurements result in a three-dimensional cloud of points with irregular spacing(Briese et al., 2007) using GPS and an IMU for direct geo-referencing.
LiDAR can produce high quality DEMs with horizontal resolution of less than one meter.It allows a rapid and cost-effective measurement of topography at high spatial resolutions overlarge areas but its cost is prohibitive for vary small or very large projects.
The two major problems are the detection and correction of systematic errors in the LiDARdata and separation of ground points from points resulting from reflections on buildings, vegeta-tion or other objects above the ground (Ziegler et al., 2000). With LiDAR a certain penetrationof the canopy can be achieved, and this permits in general the detection of both ground andtree tops (Baltsavias, 1999). The penetration rate mainly depends on type of trees (deciduousor coniferous) and season.
Some examples of issues that are related with LIDAR data sets are:
1. Errors in LiDAR DEMs do not follow a normal distribution, except over bare ground.
2. In areas covered by vegetation, the tendency for LiDAR is to yield elevations above theground due to returns off the canopy.
3. In built-up areas, there are many returns on objects above the ground.
4. LiDAR tends to measure elevations a bit below the ground on the dark asphalt surfacesthat are common to roadways and urban areas.
5. Some surface materials provide poor reflectance (Huising & Gomes Pereira, 1998).
6. It is blocked by clouds and it is not weather independent.
7. Theoretically, it is independent of the sun-light (can be operated in the night) (Brieseet al., 2007). Actually, the less the background radiance, especially from sunlight, thebetter its maximum range performance (Baltsavias, 1999).
8. Unlike photogrammetric methods, it is capable of mapping areas characterized as lowcontrast and relatively dense vegetation.
9. In coastal zones and forest areas, LiDAR is considered as a superior data collection toolover conventional photogrammetric techniques where it is extremely difficult to locateterrain points in the imagery.
17
10. Shorelines, stream channels, ridge lines and other types of breaklines within the terrainmay be missed.
11. While some fog is manageable, generally it is not usable under heavy fog or extendedcloud cover.
12. Because of the small field of view surface discontinuities like building walls are bettermodelled (less problems with obscuration).
13. LiDAR outperforms photogrammetry for the mapping of long, narrow features (e.g. roadmapping, planning and design, power-line corridor planning and tower design, coastalerosion monitoring, coastal zone management, traffic and transport, river-ways and wa-ter resources and traffic management, mapping of railway lines, fiber-optic corridors,pipelines, dikes, etc.) and of very small objects with very good reflection (e.g. powerlines), which are hardly visible in optical images, or whose measurement cannot be au-tomated (Baltsavias, 1999).
Radar and SAR Interferometry DEMs
Synthetic Aperture Radar (SAR) interferometry is a widespread technology in satellite remotesensing (Toutin & Gray, 2000; Toutin, 2000). It has the important advantage that data ac-quisition is independent of daylight, and mostly also atmospheric conditions. Depending onthe wavelength, SAR in principle allows to generate both DSMs (X- and C-bands) and DTMs(L- and P-bands), however most satellite sensors operate in the short wavelengths and deliverDSMs (Leberl, 1990; Costantini et al., 1999). Airborne SAR is less widespread, but has alsobeen used to generate wide-area DEMs, employing usually X- or C- band and P-band.
Some examples of issues that are related with Radar height data are:
1. InSAR DEMs accuracy depends on the terrain cover.
2. The degree of penetration depends on vegetation gap structure, canopy structure (multi-ple or single canopy), leaf-on versus leaf-off, wetness, ground reflectivity, and tree type(Bhang et al., 2007).
3. The penetration depth of the radar signal depends on wetness, temperature, and porosityof snow cover (Braun & Fotopoulos, 2007).
4. InSAR works fine in the presence of areas with low texture.
5. InSAR fails under high terrain slope or rapid change in surface roughness. This phe-nomenon causes DEM smoothing and thus introduces low frequency artefacts (Honikel,1998).
6. It is not affected by cloud blockage. Mapping in tropical regions that are often cloud-covered can be done more reliably with InSAR systems that penetrate clouds (Mauneet al., 2001).
18
7. The obtained images are independent of the weather conditions and sun illumination.
8. Urban mapping is problematic for InSAR due to the extremely complex scattering envi-ronment (Maune et al., 2001).
9. Densely vegetated surface can be problematic if bare surface elevations are desired.Heights measured by InSAR systems are reflective surface heights and can lie every-where within the canopy.
Geodetic DEMs
Geodetic DEMs are obtained using field measurements. Traditional topographic survey tech-niques determine the position of a point through the measurement of distance and angles. Thetraditional instruments are computerized total stations and GPS stations. Since ground surveydata tend to be very accurate, the accuracy of the resulting DEM is very high. However, as thesetechniques are expensive, labor-intensive and time consuming, their use is limited to small ar-eas or they complement other types of DEMs or they are used as reference DEMs for qualityevaluation reasons.
Contours DEMs
A cost-effective method for producing a DEM is interpolating elevations from contours of topo-graphic maps. These analogue data are digitized through manual digitization, semi-automatedline-following, or by means of automatic raster scanning and vectorization (Weibel & Heller,1991).
Contour DEMs present many artefacts and other distortions. On the other hand, in case ageometry item never intersects a contour line, no height information can be directly obtainedand associated to it. Furthermore, errors may be introduced (in drawing, generalization) anda lot of the original information is lost in the map making process. Consequently, contourdata yield DEMs of limited accuracy. In terms of efficiency, the speed of operation for mapdigitization is very slow.
19
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USA
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2
20
2.2 Error Types in DEMsDEM data contain errors which are classified into three types: blunders, which are removedor not prior to entry in the data base; systematic errors, which occur in sensors and processingmethods; measurements noise; and random errors, which are of a purely random nature andare completely unpredictable (USGS, 1997b). Although all three types may be reduced inmagnitude they cannot be completely eliminated.
2.2.1 Systematic ErrorsSystematic errors in DEMs extend across large regions of pixels and are often best identifiedusing existing data of higher accuracy or expert knowledge of the surface within the map region(USGS, 1997a). They follow some fixed pattern or rule, are generally of constant magnitudeor sign, are introduced by sensors and processing methods, and are predictable. These types oferrors cause bias or artefacts in the final product such as systematic striping (Polidori, 1991) orunder- and overshoots of the approximation method (Florinsky et al., 2002). Systematic errorscan be eliminated or substantially reduced when the cause is known.
2.2.2 BlundersFor DEM data, a blunder is a vertical error usually of major proportions often exceeding max-imum absolute error (3 σ) and as such it is easily identifiable. Moreover, a blunder is an indi-cation that the data collection process has deteriorated beyond the level of simple systematicor random errors. Spikes are spurious points such as anomalously high or low values of smallextent. Wherever detected, errors caused by blunders or spikes must be removed, since theymay introduce significant bias in the surface analysis (Wise, 2000).
Various methods were developed for detecting gross errors in DEM source data (Felícisimo,1994; Hannah, 1981; Li, 1990; Li et al., 2005; Lopez, 2000). Their detection has been basedon three alternative methods:
1. Automatic methods based on the divergent anisotropy and fractal dimensions of the DEM(Polidori, 1991; Brown & Bara, 1994),
2. Automatic methods based on anomalous slopes (Hannah, 1981; Li et al., 2005) where in afirst step, indicators of correctness are computed for individual DEM elevations based ongradient differences around each DEM point and in a second step, an iterative procedureis employed to constrain errors to the elevation of adjacent points based on correctnessindicators, and
3. Manual methods based on DEM visualisation (Carter, 1989; Wood & Fisher, 1993;Wood, 1994, 1996; Wise, 1998).
The value for the threshold for blunders is not a fixed value and changes according to thecharacteristics of each DEM. For normal distribution, it is generally accepted to be set at 3times the RMSE error.
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2.2.3 Random ErrorsRandom errors result from mistakes, such as inaccurate surveying or improper recording ofelevation information and from accidental or unknown combinations of mistakes (Wechsler,1999). The identification of these errors is beyond the control of the DEM user. Random errorsremain in the data after blunders and systematic errors are removed (USGS, 1997a).
Random errors are considered to be normally distributed and they meet the following char-acteristics (Greenwalt & Shultz, 1962):
1. Variation in sign - positive and negative errors occurring with equal frequency,
2. Small errors occur more frequently than large errors, and
3. Extremely large errors rarely occur.
2.3 DEM Quality AssessmentThe topic of DEM quality assessment has been the subject of much debate and study becausein conjunction with mass DEM production there is a growing need for procedures for qualityevaluation and correction of errors. DEM quality is measured by how accurate the elevationis at each point and how accurately the morphology is presented. In general, errors affectevery surface model because of the inexact nature of sampling a continuous surface functiondiscretely (Shearer, 1990) and inevitable measurement noise. Many studies have examined thecauses of DEM errors and several methods have been proposed for the detection of errors andestimation of their magnitude and/or spatial distribution (Polidori, 1991; Brown & Bara, 1994;Li, 1994). Two additional research topics related to the DEM errors are their visualization andcorrection.
According to (Li, 1992) and (Yue et al., 2007) the accuracy of a DEM is a result of manyindividual factors listed below:
1. The inherent sensor distortions
2. Attributes of the source data such as accuracy, density, and distribution.
3. Characteristics of the surface (surface relief, land cover, texture) and
4. The mathematical methods used for the construction of DEM surface, i.e. DEM genera-tion algorithms and interpolation techniques.
Before we proceed further with the topic of accuracy assessment, we should give the defi-nitions of three basic crucial terms:
Absolute accuracy is the closeness of an estimated, measured, or computed value to a stan-dard, or true value of a particular quantity. In mapping, a statement of absolute accuracyis made with respect to a datum, which is, in fact, also an adjustment of many measure-ments and has some inherent error. The statement of absolute accuracy is made withrespect to this reference surface, assuming it is the true value.
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Relative accuracy is an evaluation of the amount of error in determining the Z-error of onepoint or feature with respect to another. For example, the difference in elevation betweentwo points on the earth’s surface may be measured very accurately, but the stated eleva-tions of both points with respect to the reference datum could contain a large error. Inthis case, the relative accuracy of the point elevations is high, but the absolute accuracyis low.
Precision is a statistical measure of the tendency for independent, repeated measurements of avalue to produce the same result. A measurement can be highly repeatable, therefore veryprecise, but inaccurate, if the measuring instrument is not calibrated correctly. The sameerror would be usually repeated in every measurement, but none of the measurementswould be accurate. Errors do not exist only in the measurements; they also exist in thedata processing methods, e.g. wrong datum transformation.
ASPRS Accuracy Standards for Large-Scale Maps (ASPRS, 1990) indicate that verticalaccuracy (height point accuracy) is a function of horizontal accuracy. Here, quality is equal toheight accuracy. We will not consider errors in the intermediate steps in the process of DEMgeneration, but we will concentrate on the errors in the final product.
2.3.1 Global Accuracy ParametersThe quality of a DEM should denote how accurate is a height value at a certain position. It isclear that a given single value as an accuracy measure of a DEM is insufficient. A single globalmeasure assumes that the error is uniform in the area covered by the DEM, an assumptionwhich is of course not valid.
Comparison with Reference Data
Quality control of a DEM may be performed by comparison with reference data using statisticalmethods. The reference elevation data ought to be at least three times more accurate than thesample DEM data. Many of the statistical procedures assume that data are normally distributed.Unfortunately, when there are outliers in data, classical statistical methods often have very poorperformance and large deviations from the normal distribution can cause problems (Höhle &Höhle, 2009).
In order to compare two DEMs with different grid spacing the DEMs need to be resampledor one can compute the point - to - surface distance. Resampling is a process of changing theresolution of a DEM. There are two types of resampling: a) upsampling, and b) downsampling.The first one increases the resolution; whereas the second one reduces the resolution of thedataset. Resampling is an irrevocable procedure because it is a many-to-one (or one-to-manyin case of upsampling) type of operation. Upsampling in geosciences is rarely performed. Thisfact justifies that resampling will mean downsampling, whenever possible.
Statistical Accuracy Measures. The following accuracy measures for DEMs can be applied:
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Maximum height error max |4hi|
Mean error µ = 1n
∑ni=04hi
Mean absolute error µ± = 1n
∑ni=0 |4hi|
Median µ =
{4h(n+1)/2 : if n is odd
12(4hN/2 +4h1+N/2) : if n is even
Standard deviation σ = 1(n−1)
√∑ni=0(4hi − µ)2
Root mean square error RMSE = 1n
√∑ni=04h2i
Mean absolute deviation MAD0 = 1n
∑ni=0 |4hi − µ|
Median absolute deviation MAD1 = median |4hi − µ|
where,4hi denotes the difference (error) from the reference data for a point i, and n is thenumber of tested points in the sample (sample size).
For the evaluation of a DEM the most widely used measure, usually the only one, is thewell known Root Mean Square Error (RMSE) (Li, 1988; Yang & Hodler, 2000). There are notfew authors (Wood, 1996; Kraus, 2004) who have expressed objections about its validity. Inthe calculation of the RMSE there is the assumption that there is no bias in the error, which isoften invalid (Li, 1993). Moreover the RSME does not give any information about the meandeviation between the two measures of elevation. Finally, the magnitude of the RMSE value isalso influenced by the variance of a true elevation distribution and depends on the relative reliefand scale of measurements. In order to overcome the disadvantages of non robust measuresthree more robust measures are used: the median, the mean absolute deviation (MAD0) and themedian absolute deviation (MAD1). They are more robust estimators when outliers exist in thedata. In the MAD0 and MAD1, the magnitude of the distances of a small number of outliers isirrelevant (Wikipedia, 2012).
Histograms and Q-Q Plots. The distribution of errors can be visualized by a histogram ofthe errors, where the number of errors (frequency) within certain predefined intervals is plotted.Such a histogram gives a first impression of the normality of the error distribution. A betterdiagnostic plot for checking a deviation from the normal distribution is the so-called quantile-quantile (Q-Q) plot. The quantiles of the empirical distribution function are plotted against thetheoretical quantiles of the normal distribution. If the actual distribution is normal, the Q-Qplot should yield a straight line (D’Agostino et al., 1990).
Visual Inspection
The method most often used for DEM quality assessment is visual inspection. Visual methodscan be very useful for highlighting surface artefacts and can balance some weaknesses of sta-tistical methods. For example, water bodies tend to be problematic areas and generally requiresome sort of manual editing. The problem in direct visualisation of DEM heights is that the
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result is not discriminating, the surface features within a certain relative elevation thresholdremain invisible (Oksanen, 2006) and they depend largely on the expertise and experience ofthe operator. To overcome these limitations, a number of methods have been used, includingshaded relief maps, slope and aspect maps, maps based on edge detection filters (e.g. Laplacianfilter), profile and plan curvature maps, RMSE maps, modulo maps and a-spatial representa-tions, such as hummock plots (Wood & Fisher, 1993; Wood, 1996).
In the following, some visualization techniques for DEM quality assessment are shortlyexplained.
2D Raster Rendering. One of the most common ways to display DEMs is to associate eachelevation with a color or gray value. The resulting image broadly indicates topography andmight also imply the existence of blunders which appear as localized deviations in the elevationvalues.
Bi-polar Difference Maps. When two DEMs are available, one of which is known to be ofhigher accuracy than the other, a difference map is produced using simple map algebra.
Hillshading. Hillshading provides a convenient way of qualitative relief depiction (Yoëli,1967; Brassel, 1974). Landforms may be readily perceived from shaded relief displays. Awell-known problem in shaded relief images is that the choice of direction and elevation for thelight source has a major influence on the visualisation result (Imhof, 1982).
2.3.2 Local Accuracy ParametersThe use of global characterisation of DEM error is a gross generalisation of reality. The verticalaccuracy varies within a single DEM, based on the type of the surface and the land cover beingmapped. In other words, the accuracy itself is a spatial variable. The quantification of a spatialvariable (DEM quality) is much more complex than a simple calculation of RMSE. In contraryto global quality parameters, local ones describe the quality of a DEM at a high level of detail(Karel & Kraus, 2006), since they describe the quality of each grid point of a DEM.
Theoretical Mathematical Models
A formal mathematical model of DEM accuracy is a model obtained through purely theoreticalanalysis (Li, 1993). A number of attempts have been made by several scientists (Makarovic,1972; Kubik & Botman, 1976; Frederiksen, 1981; Tempfli, 1980; Frederiksen et al., 1986) toestablish mathematical models for investigating the height accuracy of DEMs in a theoreticalframework. According to (Li, 1993) these theoretical models have not found their way topractical applications. We describe some of these existing models on the next paragraphs. Theanalytical approaches are primarily used when reference data is not available (Martinoni &Bernhard, 1998).
Makarovic (1972) used Fourier analysis to assess the accuracy of DEMs, considering sam-pling and reconstruction from sinusoidal functions. The fidelity of a DEM is represented by
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the ratio of the mean value of the magnitude of the linearly constructed sinusoidal waves to theamplitude of the input waves. Makarovic (1975) then converted the fidelity figures into stan-dard deviation values. Brown & Bara (1994) used semivariograms and fractals to detect thepresence of errors in 7.5’ USGS 30 m DEMs and evaluated several types of filters for reducingthe magnitude of these errors. Their method does not require reference values.
Models based on the DEM Production Method
Depending on the specific methodology used for DEM generation, some empiric models havebeen presented in the past years, that permit the estimation of DEM height accuracy a posteriorilocally.
ALS. For DEMs produce with ALS, (Kraus & Karel, 2006) propose the following formula:
σH [cm] = ±(
6√n+ 50 tanα
)(2.1)
where σH is the standard deviation of the DEM, n is the number of points per square meter(density), and α is the surface slope.
Photogrammetry. (Kraus, 2004), has presented the following formula for stereo photogram-metry, which can be used for open terrain:
σH [m] = ±(0.15h of h+ 0.15
h
ctanα
)(2.2)
where σH is the standard deviation of the DEM, h is the height of the flight over the ground inm units, c is the focal length in mm units, and α is the surface slope.
RADAR. One of the valuable map products generated by InSAR data is the local height errormap which is generated from the correlation measurements (Maune et al., 2001). The heighterror map accuracy is assessed using the local relative height error at a point p, σhp , which isdefined as
σhp =
√√√√ 1
#(B)
∑q∈B
(hrq − htq
)2 − [ 1
#(B)
∑q∈B
(hrq − htq
)]2(2.3)
where hr is a radar height value, ht is the corresponding ground truth height value, B is the setof points in a neighbourhood of the point p (to be defined), and #B is the number of points inB. B should be a box centred at p with size equal to 5 pixels. The estimate of σhp should beconsidered valid if and only if #B is greater than or equal to 10.
In order to quantitatively assess the geometric, radiometric, and polarimetric quality of SARproducts, SARMAP (SARMAP, 2008) built a quality Assessment Tool (QAT). They deliverSAR DEMs with an accuracy map. The values of the accuracy map are derived according tothe formula:
σ = AF · (1− coherence2)/(2− coherence2) (2.4)
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where AF = R · (sin(θ)/(Bn · 4π/λ)), R is the range , θ is the local incidence angle, Bn isthe baseline normal component and λ is the wavelength. Coherence is the magnitude of aninterferogram’s pixels, divided by the product of the magnitudes of the original image’s pixels.It ranges from 0.0, where there is no useful information in the interferogram to 1.0, wherethere is no noise in the interferogram. Coherence serves as a measure of the quality of aninterferogram.
Raw data-based Measures
ALS and digital photogrammetry produce millions of height points, randomly distributed andwithout topological information. Karel & Kraus (2006) and Karel et al. (2006) give some localmeasures that assess indirectly the quality of a DEM and their calculation is based on theseprimary collected data points (initial raw data):
1. The density of the original points n. It shows the number of points per area unit. It canbe easily visualised and reveal zones that are covered by too few or no data at all.
2. The distance between each depicted grid point and its nearest source data point. It iscomputed efficiently using the Chamfer function (Borgefors, 1988).
3. The DEM curvature.
4. A weighted RMSE is used that is calculated using the discrepancies in height betweenthe original data and the DEM surface.
5. The cofactor in height of an adjusting plane through the neighbouring original points.
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Chapter 3
Fusion Processing Chain
3.1 Overview of DEM Fusion ProcessingDEM fusion requires well-defined techniques as well as a good understanding of the inputdata. There are many aspects to be looked at before being able to implement a DEM fusionapproach. The fundamental issues that should be addressed in building an effective and realisticDEM fusion system include:
1. Which types of data are the most useful.
2. Estimation of weights for the input DEMs.
3. What architecture should be used (i.e., where in the processing flow should data befused).
4. What are the necessary pre-processing steps involved.
5. Which fusion method of the data is the most successful.
These and other questions comprise a large number of parameters to be considered.The present work is built according the following crucial assumptions:
1. The input DEMs should be in raster format.
2. Currently, only two DEMs can be fused simultaneously.
3. The input DEMs should have similar resolution and especially similar accuracy.
4. The input DEMs should contain coinciding grid points (100% overlapping), which meansthat either the two grids are identical or that the resolution of one DEM is an integer ofthe resolution of the other DEM.
To perform the fusion of two DEMs, the procedure shown in Fig. 3.1 is proposed:
1. Change the input DEMs to the same coordinate system.
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2. Co-register the input DEMs.
3. Re-sample the DEMs to the required grid spacing and boundary definitions.
4. Assess the quality of the input DEMs. Weight maps are needed for each input DEMpoint.
5. Fuse the DEMs.
6. Assess the quality of the output DEM.
We fuse two DEMs, called DEM1 and DEM2, with resolutions r1 and r2, where r1 < r2, witharea of coverage S1 and S2, where S1 ⊆ S2, and we produce a new DEM, with grid spacingr1 and area of coverage S1.
The only a priori information that is given for the input DEMs is their production technol-ogy (e.g LiDAR, photogrammetry, SAR), the acquisition date (leaves on/off) and one globalmeasure of accuracy(e.g σ, RMSE). If the input surface models are available as point clouds, aregular grid is generated with grid size equal to the average point distance in order to fulfil theprerequisite of the grid format.
The fusion can function for DSM to DSM or DTM to DTM. For DSM to DTM the above-ground objects (an possibly a slightly larger area) should first be detected in the DSM and givenweights, 0 or 1, depending on whether the output should be a DTM or a DSM respectively.
Input DEMs may contain NoData values. In raster data, this is the absence of a recordedvalue. All the calculations, accuracy assessment and DEM fusion ignore the points that areassigned NoData values.
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Figure 3.1: Processing flow chart for DEM fusion.
3.2 Pre-processingThere are some issues that have to be dealt with before the fusion can be performed. Thepre-processing part includes the following steps:
1. Transformation to the same geographical coordinate system.
2. Co-registration of the input DEMs.
3. DEMs resampling.
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3.2.1 DEM Co-registration
All input data should be transformed into a common coordinate system after finding the correctmapping of one DEM onto another. The first step in fusing multiple DEMs is registering themto a common frame of reference (González & Woods, 2008). It is a prerequisite for furthersteps in order to remove the potential horizontal and vertical shifts and less probably rotationand scale differences between input DEMs before applying fusion techniques.
Registration can be performed by traditional co-registration techniques. Among meth-ods that have been reported for surface registration, the Iterative Closest Point (ICP) (Besl &Mckay, 1992) algorithm has been recognized as the standard for surface matching in computervision.
In order to co-register the two input DEMs we used the Least Squares 3D (LS3D) surfacematching method (Grün & Akça, 2005). LS3D is a rigorous algorithm that matches a “slave”3D point cloud to a “master” point cloud. LS3D estimates the transformation parameters be-tween the two DEMs using the general Gauss-Markoff model, minimizing the sum of squaresof the Euclidean distances between the surfaces. The geometric relationship between the con-jugate DEMs is defined as a 7-parameter 3D similarity transformation (X, Y, Z translations,ω, ϕ, κ rotations, and scale). This parameter space can be extended or reduced, as the situationdemands it. In most cases we use a 3-parameter transformation (X, Y, Z translations). Themathematical model is a generalization of the Least Squares image matching method, in partic-ular the method given by Grün (1985). LS3D was originally developed for the co-registrationof LiDAR point clouds and surfaces but it has also been used for inspection, comparison andvalidation studies (Akça, 2007).
For the co-registration procedure, it should be known which of the two DEMs is more“correct”. This is usually determined either from the DEMs’ resolutions or from the givenglobal accuracy measure. The apparently more accurate DEM is then used as the master towhich the other one (slave) is adjusted.
LS3D gives the user two important possibilities (a) to use control patches, e.g. for the co-registration DSM to DTM, and (b) to exclude blunders from the estimation of the 7 parameters.
After co-registration, of importance is the evaluation of the accuracy with which the DEMshave been spatially aligned relative to each other. The Euclidean distances give appropriatemetrics for the co-registration quality. The Euclidean distance provides the so-called “residualmaps”. Calculating the mean, standard deviation, maxima and minima, RMSE, and analysingthe distribution of the residuals gives an indication of the co-registration errors and often im-plicitly the combined errors of the input DEMs. The residual maps are used extensively duringthe quality assessment step.
3.2.2 DEM Resampling
The next step in the processing chain is re-sampling. The different resolutions (grid spacing)and the different boundary definitions of the two input DEMs make it difficult to establish thecorrespondence between the DEMs during fusion.
We use interpolation for densification/coarsening of the original rectangular grid, so calledre-sampling.
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Abundant literature exists on methods for interpolation of DEMs although there is no op-timal interpolation algorithm that is clearly superior to all the others and appropriate for allapplications. The quality of the resulting DEM depends on the distribution and accuracy of theoriginal data points (Weibel & Heller, 1991).
The adopted re-sampling approach is bicubic interpolation (Keys, 1981). Bicubic interpo-lation considers the 16 grid cells around it (4 × 4 cells), while computing a weighted average.Bicubic interpolation is a good combination of processing time and output quality in compari-son to the nearest neighbour (1 cell) and bilinear (4 cells) methods.
DEM resampling is not a necessity for all the DEM fusion methods, i.e. DEM fusionmethods that are based on data fitting do not require resampling.
3.3 Core Procedure
The core fusion procedure includes the following steps:
1. Quality assessment and derivation of weights.
2. DEM fusion.
3.3.1 Quality Assessment
The ability to detect errors in the input DEMs, prior to fusion, is a crucial part of the proposedprocedure. Indeed, emphasis is placed on this aspect, as without a sense of accuracy of theinput data, DEM fusion is useless. Each individual DEM is precisely evaluated by calculatinga variety of quality measures. The output products of this step are two accuracy maps, one foreach input DEM. These maps provide local accuracy values, one for every individual grid cell.Local accuracy is a value that represents the uncertainty of a height point in an absolute sensein meters. Then, weights must be derived. Absolute accuracy values have the same units as themeasured heights (m) while the relative accuracy values (weights) are ratios, and have no units.
3.3.2 DEM Fusion
In the following step, the fusion is conducted. Fusion decisions are based on the height infor-mation from both DEMs and both quality maps. The desired end result of fusion is a singleoutput DEM where problems specific to the integrated source data (in this case the two inputDEMs) are hopefully sometimes abated.
The quality assessment and DEM fusion step will be described separately in the Chapters 4and 5, respectively.
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3.4 Post-processing
3.4.1 Quality Assessment
The final step of the proposed methodology is the evaluation of the achieved results i.e thequality assessment of the produced DEM. This is a considerably complex task because of theheterogeneity of the involved source data and the different fusion methods. During qualityassessment the two crucial factors to be taken into consideration are the different accuracies ofthe input DEMs as well as the chosen approach of DEM fusion itself.
3.5 DEM Fusion Literature Survey
3.5.1 Introduction
It is only recently that DEM fusion has attracted research attention by the scientific commu-nity. Soon after the increase of the DEMs production systems, researchers began consideringDEM fusion as a necessity in order to benefit from the information load. Since the end of the1990s and throughout the 2000s DEM fusion was established as a subject through publicationspresenting DEM fusion approaches. Publications emerged from both research and industrialorganisations. In this section an overview of the literature published on DEM fusion from theend of 1990s is given. Some related research published after the start of this work is also dis-cussed. Sec. 3.5.2 is devoted to the review of methods for the fusion of DEMs that are producedusing RADAR images. The following Sec. 3.5.3 reviews the subject of fusing DEMs that havebeen produced using different technologies.
3.5.2 Fusion of InSAR DEMs
A straightforward way to fuse RADAR DEMs is to combine ascending-descending orbit (Car-rasco et al., 1997). Two ascending and descending ERS DEMs have been calculated indepen-dently and a combined DEM has been obtained from them on a coherence basis.
A weighted average of many uncorrelated topographic profiles obtained from independentSAR interferograms can reduce the impact of phase artefacts on the final fused DEM. Ferrettiet al. (1998) use a wavelet approach that allows a wave number dependent DEM weighting.
Knöpfle et al. (1998) describe the generation of DEM mosaics primarily based on interfer-ometric SAR using height error maps that represent the accuracy of the corresponding DEMs.The fusion takes into account the different prior accuracy of the DEMs and derives the qualityof the resulting DEM.
Ferretti et al. (1999) exploit multi-baseline SAR interferometry for the generation of a high-quality DEM taking into account both noise and atmospheric effects. A weighted combinationof many uncorrelated topographic profiles strongly reduces the impact of phase artifacts on thefinal DEM. The key issue is weights selection and to this end a wavelet domain approach isproposed. Taking advantage of the particular frequency trend of the atmospheric distortion, itis possible to estimate, directly from the data, noise and atmospheric distortion power for each
34
interferogram. The available DEMs are then combined by means of a weighted average, carriedout in the wavelet domain.
Slatton et al. (2002) combined spaceborne InSAR data from the ERS-1/2 platforms withmultiple sets of airborne C-band INSAR data acquired by the NASA/JPL TOPSAR platformto obtain statistically optimal high-resolution estimates of topography. They fused the INSARDEMs using a multi-scale Kalman smoothing approach. The estimated topography preservesthe spatial resolution of the TOPSAR data, while smoothing noise and providing estimateswhere there was no TOPSAR coverage.
Eineder & Holzner (2000); Eineder & Adam (2005) present a technique of a maximum-likelihood algorithm that is capable to fuse a number of heterogeneous synthetic apertureRADAR interferograms into a single DEM. The fusion process takes place in the object space,i.e., the map geometry, and considers the periodic likelihood function of each individual inter-ferometric phase sample. The interferograms may vary regarding their radar wavelength, theirbaseline, their heading angle (ascending or descending), and their incidence angle. Geometricbaseline error estimates and a priori knowledge from other estimates like existing DEMs areincorporated seamlessly into the estimation process. The algorithm is targeted to fuse an en-semble of interferometric multiangle or multibaseline observations in areas of rugged terrainor highly ambiguous data where algorithms based on phase unwrapping may fail.
3.5.3 Fusion of DEMs Produced with Different TechnologyCrosetto & Crippa (1998) proposed a two-step DEM fusion approach. They first used a SPOTderived DEM to refine the interferometric flattening. They also used prior knowledge of theterrain to improve the ghost-line unwrapping technique (Goldstein et al., 1988) of InSAR. Later,they used weighted interpolation of SPOT DEM and ERS InSAR DEMs from ascending anddescending pairs to jointly estimate the terrain.
Honikel (1998) used the synergy between stereo optical DEM and Radar DEM in the spatialfrequency domain to filter out the error prone components of SPOT DEM and ERS-1 DEM. Thefused DEM was improved in terms of RMSE and error distribution. In another study, Honikel(1999) introduced a DEM fusion process, which took advantage of the synergy between In-SAR DEM and stereo optical DEM generation, by weighting the height values in both DEMsaccording to the estimated error. The study tested the fusion approach with SPOT and ERSDEMs having very different accuracies.
Schultz et al. (1999) developed a methodology to fuse two stereoscopic DEMs. The method-ology had two key steps: (1) detection of unreliable elevation estimates, and (2) fusion of thereliable elevations into a single optimal terrain model.
Audenino et al. (2001) propose two approaches for DEM fusion. The first one is called TopDown strategy and consists in the refinement of a part of the DEM of interest for the aimedapplication, using multi-sensor and multi-resolution data. The second one is called BottomUp approach and consists in the computation of a DEM, also using multi-sensor and multi-resolution information. Parallel with this strategies applicable to any representation of theDEM, they choose to use the Delaunay triangulation. Using this modelling, they preserve thedata quality and facilitate the fusion by allowing easy deletion or addition of points. It alsopermits the integration of constraints to structure the DEMs.
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Damron (2002) uses a seven-step methodology to fuse InSAR and LiDAR DEMs withArcInfo. The first step is to import the LiDAR and InSAR DEMs into ArcInfo. The secondstep, a decision step, is where the InSAR and LiDAR datum and geoid models are examined tosee if they are in orthometric heights or ellipsoid heights. After the determination, the secondstep branches into two sub determination phases. The first branch determines the geoid modelused for the orthometric heights and extracts it from the InSAR and LiDAR DEMs. The secondbranch determines which datums were used and projects to a common 3-D projection. Thethird step performs a differencing function on the LiDAR and InSAR DEMs to determine theseparation between the two DEMs. In the third step, GPS data can be used to help in thisdetermination or just use the LiDAR and InSAR DEMs. Once the separation is determined, acorrection is applied to the LiDAR DEM or InSAR DEM to correct vertical offsets. The fourthstep is the DEM fusion. The fourth step uses the merge tool of ArcInfo to fuse the LiDARand InSAR DEMs. The fifth step is a DEM accuracy assessment. The sixth and final step is adecision step to add a geoid model to the LiDAR/InSAR DEM to provide orthometric heights tothe LiDAR/InSAR DEM. The user requires orthometric heights, the entire process is finished.
Hosford et al. (2003) described a geostatistical method for the fusion of a LiDAR and astereo-radargrammetric DEM. They characterise the LiDAR data and the differences betweensatellite and airborne data from a geostatistical point of view. They define some relief zonesbased on elevation and they perform a variographic analysis of each one of them. Variogramsare calculated for the LiDAR DEM, the radargrammetric DEM and the residual DEM. Basedon the variographic analysis of the different variables and according to the various relief zones,the two DEMs are combined with Kriging.
Mills et al. (2003, 2005) presented a fusion approach based on two component techniques,the GPS and digital small format aerial photogrammetry in order to eliminate the disadvantagesassociated with each technique individually. A least squares surface matching algorithm wasdeveloped to perform the orientation. The surface matching algorithm was employed to registerthe datasets and determine differences between the DEMs. Finally, surfaces derived from smallformat digital imagery are successfully fused with wireframe GPS surfaces.
Nokra et al. (2003) and Nokra et al. (2004) proposed a fuzzy-based fusion technique of twoDEMs issued from different techniques (Shape from Shading and Interferometry). Interfer-ometry results are more confident when the surface is inhomogeneous and in the case of bigvariations (i.e. discontinuous surface), while the Shape from Shading results are better in thecase of homogenous surface and for small variations (i.e. continuous surface). A confidencemap is constructed for each DEM. After the generation of the confidence maps for each DEM,the decision making is based on certain proposed rules. The decision is either the maximum,the minimum or the mean of the two different values of the two DEMs.
Ye et al. (2003) fuse DEMs by determining the height at each grid location in the overlap-ping region by using maximum likelihood as a fusion strategy.
Buckley & Mitchell (2004) develop a DEM integration method by detailing the error bud-gets of individual and fused DEMs. Initially, comparison of the input DEMs is performed toisolate differences. Next, objects on the terrain surface are optionally removed before a sur-face matching algorithm is employed to detect and overcome systematic effects. Finally, themodels are merged using data fitting and the point distribution is optimised to achieve an effi-cient surface representation. The methodology is used to integrate a LiDAR DEM with sparser
36
photogrammetric data.Karkee et al. (2005, 2008) fuse SRTM and ASTER DEMs. The ASTER relative DEM was
co-registered to the SRTM and converted to an absolute DEM by shifting the histogram to theaverage elevation of the SRTM DEM. The voids in the DEMs were filled through an erosiontechnique using the slope and aspect from the other DEM and the elevation of surroundingpixels. Finally, the DEMs were converted to the frequency domain and an ideal low-pass filterwas applied to the ASTER DEM and a high-pass filter with the same cut-off frequency wasapplied to the SRTM DEM to filter out errors in the respective frequency ranges. The filteredDEM spectra were then summed in the frequency domain before being converted back to thespatial domain.
Lee et al. (2005) fuse InSAR, ICESat and LiDAR DEMs presenting a data fusion schemebased on neural networks.
Reinartz et al. (2005) fused a DEM derived with optical stereo data acquired with the FrenchSPOT-5 HRS instrument and a DEM derived from C-band and X-band acquired during theSRTM mission utilizing height error maps for each DEM. In the SRTM case, the height errormap is produced by using features of coherence and density of residuals in the DEM generationprocess. For the SPOT data, the height error map was generated by using the mean standarddeviation of the three input DEMs as a lower limit and the density of the matched points afterthe region growing process as a criterion of the reliability of the DEM raster interpolation.
Warriner (2005) combine LiDAR and optical DEMs in a way that each grid post of the newDEM is derived from the most accurate source available.
Costantini et al. (2006) fused SRTM SAR-X and ERS SAR tandem data. They exploitthe information contained in the area of overlap between different DEMs in order to reducehorizontal and vertical systematic errors. The method is applied first to estimate horizontalsystematic errors, in order to correctly co-register the DEMs, and then to correct the systematicvertical errors. After removal of the systematic horizontal and vertical errors, all the correctedDEMs are averaged with suitable weights, depending on the DEMs accuracies, in order toreduce vertical random errors. Finally, if necessary, an interpolation to fill the areas where dataare completely missing is performed.
Hoffmann & Walter (2006) fused SRTM C- and X-band. They proposed a weighted averagefusion algorithm based on an analysis of the relative differences and the deviations from anabsolute reference in one test area.
Hoja et al. (2006) performed DEM fusion applying a triangulation process for the pointsof all the input DEMs. Each DEM is accompanied with a height error map which are used toweight the input data.
Podobnikar et al. (2000); Podobnikar (2005, 2006) proposes a method, defined “weightedsum of data with geomorphologic enhancement”, to sequentially combine different datasets. Hefirst combines individual datasets according to their weights and then applies a geomorphologicenhancement. The weights depend on the quality of the spatial datasets. When combining thedatasets, Podobnikar (2005) proposes to start from data of the lowest quality and finish with thebest one. Since the obtained DEM tends to be smoother than the highest quality data source,at the end he performs an enhancement by applying the geomorphologic details of the mostappropriate data source to each grid cell. In this way the final enhanced DEM is produced,which is to some extent worse than the purely weighted one, but describes the high resolution
37
variations in the shape of the terrain in a better way. Therefore the optimal solution lies inbetween statistical quality and geomorphologic accuracy.
Muller (2008) reports on experiments to assess the best method to (a) merge the most cloud-free 30 m ASTER DEM with the 90 m SRTM DEM in order to create a 30 m DEM; (b) merge astack of ASTER DEMs by cloud clearing using a fixed threshold; (c) merge a stack of ASTERDEMs by pre-screening for water and cloud features using an existing DEM. In each case,independent “ground truth” DEMs were employed to assess the quality of the input ASTERDEMs and their fused derivative products.
Hoja & d’Angelo (2009) propose three combination methods: DEM fusion utilizing heighterror maps for each DEM; DEM integration, where single point information from anotherDEM is inserted during the triangulation process; and the delta surface fill method. The deltasurface fill method adjusts the values of both DEM taking into account the edges of the gapwhere both DEM are available. Normally, merely removing a bias or a void-specific differencewill be insufficient if there are variable deltas and/or slope differences between the two surfaces.For the void filling we propose a triangulation and interpolation of points extracted from thedelta surface along the edges of the voids (Grohmann et al., 2006).
Xu et al. (2010) proposed a four step fusion process to merge a TopoSAR and a PRISMDEM: (a) Form a difference surface between two input DEMs and remove the areas with out-liers in the DEM difference; (b)Estimate the long term discrepancy of the difference surface andadjust accordingly; (c) Apply a smoothing/filtering method to filter the noise of the differencesurface while preserving the spatial signal, and then superpose the modified difference onto thePRISM DEM; (d) Generate the hybrid DEM by merging the TopoSAR DEM and the modi-fied PRISM DEM using a weighting model based on the slope information from the TopoSARDEM.
3.5.4 Conclusions
The majority of the methods described above perform DEM fusion using the method of weightedaverage. In most cases, the calculation of the weights for the input DEMs is unclear and obscure(Costantini et al., 2006; Hoja & d’Angelo, 2009). The fusion of InSAR DEMs was describedin a separate paragraph because it is based on special characteristics of the InSAR data whichthey cannot be extended for DEMs produced from LiDAR or optical data.
In order to summarize the DEM fusion bibliography research we list here the different DEMfusion methods that scientists used through the years:
1. Weighted average (Ferretti et al., 1998; Honikel, 1998; Knöpfle et al., 1998; Ferretti et al.,1999; Podobnikar, 2005; Reinartz et al., 2005; Podobnikar, 2006; Costantini et al., 2006;Hoja & d’Angelo, 2009; Xu et al., 2010)).
2. Combination of ascending and descending orbits for InSAR DEMs (Carrasco et al.,1997).
3. Best area selection (Pesci et al., 2007; Fabris et al., 2010).
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4. Maximum-likelihood methods (Ye et al., 2003; Eineder & Holzner, 2000; Eineder &Adam, 2005), i.e. Kalman filtering (Slatton et al., 2002) or data fitting (Audenino et al.,2001; Damron, 2002; Mills et al., 2003, 2005; Ye et al., 2003; Warriner, 2005).
5. Weighted interpolation (Crosetto & Crippa, 1998).
6. Fuzzy-based fusion (Nokra et al., 2003, 2004).
7. Neural networks (Lee et al., 2005).
Finally, we list the different ways the accuracy maps were calculated in the studies that useas DEM fusion method the weighted average:
1. Signal properties of InSAR DEMs (Ferretti et al., 1998; Honikel, 1998; Ferretti et al.,1999; Nokra et al., 2003, 2004; Reinartz et al., 2005).
2. Phase accuracy, imaging geometry and atmospheric distortions for InSAR DEMs.Thetotal height error is the root sum square of these height error contributions (Knöpfleet al., 1998; Ferretti et al., 1999).
3. DEM slope information (Xu et al., 2010).
4. Deviation from a reference DEM (Hoffmann & Walter, 2006; Podobnikar, 2006).
5. Self-consistency of optical DEMs (Schultz et al., 1999, 2002).
6. Shape from shading properties (Nokra et al., 2003, 2004).
7. Density of the matched points for optical DEMs (Reinartz et al., 2005).
8. Coherence maps (Euillades et al., 2005).
9. Variographic analysis (Hosford et al., 2003).
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Chapter 4
Quality Characterisation of DEMs andDerivation of Weights
For a reasonable fusion of DEMs, a height error map for each DEM giving the local estimate ofaccuracy is obligatory. Height error maps can be produced taking into account different qualitycriteria.
The focus of this research is mainly the relative variation of elevation errors for qualitycriteria, and not the evaluation and comparison of absolute elevation accuracy.
We propose a data-driven method, which allows one to derive local measures of DEM qual-ity (and thus also fusion weights) for each point or segment of a DEM, if no such informationis available. To this end we use input measures that are derived directly from the DEMs suchas geomorphological characteristics of the terrain (slope, aspect, roughness) and edginess aswell as optionally semantic information such as land-cover maps. Using existing high-qualityground-truth DEMs as reference, we learn regression functions relating the available geomor-phological characteristics to the DEM quality, which then allow one to estimate the local qualityof a new DEM.
The fusion is accomplished with the support of weight maps that reflect the estimated rel-ative accuracy of the two DEMs at every single grid point. In some cases, DEM providersdeliver such error maps, which can then be directly used for fusion. In most, cases, however,these maps are either not available or not reliable, or coarse, e.g. three classes based on theslope, and the weights need to be estimated from the data.
The data sources used in this chapter are a SPOT Reference 3D DEM with 30m grid spac-ing, a high resolution SPOT5 orthoimage and a LiDAR DEM with 2m grid spacing. The chosenstudy area is Thun, Switzerland (Figs. 4.1 and 4.2). More details on the studied datasets aregiven in Chapter 6.
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(a) Gray level coded DEM.
(b) Hill shading with azimuth of the sun to the east of north 315◦and altitude of the sun abovethe horizon 60◦.
Figure 4.1: SPOT DEM.
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(a) SPOT orthoimage.
(b) LANDSAT ETM+ L1T, true color, 16 March 2004.
Figure 4.2: Thun satellite images.
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4.1 Quality CriteriaThe parameters to quantify the quality of a DEM vary. Many studies (Toutin & Gray, 2000;Crosetto & Crippa, 1998) have shown that the relief is one of the principal parameters that havea significant impact on the DEM accuracy.
4.1.1 Geomorphological ParametersA straightforward method for DEM interpretation is the derivation of geomorphological param-eters. Geomorphology has been described as “a set of measurements that describe topographicform well enough to distinguish topographically disparate landscapes” (Pike, 1988).
Geomorphological characteristics are used in both simple descriptive studies as well asin complex mathematical analyses and predictions. Simple geomorphological characteristicsinclude slope and aspect. More complex characteristics include curvature, measures of the con-vexity or concavity and roughness measures. Speight (1974) described over 20 attributes thatcan be used to depict landforms. Moore & Hutchinson (1991) and Moore et al. (1993) alsodescribed terrain attributes and divided them into categories, namely primary and secondary orcompound attributes. Primary attributes are computed directly from the DEM and secondaryattributes involve combinations of primary attributes and constitute physically-based or empiri-cally derived indices that can characterize the spatial variability of specific processes occurringin the landscape (Moore & Hutchinson, 1991; Moore et al., 1993). Primary attributes includeslope, aspect, plan and profile curvature. Most of these topographic attributes are calculatedfrom the directional derivatives of a topographic surface. The complete mathematical repre-sentation of most attributes and the methods for calculating them can be found in (Moore &Hutchinson, 1991; Moore et al., 1993; ESRI, 1993; Gallant & Wilson, 1996; Wilson & Gallant,2000).
Among the geomorphological attributes, those of most interest to us in terms of the influ-ence on the DEM accuracy include slope, aspect and roughness.
Slope and Aspect. At a given point on a surface z = f(x, y), the slope S and aspect A aredefined as a function of gradients at X and Y directions:
S = arctan√f 2x + f 2
y (4.1)
A = 270o + arctan
(fyfx
)− 90o
fx|fx|
(4.2)
where fx and fy are the gradients at W-E and N-S directions, respectively.From the above equations, it is clear that the key for slope and aspect computation is the
calculation of fx, and fy. Using a grid-based DEM, the common approach is to use a moving3 × 3 window (Fig. 4.3) to derive finite differential or local surface fitting polynomial for thecalculation. Jones (1998) describes eight algorithms for calculating slope of a DEM. Mostof these methods differ only in the number of grid cells used and the weightings applied toeach value. In a scientific study, it is important to know exactly what method is used when
44
calculating slope, and exactly how slope is defined. We used a third-order finite differencealgorithm (Horn, 1981; Wood, 1996):
fx = (Z9 − Z7 + Z6 − Z7 + Z3 − Z1)/6g (4.3)fy = (Z1 − Z7 + Z2 − Z8 + Z3 − Z9)/6g (4.4)
where g is the grid cell size.Many authors regard slope as the most essential surface characteristic of our planet (Evans,
1972). Slope is a basic approximation of the surface gradient, and its accuracy depends primar-ily on three factors: a) the distance between two points, b) local variability of the terrain (Siska& Hung, 2004) and c) the accuracy of the height points. The lower the slope value, the flatterthe terrain; the higher the slope value, the steeper the terrain.
Z1 Z2 Z3
Z4 Z5 Z6
Z7 Z8 Z9
Figure 4.3: 3X3 Height matrix.
Slope is measured as angle of slope or percent slope. This means that a 45◦is a slope of100% and a 90◦slope is infinity. The angle of the aspect ranges from 0o to 360o. 90o is North,180o is West, 270o is South and 360o is East.
Many researchers noticed that elevation error increases with increasing slope (Bolstad &Stowe, 1994) since planimetric errors cause vertical errors at terrain with steep slopes andbuildings. (Gong et al., 2000) noted a general, but inconsistent, relationship between increasingelevation error and increasing slope angle. (Hodgson et al., 2003) also identified a significantmonotonic relationship between the mean absolute elevation error and increasing slope for aLiDAR-derived DEM.
From our investigation several general remarks can be developed. Elevation accuracy de-creases consistently as slopes increase irrespectively of the aspect. In general, elevation accu-racy and slope are almost linearly correlated. The steeper the slopes are, the more correlatedthe aspect results are: the steeper the slope, the larger or more pronounced the elevation accu-racy variations are for the different aspects. In other words, elevation accuracy is significantlydependent on the aspect of the slopes for the steepest terrain, but less for the flattest terrain(Toutin, 2002). Aspect, in relation with significant slopes, relates mainly to illumination shad-ows that cause DEM errors, when the DEM is generated by image matching. It also relatesto DEM errors produced by SAR interferometry and airborne laser scanning, but to estimate
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these relations one needs more detailed knowledge of the data acquisition parameters (e.g.flight path, viewing angle).These results strongly confirm preliminary studies (Toutin, 2002;Crosetto & Crippa, 1998) showing that the relief is one of the principal parameters that has asignificant impact on the DEM accuracy. There is a combined correlation between elevationaccuracy and terrain slopes and aspects.
Figs. 4.4 and 4.5 show the slope and the aspect maps, respectively, of the SPOT DEM.
Figure 4.4: Slope map the SPOT DEM. The bar unit is in degrees.
Roughness. Roughness describes the general variability of the terrain on the earth’s surface(Siska & Hung, 2004) and can be measured in different ways, e.g. using the standard deviationor fractal dimension. We describe three approaches to characterize terrain roughness withgeometric quantities: the local standard deviation, the topographic ruggedness index and thelocal entropy.
In the first approach, the terrain roughness is parametrized by the standard deviation ofthe height values of all the points within a fixed size window. We calculate the residuals of afitting plane with orthogonal regression (Fig. 4.6). Orthogonal fitting means that the orthogonaldistances from plane to points are minimized (Hollaus & Höfle, 2010). The unit of the soderived terrain roughness parameter is in meters.
For the example in Fig. 4.6 we use a 5 × 5 window (25 3D (x, y, z) grid points) randomlychosen from the SPOT DEM. Next, we fit a plane to the data using PCA. The coefficients forthe first two principal components define vectors that form a basis for the plane. The thirdprincipal component is orthogonal to the first two, and its coefficients define the normal vectorof the plane. Because the first two components explain as much of the variance in the data
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Figure 4.5: Aspect map of the SPOT DEM.
as is possible with two dimensions, the plane is the best 2D linear approximation of the data.Equivalently, the third component explains the remaining amount of variation in the data, andcan be also viewed as the error term in the regression. The eigenvalues from the PCA define theamount of explained variance for each component. The first two coordinates of the principalcomponent scores give the projection of each point onto the plane, in the coordinate system ofthe plane. To get the coordinates of the fitted points in terms of the original coordinate system,we multiply each principal component coefficient vector by the corresponding score, and addback in the mean of the data. The residuals are simply the original data minus the fitted points.The perpendicular distance from each point in the window to the plane, i.e., the norm of theresiduals, is the dot product of each centred point with the normal to the plane. The fitted planeminimizes the sum of the squared errors. To visualize the fit, we plot the plane, the originaldata, and their projection to the plane. Blue points are above the plane, red points are below.
(Riley et al., 1999) in order to express the amount of elevation difference between adjacentcells of a digital elevation grid, developed a measure called topographic ruggedness index(TRI). The TRI measure expresses the difference in elevation values from a center cell and theeight cells immediately surrounding it. Then, it squares each of the eight elevation differencevalues to make them all positive and averages the squares. The topographic ruggedness indexis then derived by taking the square root of this average, and corresponds to average elevationchange between any point on a grid and it’s surrounding area.
The entropy E(x, y) quantifies the randomness of the height estimation. It is defined as
E(x, y) = −∑
(p× log p) (4.5)
47
Figure 4.6: Standard deviation of DEM points. Principle of orthogonal regression plane fitting.
where p are the probability densities of the heights, approximated by histogram counts.Each output grid cell contains the entropy value of the surrounding n × n neighbourhood.
It is low when the values of the window have similar height values. It is high when the entropyvalues are close to 1. Entropy is a unitless measure.
There is a close relationship between roughness and DEM quality. Rougher areas are typi-cally less accurate and more susceptible to errors.
Fig. 4.7 shows the roughness of the SPOT DEM, calculated using the three methods de-scribed above.
4.1.2 Edginess
Feature specific elements such as edges contribute significantly to DEM quality. Edges areimportant features as they describe changes on the terrain (Lichtenstein & Doytsher, 2004) andthey highlight terrain surface with more significant information than other points (Li et al.,2005). Therefore, data fusion should be conducted in such a way that such critical elements arekept on the final fused DEM.
Extraction of edges is feasible when additional image data are available. Then, edge extrac-tion is done using classic edge detection algorithms i.e Canny, Sobel, Laplacian of Gaussian.
Fig. 4.8 illustrates the edge maps that were generated applying the Canny and the LoGedge dectors on the SPOT high resolution orthoimage. The Canny algorithm finds edges bylooking for local maxima of the gradient of the input image. The gradient is calculated usingthe derivative of a Gaussian filter. The method uses two thresholds, one to detect strong andanother for weak edges, and includes the weak edges in the output only if they are connectedto strong edges. This method is therefore less likely than the others to be biased by noise,and more likely to detect true weak edges (Canny, 1986). The Laplacian of Gaussian method,
48
proposed by (Marr & Hildreth, 1980), finds edges by looking for zero crossings after filteringthe input image with a Laplacian of Gaussian filter.
Edges appear also in the DEMs. Applying the edge detection algorithms in DEMs weextract segments that really follow the terrain topography. This leads to the extraction of theexact shape of the geomorphologic features that should be preserve during fusion.
4.1.3 Further CriteriaLand Cover
When one begins to study the error distribution for a DEM in detail, it is obvious that theaccuracy varies within the dataset due to variation in land cover. The most common landcover types identified for DEM accuracy assessment purposes are: open ground, weeds andcrops, water bodies, forest and urban. The accuracy of the height point of each land covertype is mostly related to the production technology of the DEM, i.e. LiDAR stands as themost accurate and comprehensive technology to produce highest resolution DEMs of coastalenvironments (for more information see Sec. 2.1.3).
Texture
In case that the input DEMs have been produced using photogrammetric methods the textureof the aerial or satellites images could also be used as accuracy indicator. Photogrammetricmethods fail where there is a lack of texture. Areas with absence of sufficient texture can beidentified using texture segmentation methods i.e. local range, local standard deviation, localentropy etc.
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(a) Local standard deviation.
(b) Topographic ruggedness index.
(c) Entropy.
Figure 4.7: Roughness maps of the SPOT DEM. For (a)-(b) the bars units are in meters, (c)unitless.
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(a) Canny edge detector.
(b) LoG edge detector.
Figure 4.8: Edge detection on the SPOT orthoimage.
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4.2 Derivation of Quality Values for DEM Points based onSingle Criteria
In order to derive a DEM error distribution pattern we create a so-called residual map, usinghigh quality reference data (Hebeler & Purves, 2009). Residual maps are generated by applyingan algebraic operation of pixel by pixel subtraction between a DEM and the reference DEM.The input DEMs are first transformed to the same coordinate system and then they are co-registered. The residual map contains the absolute height differences. This operation resultsin a numerical image representing Re, the pixel-wise absolute residual value. In this way, theresidual map provides an understanding of error sources and forms a basis for further analysis(Van Niel et al., 2004). Fig. 4.9 shows the residuals between a high quality LiDAR DEM andthe SPOT DEM.
Figure 4.9: Absolute height residual map of the SPOT DEM in meters.
The values of the residual map are compared with the corresponding values of the geomor-phological characteristic for each input DEM, and the error statistics of the elevation differencesare then computed. We exploit that the residuals vary (increase or decrease) in a non-randompattern with the quality criteria. A relation between error and quality measure value is empiri-cally estimated.
Notice that we have one response variable, the expected residual (absolute height error), andone predictor variable, the geomorphological characteristic. The main objective is to analysethe data and define a model that predicts the response variable. That is, we try to describe thetrend line, or the mean response of y (expected residual), as a function of x (geomorphologicalcharacteristic). In mathematical terms this is parametric curve fitting with bivariate data.
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Parametric fitting involves finding coefficients (parameters) for the model that one fits tothe data. The data is assumed to be statistical in nature and is divided into two components:
data = deterministic component + random component (4.6)
The deterministic component is given by a parametric model and the random component isoften described as error associated with the data:
data = parametric model + error (4.7)
The model is a function of the independent (predictor) variable and one or more coefficients.The error represents random variations in the data that follow a specific probability distribution.
In order to specify the “best” parametric model we want to fit to the data we experimentedamong various models (exponential, polynomial, Gaussian, power functions. fourier series andsum of sines) and we choosed the one which give the smallest RMSE.
In all the cases, described below, outliers are detected and removed prior to fitting. Wefind outliers following the simple rule described in Sec. 2.2.2. Briefly, we remove the residualsmore than three times the RMSE (Re > 3RMSE).
Based on several experimental tests we concluded that the expected residuals are well-approximated using a sum of sines model as a function of the slope. The sum of sines is givenby the equation:
y =n∑i=1
ai sin (bix+ ci) (4.8)
where x is the slope, a is the amplitude, b is the frequency, and c is the phase constant foreach sine wave term. n is the number of terms in the series and 1 ≤ n ≤ 8 (MATLAB, 2011).
In Fig. 4.10(a) we fit an eight-term sum of the sine model. Table 4.1 contains the 24 param-eters of the model. We notice that elevation accuracy decreases consistently as slope increases.Fig. 4.10(b) shows the distribution of the slope values.
In case of aspect the residuals follow a Gaussian model, verified by multiple tests, whichis given by the equation:
y =n∑i=1
aie
[−(
x−bici
)2]
(4.9)
where x is the aspect, a is the amplitude, b is the centroid (location), c is related to the peakwidth, n is the number of peaks to fit, and 1 ≤ n ≤ 8 (MATLAB, 2011).
In Fig. 4.11(a) we fit an eight-term Gaussian model. Table 4.2 contains the 24 parametersof the model. Fig. 4.11(b) shows the distribution of the aspect values. A fairly good fit isfound between the residuals and the Gaussian model. The relationship shows a cyclic effect(Fig. 4.11(a)) of the slope aspect on the DEM error. This relationship may be attributed to thelow surface texture in the slope that is oriented parallel to the direction of the sunlight whichmay lead to poor elevation estimates.
In case of roughness, calculated using the entropy method, the expected residuals follow alinear polynomial model, empirically found with multiple tests which is given by the equation:
y =n+1∑i=1
pixn+1−i (4.10)
53
where x is the roughness, pi are the coefficients of the polynomial model, n+ 1 is the order ofthe polynomial, n is the degree of the polynomial, and 1 ≤ n ≤ 9.
The order gives the number of coefficients to be fit, and the degree gives the highest powerof the predictor variable (MATLAB, 2011).
In Fig. 4.12(a) we fit a seven degree Gaussian model. Table 4.3 contains the 8 coefficientsof the model. Fig. 4.12(b) shows the distribution of the roughness values.
(a) Slope curve fitting.
(b) Slope values distribution.
Figure 4.10: (a) Residuals between the LiDAR and the SPOT DEM versus slope (mean everydegree). (b) The slope histogram with bins of one degree.
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Table 4.1: The parameters of the sum of sines model for fitting the slope to the Z residuals.
i 1 2 3 4 5 6 7 8aiaiai 43.56 32.65 0.61 1.43 0.56 0.25 0.22 0.09bibibi 0.03 0.04 0.13 0.18 0.30 0.28 0.44 0.55cicici 0.60 3.81 2.25 4.25 4.57 1.50 3.85 5.00
(a) Aspect fitting.
(b) Aspect values distribution.
Figure 4.11: (a) Residuals between the LiDAR and the SPOT DEM versus aspect (mean everydegree). (b) The aspect histogram with bins of one degree.
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Table 4.2: The parameters of the Gaussian model for fitting the aspect to the Z residuals.
i 1 2 3 4 5 6 7 8aiaiai 0.83 3.23 5.25 1.23 7.23 0.35 5.86 1.01bibibi 11.28 353.60 140.00 246.60 23.02 106.20 293.50 212.10cicici 14.28 22.28 64.36 18.41 67.95 10.26 110.80 20.85
(a) Roughness curve fitting.
(b) Roughness values distribution.
Figure 4.12: (a) Residuals between the LiDAR and the SPOT DEM versus roughness (meanevery 0.01). (b) The roughness histogram, with bins of 0.01.
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Table 4.3: The coefficients of the polynomial model for the fitting of roughness.
i 1 2 3 4 5 6 7 8pipipi -0.09 -0.20 0.68 1.47 -1.12 -1.65 4.54 9.53
4.3 Quality Characterisation through Combination of Crite-ria
We have explored the case of using more geomorphological parameters simultaneously. Inthis case we derive weights for a single DEM through combination of criteria applying eithersurface fitting (combination of two criteria) or volume fitting (combination of three criteria).The combination of several criteria often gives better results than individual criteria because itcaptures the correlations.
In all the cases, described below, the outliers are detected and removed, as described earlier.
4.3.1 Combination of Two Criteria
We combine two geomorphological characteristics applying surface fitting. The surface fittingis done using a locally weighted smoothing quadratic (loess) regression. Smoothing regressioninvolves drawing a smooth surface on a scatter diagram to summarize a relationship, in a fash-ion that makes few intitial assumptions about the form or strength of the relationship (Cleveland& Loader, 1996). It is a special case of non-parametric regression, in which the objective isto represent the relationship between a response variable and one or more predictor variables,again in way that makes few assumptions about the form of the relationship. This kind ofmethod is appropriate for our investigation since we have no a priori notion about the relation-ship of the geomorphological characteristics with the expected residuals. Another advantageof the locally weighted smoothing regression is that it depicts the “local” relationship betweena response variable and a predictor variable over parts of their ranges, which may differ from a“global” relationship determined using the whole data set.
A quadratic smoother is a procedure for drawing a smooth surface through a scatter dia-gram. Like linear regression (in which the “surface” is a rectangular plane), the smooth surfaceis drawn in such a way as to have some desirable properties. In general, the properties are thatthe surface should be smooth, and that locally the surface minimizes the variance of the predic-tion error. The quadratic smoother is known as “loess” surface. The name “loess” is derivedfrom the term “locally weighted scatter plot smooth”, as the method uses locally weightedregression to smooth data (Cleveland & Loader, 1996). This method makes no assumptionsabout the form of the relationship, and allows the form to be discovered using the data itself. Inthe loess method, weighted least squares is used to fit quadratic functions of the predictors atthe centres of neighbourhoods. The radius of each neighbourhood is chosen so that the neigh-bourhood contains a specified percentage p of the data points. The fraction of the data, calledthe smoothing parameter or span, in each local neighbourhood controls the smoothness of theestimated surface.
In Fig. 4.13(a) we combine the geomorphological characteristics of aspect and slope by a
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locally weighted smoothing quadratic regression computed with span 25%. The slope (x-axis)is normalized by mean 27.7o and standard deviation 16.7o. The aspect (y-axis) is normalizedby mean direction for circular data 34.1o and circular standard deviation 75.1o. The circularmean and standard deviation are calculated using Fisher statistics (Fisher, 1995).
(a) Loess surface fitting.
(b) Residuals contour plot.
Figure 4.13: (a) Loess surface fitting. Residuals between the LiDAR and the SPOT DEM (z-axis) versus slope (x-axis) and aspect (y-axis). (b) Contour plot.
In Fig. 4.14(a) we combine the geomorphological characteristics of slope and roughnessby a locally weighted smoothing quadratic regression computed with span 25%. The slope(x-axis) is normalized by mean 27.7o and standard deviation 16.7o. The roughness (y-axis) is
58
normalized by mean 0.6 and standard deviation 0.2.
(a) Loess surface fitting.
(b) Residuals contour plot.
Figure 4.14: (a) Loess surface fitting. Residuals between the LiDAR and the SPOT DEM (z-axis) versus slope (x-axis) and roughness (y-axis). (b) Contour plot.
In Fig. 4.15(a) we combine the geomorphological characteristics of roughness and aspectby a locally weighted smoothing quadratic regression computed with span 25%. The rough-ness (x-axis) is normalized by mean 0.6 and standard deviation 0.2. The aspect (y-axis) isnormalized by mean 34.1o and standard deviation 75.1o.
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(a) Loess surface fitting.
(b) Residuals contour plot.
Figure 4.15: (a) Loess surface fitting. Residuals between the LiDAR and the SPOT DEM (z-axis) versus roughness (x-axis) and aspect (y-axis). (b) Contour plot.
4.3.2 Combination of Slope Aspect, and Roughness
We combine three geomorphological characteristics applying a 4D interpolation algorithm. Thegiven scattered dataset is defined by locations X(x, y, z) where x is the slope, y is the aspect,and z is the roughness (Fig. 4.16). The corresponding values V of the locations X are theexpected residuals. The scattered dataset can be interpolated using a trilinear interpolation of
60
X . This produces a volume of the form V = F (X). Then, the volume V can be evaluatedat any query location QX , using QV = F (QX), where QX lies within the convex hull ofX . The interpolant F always goes through the data points specified by the sample MATLAB(2011). Trilinear interpolation is a method of multivariate interpolation on a 3-D regular grid.It approximates the value of an intermediate point (x, y, z) within the local axial rectangularprism linearly, using data on the lattice points.
Figure 4.16: Colour coded residuals between the LiDAR and the SPOT DEM versus slope (x-axis), aspect (y-axis) and roughness (z-axis).
4.4 Derivation of WeightsConsider the fusion of two co-registered DEMs, H1 and H2. We derive quality values for bothDEMs applying the methodology described above and after mapping we obtain two accuracymaps Q1 and Q2 that contain the expected residuals. The expected residuals are an absoluteaccuracy measure which denote the error in meters at each grid point. The weight maps W1
and W2 are respectively
W1 =1
Q21
W2 =1
Q22
(4.11)
The smaller the measurement error for a particular experimental point, the larger the weightof that point. The values of each weight map are normalised at each overlapping point in theinterval [0, 1] thus yielding two new normalised weight maps Wn1 and Wn2.
Wn1 =Q2
2
Q21 +Q2
2
Wn2 =Q2
1
Q21 +Q2
2
(4.12)
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If the error of height point is zero, then the weight would not proceed as that value is notallowed because of the 1/zero or 1/very-small-number division. The only way to override thisis to set a different value, such as setting it to a small number (i.e. 0.1, to indicate the error isalmost negligible on that point, if other error values are much larger).
In case that accuracy maps for the input DEMs are given from the DEM producer and thevalues of the given accuracy map do not denote the error in meters at each grid point a similarapproach is adopted in order to extract the weight maps.
4.5 ConclusionsFor DEM fusion weights are required, which govern the relative influence of the two inputDEMs at a given raster location. These weights are critical for proper DEM fusion. We haveanalysed the residuals of different DEMs with regard to LiDAR ground truth data. It turns outthat in fact clear correlations exist between height errors and certain surface properties.
Above, geomorphological parameters were described using a 3 × 3 neighbourhood. Thisapproach works well for smooth and non-flat areas. However, for high resolution data rep-resenting relatively flat areas with small differences in elevations or noisy surfaces, the smallneighbourhood may not be sufficient to adequately capture the geometry of land-surface fea-tures (Mitášová et al., 1995). Surface characteristics (slope, aspect and roughness) can be mea-sured within different sizes of kernel windows. Then, the geomorphological parameters canbe computed using an explicit form of the function derivatives, usually simultaneously withinterpolation.
Composite measures that incorporates multiple individual quality measures to provide asingle score perform better than individual quality measures. Moreover, the advantage of usingcomposite measures is comprehensiveness since the use of individual quality measures comesat a loss of information. No prior knowledge is needed about the influence of each individualcharacteristic to the overall accuracy.
More than three individual quality measures could be combined in order to form a compos-ite quality measure using multivariate interpolation methods (Bajaj, 1993; Lalescu, 2009).
The trends that geomorphological characteristics follow in relation to the height errors arestable when using different resolution DEMs.
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Chapter 5
DEM Fusion
This chapter describes techniques to fuse DEMs. A thorough literature review is initially pre-sented, followed by the description of the mathematical models that we used in order to fuseDEMs. Particular attention is given to the method that we introduced for this reason, the sparserepresentations. Finally, the last step of the DEM fusion, the quality assessment of the fusedDEM, is described.
5.1 Mathematical Models of FusionIn Sec. 3.5, existing approaches for DEM fusion have been presented. Four mathematicalmodels that we tested for DEM fusion will be described and discussed here.
5.1.1 Averaging
Averaging is commonly called in mathematics as “arithmetic mean”. It is computed as the sumof all the numbers in the series divided by the count of all numbers in the series. Given a set ofsamples xi, the arithmetic mean is
x =
∑ni=1 xin
(5.1)
5.1.2 Data Fitting
The measured terrain data are processed to DEM using spatial interpolation. There are manymethods of interpolating randomly spaced point data. Some of these method are global whileothers are local. Global methods utilize all the known values to evaluate an unknown value,while in local methods only a specified number of nearest neighbours are used to evaluatean unknown value. Detailed information about these methods can be found in literature, e.g.(Mitas & Mitášová, 1999). The quality of the resulting DEMs depends mainly on the qualityof input data and the interpolation method.
Some details are given below for three representative interpolation methods: the inversedistance weighted interpolation (IDW), the Kriging and the regularized spline with tension
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interpolation method. The last one is the fitting method that was selected to be compared withthe other DEM fusion methods on the experimental results presented in Chapter 6.
Inverse Distance Weighted Interpolation
The inverse-distance weighted procedure is easy to program and fairly accurate under a widerange of conditions (Lam, 1983). Using this method, the height at each unknown location isgiven by:
Pi =
∑Gj=1 Pj/D
nij∑G
j=1 1/Dnij
(5.2)
where Pi is the height value at the location i; Pj is the height value at the sampled locationj; Dij is the distance from i to j; G is the number of sampled locations; and n is the inverse-distance weighting power.
The value of n, in effect, controls the region of influence of each of the sampled locations.As n increases, the region of influence decreases until, in the limit, it becomes the area which iscloser to point i than to any other. When n is set equal to zero, the method is identical to simplyaveraging the sampled values. Usually, the value of n is set arbitrarily. The most commonchoice is n = 2.
Kriging Interpolation
Kriging is a geostatistical gridding method that produces visually appealing maps from irreg-ularly spaced data. It is a method of interpolation which predicts unknown values from dataobserved at known locations. Kriging is a very flexible gridding method that can be custom-fitto a data set by specifying the appropriate variogram model. In the Kriging method, a weightedmean of the sampled values is evaluated for each point, such that the estimation error is mini-mized by solving the following set of equations simultaneously (Davis, 1986):
Pi =G∑j=1
WjPj (5.3)
G∑j=1
Wj = 1 (5.4)
G∑j=1
Wjγ (Dkj) + λ = γ (Dkj) k = 1, 2, ...G (5.5)
where Wj is the weight for location j; λ is a variable added to minimize the estimation error;and γ (Dkj) is the semivariogram value between points k and j.
Regularized Spline with Tension
In order to derive high resolution DEMs we used a method of interpolation by regularizedspline with tension (RST) (Mitášová & Hofierka, 1993). The RST interpolation method is one
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of the best interpolation methods available in current GIS (McCauley & Engel, 1997; Hofierkaet al., 2002). It is a variational method based on the assumption that the approximation func-tion should pass as closely as possible to the given data points and should be as smooth aspossible (Mitášová et al., 1995). A tension parameter controls the distance over which a givenpoint influences the resulting surface from a membrane to a thin plate. A smoothing parame-ter controls the deviation of the resulting surface from the individual data points, a procedurethat is necessary for processing noisy data. The accuracy of interpolation by RST can be as-sessed by various statistical methods evaluating interpolation errors (residuals) in input pointsor predictive error of interpolation in areas outside input points using cross-validation technique(Hofierka et al., 2002; Mitášová et al., 1995; Neteler & Mitášová, 2002). Optimum parametersare found empirically by visual analysis or by minimizing the cross-validation error (Mitášováet al., 1995).
The RST interpolation method is implemented in GRASS version 6.4.0 (http://grass.osgeo.org/) as s.surf.rst command. A specific feature of this method and its im-plementation in GRASS is a set of four interpolation parameters providing flexibility in dataprocessing. These parameters give the user a capability to process “difficult” datasets (e.g.photogrammetric, GPS and LiDAR data), that can be hardly effectively processed by otherconventional methods (Cebecauer et al., 2002). These parameters are: tension, smoothing,anisotropy, and minimum and maximum distance between points. These parameters can beselected empirically, or automatically, by minimisation of the predictive error estimated by across-validation procedure (Hofierka et al., 2002).
5.1.3 Weighted AverageProbably the most obvious fusion algorithm is to compute an output value at each grid nodeby weighted averaging of the inputs. This approach is used frequently (Costantini et al., 2006;Reinartz et al., 2005; Schultz et al., 1999; Xu et al., 2010).
Formally, the weighted mean of a non-empty set of data {x1, x2, ..., xn} with non-negativeweights {w1, w2, ..., wn}, is the quantity
x =
∑ni=1wixi∑ni=1wi
=w1x1 + w2x2 + ...+ wnxn
w1 + w2 + ...+ wn. (5.6)
Therefore, data elements with a high weight contribute more to the weighted mean than doelements with a low weight. The weights cannot be negative. Some may be zero, but not all ofthem since division by zero is not allowed (see Sec. 4.4). The formulas are simplified when theweights are normalized such that they sum up to 1, i.e.
∑ni=1wi = 1.
5.1.4 Sparse RepresentationsWe propose a novel approach that performs DEM fusion using sparse representations (Yanget al., 2010).
In the past decades, sparse representation has become an important tool for image de-noising, compression, and super-resolution (Elad, 2010). Sparse representation describes natu-ral signals, such as images, by a sparse linear combination of a few atoms, which are members
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of an over-complete dictionary (Mallat & Zhang, 1993; Bruckstein et al., 2009). The numberof atoms in the dictionary is larger than the signal dimension. This means that the dictionaryis over-complete, so there are numerous ways to represent the signal, among which sparserepresentation refers to the one with the fewest atoms to represent the local signal.
In the following, we describe the problem statement. Consider two noisy measurements yland yh of a height field x, possibly at different resolutions, e.g., low and high respectively. Weassume that the measurements have been produced from the original DEM x by the followingmodel:
yh = x+ εh and yl = Lx+ εl, (5.7)
where εh, εl are noise vectors and L is an unknown downsampling operator.In the case when the two measurements yl and yh are at the same resolution, the operator L
equals to 1.The problem addressed in this dissertation is to fuse the noisy measurements yl and yh
in order to recover the original DEM x. If we had only yh in our disposal, we would havea denoising problem. If we had only yl in our disposal, we would have a super-resolutionproblem. The hope is that the redundancy of the measurements will offer robustness againstnoise and result in an accurate estimation of x. The problem, as stated above, can be seen asa denoising problem – in the case of different resolutions, with simultaneous super-resolutionfor the coarser signal.
Problem Formulation. In order to achieve robustness without over-smoothing, we pose thefusion problem in the framework of sparse representations. Sparse representations have beenshown to result in state-of-the-art performance in image denoising (Elad & Aharon, 2006) andsuper-resolution problems (Yang et al., 2010). In what follows, yh ∈ RNh , yl ∈ RNh andx ∈ RNh denote patches extracted from the corresponding signals in Eq. 5.7. We also workwith local DEM patches to achieve computational efficiency and to ensure a moderately sizeddictionary able to capture the variations in terrain shape. In what follows, yh, yl, and x thusdenote local patches in the corresponding terrain models in Eq. 5.7.
We assume that x can be represented as a sparse linear combination of elements from adictionary D (i.e., local terrain shapes). The dictionary is a basis set spanning the signal spaceand is typically overcomplete, that is, it contains more elements than the dimension of the signalspace. The elements of the dictionary are called atoms. We say that x is sparsely representedover D if x = Dα0, where α0 ∈ RN is a sparse coefficient vector with most entries zero andvery few non-zero entries (Fig. 5.1). N denotes the size of the dictionary whose atoms areorganized as columns of D. The sparsity of α0 implies that already a few atoms are sufficientfor obtaining a good approximation of x.Under this representation, the generative model inEq. 5.7 can be re-written as
yh = D︸︷︷︸:=Dh
α0 + εh and yl = LD︸︷︷︸:=Dl
α0 + εl, (5.8)
where we have defined a high-resolution dictionary Dh and a low-resolution dictionary Dl.Both dictionaries are coupled via the relation Dl := LDh.
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The key observation in Eq. 5.8 is that the same sparse coefficient vector α0 is involved inboth measured DEMs yh and yl. This leads to the following optimization problem in order torecover x from the measured yl, yh.
Optimization Problem. Assume for a moment that Dh and Dl are available. We postponethe discussion of how to determine these two dictionaries until the end of this section. Giventhe two dictionaries Dh and Dl and the measurements yl and yh, we would like to recoverthe sparse coefficient vector α0. Once α0 has been computed, one can simply recover x bycomputing Dhα0. The redundancy of the measurements can be exposed towards robustnessagainst noise. For the estimation of α0 we formulate the following optimization problem:
minα∈RN
‖Dlα− yl‖22︸ ︷︷ ︸low resolution
+ ‖Dhα− yh‖22︸ ︷︷ ︸high resolution
+ τ‖α‖1︸ ︷︷ ︸sparsity term
(5.9)
The first two terms correspond to the reconstruction error with respect to the observed DEMsyl and yh. The third regularisation term is associated with the `1 norm of the candidate solutionvector α. It is well known that the minimization of the `1 norm encourages a sparse solution1(see, e.g., (Tibshirani, 2011)). Since the true coefficient vector α0 that we seek to recoveris sparse, we would like our estimated solution α to be sparse as well. The parameter τ > 0controls the trade-off between data fitting and sparsity.
The current formulation (Eq. 5.9) implicitly assumes that both data terms have the sameimportance. However, this is not typically the case with DEMs, since the two inputs have,at each point, different accuracy. It is therefore beneficial to include weights in the problemformulation that will reflect such prior knowledge. We therefore modify the optimization toinclude weight vectors wl, wh:
minα∈RN
‖√wl � (Dlα− yl)‖22 + ‖
√wh � (Dhα− yh)‖22 + τ‖α‖1. (5.10)
Here, � denotes component-wise multiplication.
Consistency among Neighbouring Patches. Solving Eq. 5.10 for each patch independentlywould result in blocking artefacts along the patch borders. To remedy this problem, we intro-duce overlap between patches and impose consistency between neighbouring patches. Morespecifically, let P denote an operator that extracts the overlap region between the current work-ing patch and the patches that have been computed before. Furthermore let yp denote a vectorthat collects the values of the fused DEM in the overlap region. Minimizing the discrepancy‖PDhα− yp‖22 between overlapping patches will impose consistency and ensure smooth tran-sitions. Introducing this term into Eq. 5.10, we reach the final formulation of our optimizationproblem:
minα∈RN
‖√wl � (Dlα− yl)‖22 + ‖
√wh � (Dhα− yh)‖22
+β‖PDhα− yp‖22 + τ‖α‖1, (5.11)
1In fact, it is the basis of the sparse representation framework that the computationally inconvenient `0 “norm”(number of non-zero elements) can be replaced by the `1 norm.
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where we have introduced the parameter β > 0 to control the influence of the patch overlapfactor.
Equation (5.11) can be written in compact form:
minα∈RN
‖Dα− y‖22 + τ‖α‖1, where
D =
√wl �Dl√wh �Dh√βPDh
and y =
√wl � yl√wh � yh√βyp
. (5.12)
The problem in Eq. 5.12 is a convex `1-regularized least-squares problem that can be solved toglobal optimality. Since optimization problems of this form constitute the main computationalkernel of compressed sensing applications, there exists a wide selection of algorithms for theirsolution. Here, we use the Orthogonal Matching Pursuit (OMP) (Mallat, 1999), because of itslow complexity, simple implementation and computational efficiency. Problem in Eq. 5.12 issolved for each patch with OMP. Here, we omit details on OMP and refer the interested readerto the original publication.
Dictionary construction. The proposed framework requires dictionaries Dh, Dl, whichmust be acquired from training data. Different learning techniques could be used to obtain a setof atoms from available high-quality DEMs. The dictionary should ideally include all possiblesurface forms. We have experimented with different methods and found that the best resultsare obtained by simple random sampling of patches from high resolution DEMs, followed byclustering to remove very similar samples. This is similar to the approach used in (Yang et al.,2010).
Hence, for the construction of Dh, we use a training set of high resolution DEMs of highquality. If Dl is of lower resolution, its atoms are obtained by downsampling the correspondingatoms inDh with bicubic interpolation. The preparation of the dictionaries is off-line and needsto be performed only once. Our empirical results in Chapter 6 demonstrate that the dictionariesconstructed with the above procedure are well suited for successfully representing real terrainpatches using very few (less than ten) atoms.
Notice that two steps in this algorithm make the fusion scheme be shift-invariant. Firstly, theover-completeness of the dictionary makes the sparse representation shift-invariant (Daneshvar& Ghassemian, 2010). Secondly the “sliding window” operation is a shift-invariant scheme.
Fusion Parameters Optimization
According to the mathematical formulation of the sparse representations fusion algorithm, wehave to set the following parameters:
1. The overlapping parameter β.
2. The number of the non-zero atoms used in OMP.
3. The patch size and.
4. The number of patches in the dictionary.
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(a)
=
a0
(b)
+a1
(c)
+a2
(d)
Figure 5.1: Reconstruction of a 9 × 9 patch. (a) from three non-zero atoms (b)-(d), where[a0, a1, a2] are the sparse non-zero coefficient elements.
The size of the “sliding window” (patch size) is an important parameter. The larger thewindow, the bigger the vector transformed from the patch. The size of the corresponding dic-tionary also increases. Thus, the process of the OMP becomes slower. As the size of the“sliding window” decreases, the process of the OMP becomes faster, but the information con-tained in the patches would be not sufficient. It may miss some of the important features of thesource DEMs.
In order to fine tune these parameters we performed numerous tests using real world datasetsand we compared for each test the produced results with available high quality reference data.Best results have been achieved with the following set of parameters. The overlapping param-eter β is set within the interval [1.0, 1.5] (Fig. 5.2(b). The number of non-zero atoms used inOMP is set between 10 and 20 (Fig. 5.2(a). Below 10 the results are not reliable and above 20the processing time increases while the results do not improve any further. The minimum patchsize should not be smaller than 3× 3 and it should not be bigger than 21× 21 because then theprocessing window becomes too complicated and it is more difficult to find a suitable combi-nation of non-zero sparse atoms to reconstruct it (Fig. 5.2(c)). Fig. 5.2(c) shows that the patchsize does not influence significantly the accuracy of the results. A “good” dictionary shouldcontain atoms that describe every possible geomorphological structure. This ensures that OMPcan find a well fitting combination of atoms for every patch.
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(a) Number of atoms.
(b) beta.
(c) Patch size.
Figure 5.2: Fusion parameters optimization.
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5.2 Quality Assessment of the Fused DEMWe assessed the quality of the fused DEMs with two methods: (a) comparison with referencedata and (b) calculating a relative accuracy map.
Comparison with a Reference DEM. Accuracy assessment of the output DEM is carriedout by means of a comparison with a reference DEM. Because the grid points of the referenceDEM usually do not coincide with the ones of the output DEM, an interpolation is necessary.A bicubic interpolation is used.
Relative Accuracy Map. After the quality assessment step all the input DEMs have an ac-curacy map with a quality value on each grid point. Thus, what is needed is the propagation ofthe quality of the input DEMs through the fusion to the output DEM. We consider the case ofthe weighted average of two input DEMs. Each grid point is denoted with a quality parameterσ1 and σ2, respectively. The height values of the resulting fused DEM, derived from both inputDEMs, are
z =σ22
σ21 + σ2
2
z1 +σ21
σ21 + σ2
2
z2 (5.13)
where z1, z2 are the height values of the input DEMs.Notice that the terms with the smallest uncertainty carry the most weight. Each measure-
ment is weighted by 1/σ2. The uncertainty in the weighted mean, the quality parameter of theoutput DEM, can be evaluated using standard error propagation (Heuvelink, 1998) which gives
σ =σ1 · σ2√σ21 + σ2
2
(5.14)
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Chapter 6
Results and Discussion
In this chapter experimental results, carried out upon DEMs coming from real data, are pre-sented. DEMs of varying resolution and produced with different technology (four InSAR andfour photogrammetric DEMs) have been fused. Six examples are described. In three examples,fusion is carried out among DEMs acquired with different sensors (radar and optical images) atdifferent resolutions; and in the last three examples, DEMs originated from the same technol-ogy (radar) are used. In addition, the examples adopt different strategies to compute the fusionweights and different DEM fusion techniques. Depending on the accuracy information for eachinput DEM, a different approach is chosen and tested.
The validation was performed by comparing the input and the obtained DEMs after thefusion with a high quality reference LiDAR DEM provided by Swisstopo. All DEMs are ingrid format and they have already been previously co-registered to the reference LiDAR DEMusing the LS3D software (Akça, 2007).
For the working environment, available open-source software and free development toolswas used. Implementation has been carried out mainly in MATLAB R©(Version 7.12.0.635 /R2011a).
MATLAB functionality has been extended with the following two open-source toolboxes:
TopoToolbox. It provides a set of Matlab functions that support the analysis of relief and flowpathways in DEMs (Schwanghart & Kuhn, 2010).
OMPBox v10. Implementation of the Batch-OMP and OMP-Cholesky algorithms for quicksparse-coding of large sets of signals (Rubinstein et al., 2008).
An additional MATLAB toolbox, called export_fig, was used for data export (Woodford,2011). export_fig is a MATLAB toolbox, written by Oliver Woodford, for exporting figuresfrom MATLAB to standard image and document formats nicely. Quantum GIS (from version1.2 upwards) has been chosen as the visualisation and inspection tool which offers in additionan easy integration with GRASS GIS on a GNU/Linux operating system. For the transitionof the DEM data in the Swiss reference frame LV03 with orthometric heights LHN95 theREFRAME software from Swisstopo has been used.
For data import and export from and to MATLAB or to Quantum GIS, the ArcInfo ASCIIGrid format has been adopted.
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Figure 6.1: Overlap area of Thun. ALOS DEM.
All tests are done using a computer with Intel Core i7, Q720, 1.6 GHz CPU and 8 GB RAMusing only one core, and unoptimized Matlab code.
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6.1 Test Area and Test Data
A test areas is chosen, where many different data sets are available and much knowledge existsfrom previous studies. The test site is located at Thun, Switzerland, and characterized by areaswith different morphology and land cover, i.e steep mountains, smooth hilly regions and flatareas, both rural and urban.The elevation range is more than 1600 m, varying from 530 m to2190 m. The land cover is extremely variable with both dense and isolated buildings, openareas, forests, rivers and a lake. Fig. 6.1 shows the area of DEM overlap. The size of theoverlapping area among the various DEMs is approximately 10 km×12 km.
6.1.1 InSAR DEMs
ALOS PALSAR-1 DEM
15 m grid spacing. Image acquisition date: master 19.06.2006 and slave 04.08.2006. FrequencyL-Band (ca. 23 cm wavelength). It is produced by Sarmap SA. The overall accuracy is 20 mand has been estimated using the LiDAR DEM.
ERS-1 DEM
25 m grid spacing. Image acquisition date: late autumn to early spring time, obtained from1995 to 1998. Frequency C-Band (ca. 6 cm wavelength). It is produced by Sarmap SA. Theoverall accuracy is 10.7 m and it has been estimated using the LiDAR DEM.
The ALOS and the ERS-1 DEMs are delivered with an accuracy map. The values on theaccuracy map are derived according to the Eq. 2.4.
SRTM bands C and X DEMs
We utilized both the processed SRTM3 band-C scenes (90 m grid spacing) of the ConsultativeGroup for International Agricultural Research (CGIAR) Version 4 (http:// srtm.csi.cgiar.org/,accessed 5 May, 2012) whose data gaps were filled by means of ancillary elevation data (Reuteret al., 2007) and the SRTM band-X scenes (30m grid spacing) available free of charge fromDLR for scientific purposes. More information about this product is given in Sec. 2.1.1. Un-fortunatelly, the SRTM band-X scene for our area of study contains a big area without any data(void).
6.1.2 Optical DEMs
ASTER G-DEM
30 m grid spacing. It is the Version 2 of the free distributed global elevation model ASTERG-DEM. More information about this product is given in Sec. 2.1.1.
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IKONOS DEM
4 m grid spacing. Imaging acquisition date: December 2003, Images: two IKONOS imagetriplets, the image matching has been done with the SAT-PP (SAT-PP, 2012) software and theestimated accuracy (RMS) is 1−2 m in open areas, about 3 m on the average in the whole area,excluding vegetation and 8 m in vegetated areas.
ALOS PRISM DEM
5 m grid spacing. Imaging acquisition date: 21.09.2006, Number of PRISM images: 1 scenewith forward, nadir & backward images, the lakes are defined as water area with a given fixedheight. The RMSE for three test areas are for alpine areas: 6.7 m-7.2 m, for open areas: 4.7m, for tree areas: 7.9 m-12.8 m and for city areas: 5.0 m-5.6 m (Grün et al., 2007). The imagematching has been done with the SAT-PP software (SAT-PP, 2012). This software includes aset of algorithms for processing of high resolution imagery (HRSI) and it has been developedby our group at ETH Zurich.
SPOT Reference3D DEM
30 m grid spacing. Image acquisition date: 30.09.2002. It is produced by SpotImage usingimage matching . The given absolute elevation accuracy for flat or rolling terrain (slope≤20%)is 10 m, for hilly terrain (20 %≤ slope≤ 40%) is 18 m and for the mountainous terrain (slope>40%) is 30 m. The Reference 3D DEM is delivered with a High Resolution orthoimage (SPOT5 sensor) with 5 m ground pixel size.
6.1.3 Reference DEMLiDAR DOM Swisstopo
2 m grid spacing. The airborne LiDAR data were acquired for the Swisstopo in Spring 2000with a mean density of 1-2 points per m2, depending on the terrain, and with first and last pulserecorded. The accuracy (1 σ) of the derived DEMs is 0.5 m and 1.5 m for vegetated areas.
The LiDAR Swisstopo DEM has been adopted as the reference DEM in view of its signifi-cantly higher accuracy, and generally also resolution, as compared to the other DEM generationtechnologies. The height points of the input and the fusion derived DEMs were compared toelevations interpolated from the gridded LiDAR DEM via bilinear interpolation. The resultingdiscrepancies in elevation are summarized in Tables 6.3, 6.5, 6.7, 6.9, 6.11, 6.13, 6.15, 6.17and 6.19.
Table 6.1 summarizes themain characteristics of the used DEMs. The accuracy is the RMSEcalculated using the LiDAR DEM as reference after the co-registration. Although in principle,the SRTM-X is more accurate than the SRTM-C in our case we observe the opposite. Thereason is that there are no SRTM-X data in the biggest part of the study area Fig. 6.4(c). SRTM-X data exist only on the west-south part of the test area which is characterized by steep slopes.
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Table 6.1: Characteristics of the used DEMs. The accuracy values of the DEMs are estimatedfor the test area using the LiDAR DEM as reference.
Grid reso-lution (m)
Technology Accuracy(RMSE)(m)
Date
ALOS PALSAR 15 InSAR 20 Summer 2006SPOT Reference3D 30 Photogrammetry 15 30.09.2002PRISM 5 Photogrammetry 16.5 21.09.2006IKONOS 4 Photogrammetry 19 December 2003ERS-1 25 InSAR 13 1995-1998ASTER G-DE 30 Photogrammetry 14.5 1999-2009SRTM-C 90 InSAR 20 February 2000SRTM-X 30 InSAR 56 February 2000Lidar DO 2 Lidar 0.5-1.5 Spring 2000
6.2 Test Results
6.2.1 Co-registration ResultsFirst all DEMs were transformed to the Swiss coordinate system LV03 LN02 (Swisstopo, 2008)like the reference Swisstopo LiDAR DEM. Then, all the DEMs are co-registered to the refer-ence LiDAR DEM through co-registration using three translations and the corresponding Zresidual maps are generated (see Figs. 6.2 and 6.4). The residual maps were computed bysubtracting the individual DEMs (input and output) from the reference LiDAR DEM (L) andseveral statistics measures were computed. For this reason, each time the reference LiDARDEM was interpolated using bicubic interpolation at the resolution of the input and outputDEMs. Errors that are bigger than 20 m are coded with the same color. Table 6.2 shows theresults of the co-registration.
Higher errors are observed in the areas with higher elevations, steeper slopes and intenseroughness. In the case of the DEMs produced using image mathcing this effect is probablycaused because of the poor stereo-matching in the higher elevation area due to partial cloudcover (e.g. IKONOS DEM). We note that errors in height can in principle also result fromdifferent geoid corrections.
6.2.2 DEM Fusion ResultsThree LiDAR Swisstopo DOM DEMs are used to learn the dictionary. A dictionary of 1500patches of size 9× 9 grid cells was generated with elements drawn randomly from the LiDARDEM. The dictionary was filtered using a K-means clustering algorithm with an Euclideanabsolute distance measure of 10 m (each centroid is the mean of the points in that cluster). Theclustering reduced the dictionary to approximately 1200 patches.
In all fusion examples that we describe below we set the overlap parameter β to 1, thenumber of the non-zeros atoms used in OMP was set to 10 and we used a 9 × 9 patch size
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processing window.In all examples, the output fused DEM has the size of the rectangle comprising all given
DEMs (the two input DEMs and the reference LiDAR DEM).The statistics included in all tables that follow are:
1. Min: the minimum value,
2. Max: the maximum value,
3. Mean: the mean value,
4. Median: the median value,
5. STD: the standard deviation,
6. RMSE: the root mean square error,
7. MAD0: the mean absolute deviation from the mean, and
8. MAD1: the median absolute deviation from the median.
All units in the following tables are in meters.The explanations of the abbreviations referring to the weights of the input DEMs, included
in all the tables that follow, are:
1. AM: the weight map is calculated using the given (from the producer) accuracy maps,when those exist,
2. S: the weight map is calculated using the slope applying curve fitting,
3. A: the weight map is calculated using the aspect applying curve fitting,
4. R: the weight map is calculated using the roughness (entropy) applying curve fitting,
5. S/A: the weight map is calculated using the slope and the aspect applying surface fitting,
6. S/R: the weight map is calculated using the slope and the roughness (entropy) applyingsurface fitting,
7. A/R: the weight map is calculated using the aspect and the roughness (entropy) applyingsurface fitting,
8. S/A/R: the weight map is calculated using the slope, the aspect and the roughness (en-tropy) applying volume fitting.
The DEM fusion method which gives the best result, the smallest RMSE, is highlighted inyellow color.
At the end, we performed a more detailed analysis of the results in relation to the slope.The slope classes were obtained by processing the LiDAR DEM. For the calculations we useda 5× 5 pixel window. The eight slope classes are:
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1. Slope≤0.5%2. 0.5%<Slope≤2%3. 2%<Slope≤5%4. 5%<Slope≤10%
5. 10%<Slope≤15%6. 15%<Slope≤30%7. 30%<Slope≤60%8. Slope>60%
Table 6.2: Co-registration results of the input DEMs to with the reference LiDAR DEM. σ0 isthe σ a-priori, σ0σ0σ0 is the σ a-posteriori and Tx, Ty, and Tz are the three translations.All units are in meters.
Slave DEM Resolution σ0σ0σ0 σ0σ0σ0 Iterations Tx Ty Tz
ALOS 15 5 13.4 6 11.4 -0.2 1.0ASTER 30 5 9.7 6 -41.0 -73.8 0.5ERS 25 5 8.9 4 -16.8 13.7 1.2IKONOS 4 2 5.5 5 1.4 -5.4 0.6PRISM 5 2 6.2 7 0.4 -8.5 -51.6SPOT 30 10 8.9 7 -31.5 -72.0 3.1SRTMC 90 10 9.3 5 44.4 28.0 1.3SRTMX 30 10 20.8 6 -11.7 -16.8 -50.4
6.2.3 Example 1: ALOS – IKONOSA grid was generated for the input DEMs at 4 m intervals according to the spatial resolution ofthe IKONOS DEM. After resampling, the input DEMs have a grid size of 3000 × 4375. Theprocessing time required for the fusion was using SR was 210 minutes while using WA was19.6 seconds. The weights for the input DEMs for the performed tests are calculated using thegiven accuracy map of the ALOS DEM, geomorphological characteristics and edginess (Table6.3). The edginess is calculated extracting the edges on the SPOT orthoimage using a Cannydetector. The othoimage is first resampled at 4 m. We use the edginess information for theIKONOS DEM which was produced using image matching. On points where edge elementsexist we give maximum weight value. Fig. 6.6 shows a detail of the Z difference images of thethree DEMs. The ALOS and IKONOS DEMs contain both big blunders. Both blunders areintroduced to the fused DEM but their areas are reduced.
The DEM fusion method which performs best is the weighted average using as weights theaccuracy map given from SARMAP for the ALOS DEM and the composite measure of slope,aspect and roughness for the IKONOS DEM. In Table 6.3, we see that compared to the ALOSDEM, the fusion achieved up to 45% improvement in RMSE and similarly, as compared to theIKONOS DEM, the fusion improved the RMSE by 41%.
6.2.4 Example 2: IKONOS – PRISMA grid was generated for the input DEMs at 5 m intervals according to the spatial resolutionof the PRISM DEM. After resampling, the input DEMs have a grid size of 2400 × 3500. The
79
Tabl
e6.
3:St
atis
tical
resu
ltsof
the
fusi
onof
the
AL
OS
and
IKO
NO
SD
EM
sfo
rthe
com
plet
ear
ea.A
llun
itsar
ein
met
ers.
Wei
ghts
AL
OS
Wei
ghts
IKO
NO
SM
etho
dM
inM
axM
ean
Med
ian
STD
RM
SEM
AD
0M
AD
1
IKO
NO
S-
--
-297
.513
86.2
0.4
-0.4
18.8
18.8
5.2
1.5
AL
OS
--
--2
20.3
805.
00.
6-0
.719
.920
.010
.76.
1
Fusi
onD
EM
--
AV-1
90.8
701.
60.
9-0
.013
.513
.56.
83.
5-
-D
F-7
85.5
1384
.2-0
.3-0
.319
.919
.95.
31.
5
SS
WA
-192
.382
9.5
1.0
-0.0
13.3
13.3
6.7
3.4
SR-2
16.3
558.
30.
9-0
.011
.611
.76.
53.
4
AA
WA
-210
.154
0.7
0.7
-0.1
12.1
12.1
6.3
3.3
SR-2
17.9
732.
80.
7-0
.111
.711
.76.
23.
3
RR
WA
-212
.261
9.9
0.9
012
.912
.96.
63.
5SR
-209
.646
0.8
0.8
011
.511
.56.
43.
4
AM
S/A
/RW
A-2
15.3
429.
20.
5-0
.211
.111
.15.
62.
7SR
-208
.173
4.7
0.6
-0.2
11.1
11.2
5.6
2.7
S/A
/RS/
A/R
&E
dges
WA
-186
.773
2.6
0.7
-0.1
11.9
11.9
6.3
3.2
SR-2
21.7
751.
20.
7-0
.111
.711
.76.
33.
2
S/A
/RS/
A/R
WA
-186
.734
1.0
0.7
0.0
11.6
11.6
6.4
3.3
SR-2
27.5
743.
30.
7-0
.011
.811
.86.
43.
3
80
Tabl
e6.
4:Fu
sed
DE
MR
MSE
(m)i
nre
latio
nto
slop
e.Fu
sion
ofth
eA
LO
San
dIK
ON
OS
DE
Ms.
Mea
nR
MSE
per
Slop
eC
lass
DE
MW
eigh
tsA
LO
SW
eigh
tsIK
ON
OS
Met
hod
Slop
eC
lass
esin
%
0-0.
50.
5-2
2-5
5-10
10-1
515
-30
30-6
0>6
0A
LO
S-
--
17.7
9.4
10.7
13.7
17.5
22.5
32.2
60.7
IKO
NO
S-
--
12.3
6.7
14.0
13.2
16.0
23.5
84.9
185.
2
Fusi
onD
EM
--
AV8.
87.
69.
914
.717
.721
.637
.693
.8-
-D
F3.
52.
95.
08.
811
.020
.161
.223
3.5
SS
WA
12.4
6.1
8.6
13.4
17.1
19.5
32.7
91.6
SR9.
54.
97.
511
.915
.420
.435
.784
.6
AA
WA
9.1
5.2
7.1
10.9
14.6
19.7
38.4
111.
7SR
9.8
4.6
6.7
10.7
14.3
19.8
40.3
123.
9
RR
WA
10.9
5.8
7.8
11.5
14.8
20.1
38.7
101.
7SR
8.7
5.0
7.3
11.2
14.4
19.7
38.2
98.8
AM
S/A
/RW
A7.
14.
97.
110
.914
.219
.839
.899
.9SR
8.8
4.4
6.4
10.2
13.5
18.7
40.4
135
S/A
/RS/
A/R
&E
dges
WA
9.6
6.0
8.1
11.4
14.8
19.0
30.6
69.9
SR10
.34.
96.
710
.313
.818
.633
.395
.6
S/A
/RS/
A/R
WA
7.5
5.7
8.5
12.5
16.3
20.9
32.8
74.6
SR9.
15.
27.
711
.915
.621
.036
.410
8.5
81
processing time required for the fusion was using SR was 118 minutes while using WA was6.2 seconds. The weights for the input DEMs for the performed tests are calculated usinggeomorphological characteristics and edginess (Table 6.5). The othoimage is first resampledat 5 m. On points where edge elements exist we give maximum weight value. Fig. 6.8 showsa detail of the Z difference images of the three DEMs. The big blunder of the PRISM DEMis introduced into the final fused DEM which supports the choice of the weights but its area isreduced significantly.
The DEM fusion method which performs best is the sparse representations using as weightsfor both DEMs the aspect. In Table 6.5, we see that compared to the PRISM DEM, the fusionachieved up to 15% improvement in RMSE and similarly, as compared to the IKONOS DEM,the fusion improved the RMSE by 25%.
6.2.5 Example 3: ALOS – SPOT
A grid was generated for the input DEMs at 15 m intervals according to the spatial resolutionof the ALOS DEM. After resampling, the input DEMs have a grid size of 800 × 1167. Theprocessing time required for the fusion was using SR was 5.5 minutes while using WA was1.2 seconds. The weights for the input DEMs for the performed tests are calculated using thegiven accuracy map of the ALOS DEM, geomorphological characteristics and edginess (Table6.7). The othoimage is first resampled at 15 m. On points where edge elements exist wegive maximum weight value. Fig. 6.10 shows a detail of the Z difference images of the threeDEMs. The errors of the ALOS and SPOT DEMs are reduced significantly in the final fusedDEM which supports the choice of the weights.
The DEM fusion method which performs best is the sparse representations using as weightsthe slope for the ALOS DEM and the composite measure of slope and edginess for the SPOTDEM. In Table 6.7, we see that compared to the ALOS DEM, the fusion achieved up to 36%improvement in RMSE and similarly, as compared to the SPOT DEM, the fusion improved theRMSE by 15%.
6.2.6 Example 4: ALOS – ERS
A grid was generated for the input DEMs at 15 m intervals according to the spatial resolutionof the ALOS DEM. After resampling, the input DEMs have a grid size of 800 × 1167. Theprocessing time required for the fusion was using SR was 5.9 minutes while using WA was 1.8seconds. The weights for the input DEMs for the performed tests are calculated using the givenaccuracy map of the ALOS DEM, and geomorphological characteristics (Table 6.9). Fig. 6.12shows a detail of the Z difference images of the three DEMs. The errors of the ALOS andERS DEMs are reduced significantly in the final fused DEM which supports the choice of theweights.
The DEM fusion method which performs best is the sparse representations using as weightsfor both DEMs the accuracy maps given from SARMAP. In Table 6.9, we see that comparedto the ALOS DEM, the fusion achieved up to 39% improvement in RMSE and similarly, ascompared to the ERS DEM, the fusion improved the RMSE by 4%.
82
Tabl
e6.
5:St
atis
tical
resu
ltsof
the
fusi
onof
the
IKO
NO
San
dPR
ISM
DE
Ms
fort
heco
mpl
ete
area
.All
units
are
inm
eter
s.
Wei
ghts
PRIS
MW
eigh
tsIK
ON
OS
Met
hod
Min
Max
Mea
nM
edia
nST
DR
MSE
MA
D0
MA
D1
PRIS
M-
--
-370
.314
4.2
-1.6
0.3
16.6
16.7
6.9
3.5
IKO
NO
S-
--
-292
.013
86.3
0.4
-0.4
18.8
18.8
5.3
1.5
Fusi
onD
EM
--
AV-3
31.1
695.
0-0
.40.
316
.516
.56.
22.
4-
-D
F-1
336.
612
89.3
-10.
320
.520
.56.
02.
2
SS
WA
-334
.077
7.1
-0.4
0.3
16.2
16.2
6.1
2.4
SR-3
33.1
427.
5-0
.70.
314
.614
.65.
92.
4
AA
WA
-325
532
-0.4
0.2
14.8
14.8
5.9
2.3
SR-3
26.3
375.
9-0
.50.
214
.114
.15.
82.
3
RR
WA
-333
.458
8.2
-0.5
0.3
16.0
16.0
6.1
2.4
SR-3
35.5
332.
6-0
.70.
314
.614
.65.
92.
3
S/A
/RS/
A/R
&E
dges
WA
-338
758.
1-0
.60.
215
.015
.05.
82.
3SR
-333
.037
3.9
-0.7
0.2
14.3
14.4
5.7
2.3
S/A
/RS/
A/R
WA
-338
.036
3.8
-0.6
0.3
14.6
14.6
5.8
2.3
SR-3
33.7
260.
1-0
.70.
314
.314
.35.
72.
3
83
Tabl
e6.
6:Fu
sed
DE
MR
MSE
(m)i
nre
latio
nto
slop
e.Fu
sion
ofth
ePR
ISM
and
IKO
NO
SD
EM
s.
Mea
nR
MSE
per
Slop
eC
lass
DE
MW
eigh
tsIK
ON
OS
Wei
ghts
PRIS
MM
etho
dSl
ope
Cla
sses
in%
0-0.
50.
5-2
2-5
5-10
10-1
515
-30
30-6
0>6
0IK
ON
OS
--
-23
.47.
815
.913
.715
.019
.252
.917
2.9
PRIS
M-
--
5.7
4.2
6.1
11.0
18.4
40.2
94.2
33.9
Fusi
onD
EM
--
AV15
.05.
17.
111
.213
.625
.271
.110
0.3
--
DF
18.8
3.5
4.0
5.7
7.8
10.8
23.8
99.3
SS
WA
23.2
4.2
5.6
9.4
12.7
23.3
74.3
101.
1SR
10.4
3.4
4.9
8.4
12.1
22.0
68.1
99.5
AA
WA
11.1
3.4
4.9
8.3
11.7
21.9
70.0
119.
0SR
10.2
3.1
4.7
7.9
11.4
20.7
65.3
111.
0
RR
WA
20.4
4.1
5.5
9.0
12.4
23.0
74.4
97.8
SR10
.43.
34.
98.
311
.921
.769
.797
.0
S/A
/RS/
A/R
&E
dges
WA
20.4
4.0
5.2
8.2
10.9
18.0
60.7
109.
1SR
12.8
3.6
4.7
7.5
10.8
17.3
57.9
117.
2
S/A
/RS/
A/R
WA
16.2
3.6
5.1
8.2
11.1
18.8
67.1
117.
4SR
12.8
3.5
4.9
8.1
11.3
19.6
65.5
121.
8
84
Tabl
e6.
7:St
atis
tical
resu
ltsof
the
fusi
onof
the
AL
OS
and
SPO
TD
EM
sfo
rthe
com
plet
ear
ea.A
llun
itsar
ein
met
ers.
Wei
ghts
AL
OS
Wei
ghts
SPO
TM
etho
dM
inM
axM
ean
Med
ian
STD
RM
SEM
AD
0M
AD
1
SPO
T-
--
-294
.321
0.9
0.4
-0.1
15.0
15.0
8.5
4.2
AL
OS
--
--2
14.3
797.
50.
7-0
.719
.920
.010
.76.
1
Fusi
onD
EM
--
AV-1
70.3
411.
10.
50.
314
.014
.08.
34.
7-
-D
F-9
44.2
810.
10.
30.
019
.119
.110
.35.
9
SS
WA
-180
.244
6.5
0.6
0.4
13.8
13.8
8.2
4.7
SR-1
84.3
186.
00.
50.
312
.912
.98.
04.
6
AA
WA
-171
.419
6.2
0.5
0.4
13.2
13.2
8.2
4.7
SR-1
77.3
195.
10.
40.
413
.013
.08.
04.
6
RR
WA
-173
.440
0.3
0.5
0.4
14.0
14.0
8.4
4.8
SR-1
77.0
193.
90.
40.
413
.213
.28.
24.
7
SS
&E
dges
WA
-180
.244
6.5
0.6
0.3
13.6
13.6
8.1
4.6
SR-1
82.9
189.
90.
50.
312
.812
.87.
94.
5
AM
SW
A-1
69.8
539.
90.
60.
514
.514
.58.
55
SR-1
70.8
198
0.4
0.4
13.4
13.4
8.3
4.9
S/A
/RS/
A/R
WA
-182
.019
6.0
0.4
0.4
13.1
13.1
8.1
4.8
SR-1
79.3
197.
10.
40.
413
.013
.08.
04.
7
85
Tabl
e6.
8:Fu
sed
DE
MR
MSE
(m)i
nre
latio
nto
slop
e.Fu
sion
ofth
eA
LO
San
dSP
OT
DE
Ms.
Mea
nR
MSE
per
Slop
eC
lass
DE
MW
eigh
tsA
LO
SW
eigh
tsSP
OT
Met
hod
Slop
eC
lass
esin
%
0-0.
50.
5-2
2-5
5-10
10-1
515
-30
30-6
0>6
0A
LO
S-
--
20.7
7.4
8.5
11.3
14.5
18.7
29.0
77.5
SPO
T-
--
13.3
6.7
7.3
9.2
11.1
13.3
19.9
45.1
Fusi
onD
EM
--
AV10
.65.
67.
19.
812
.415
.522
.247
.0-
-D
F11
.29.
07.
610
.212
.716
.526
.255
.1
SS
WA
12.4
5.6
6.9
9.5
12.0
14.9
21.6
43.4
SR4.
74.
96.
38.
511
.014
.422
.043
.9
AA
WA
6.4
5.1
6.4
9.3
11.9
15.2
21.8
40.3
SR5.
44.
46.
08.
411
.014
.522
.045
.6
RR
WA
10.7
5.5
6.6
9.3
11.9
15.4
22.8
46.7
SR5.
34.
56.
28.
611
.014
.722
.344
.3
SS
&E
dges
WA
11.7
5.7
6.8
9.2
11.6
14.5
21.0
40.9
SR5.
24.
86.
18.
210
.614
.121
.542
.2
AM
SW
A14
.05.
66.
99.
612
.215
.722
.849
.0SR
4.3
5.2
6.6
8.9
11.3
15.0
22.6
44.3
S/A
/RS/
A/R
WA
4.7
5.3
6.9
9.5
11.8
14.9
21.9
40.5
SR3.
35.
06.
48.
811
.214
.822
.651
.1
86
Tabl
e6.
9:St
atis
tical
resu
ltsof
the
fusi
onof
the
AL
OS
and
ER
SD
EM
sfo
rthe
com
plet
ear
ea.A
llun
itsar
ein
met
ers.
Wei
ghts
AL
OS
Wei
ghts
ER
SM
etho
dM
inM
axM
ean
Med
ian
STD
RM
SEM
AD
0M
AD
1
AL
OS
--
--2
14.3
797.
60.
7-0
.719
.920
.010
.76.
1E
RS
--
--1
18.8
223.
01.
0-1
.112
.712
.87.
93.
9
Fusi
onD
EM
--
AV-1
39.4
404.
70.
8-0
.913
.813
.88.
34.
5-
-D
F-8
42.3
802.
60.
4-0
.816
.816
.89.
45.
2
AM
AM
WA
-114
.436
5.9
1.0
-0.9
12.9
12.9
8.0
4.4
SR-1
12.2
220.
70.
9-0
.912
.312
.37.
84.
3
SS
WA
-137
.445
2.1
0.9
-0.9
13.6
13.6
8.2
4.4
SR-1
26.8
226.
20.
8-0
.912
.712
.78.
04.
4
AA
WA
-130
.322
8.1
0.8
-0.9
13.0
13.0
8.1
4.5
SR-1
27.0
226.
50.
8-0
.912
.812
.88.
04.
4
RR
WA
-141
.037
0.4
0.8
-0.9
13.6
13.7
8.2
4.5
SR-1
30.7
225.
10.
8-0
.912
.912
.98.
04.
5
S/A
S/A
WA
-133
.422
6.4
0.9
-0.9
12.9
12.9
8.1
4.5
SR-1
23.1
225.
30.
9-0
.912
.712
.78.
04.
4
S/R
S/R
WA
-132
.546
4.2
0.9
-0.8
13.9
13.9
8.4
4.8
SR-1
24.0
223.
40.
8-0
.813
.013
.08.
24.
8
S/A
/RS/
A/R
WA
-143
.522
7.8
0.7
-0.9
12.6
12.6
7.8
4.3
SR-1
31.4
222.
00.
8-0
.912
.512
.67.
74.
2
87
Tabl
e6.
10:F
used
DE
MR
MSE
(m)i
nre
latio
nto
slop
e.Fu
sion
ofth
eE
RS
and
AL
OS
DE
Ms.
Mea
nR
MSE
per
Slop
eC
lass
DE
MW
eigh
tsA
LO
SW
eigh
tsE
RS
Met
hod
Slop
eC
lass
esin
%
0-0.
50.
5-2
2-5
5-10
10-1
515
-30
30-6
0>6
0A
LO
S-
--
20.7
7.4
8.5
11.3
14.5
18.7
29.0
77.5
ER
S-
--
11.1
5.5
7.2
9.1
10.8
13.2
18.8
39.4
Fusi
onD
EM
--
AV10
.65.
47.
19.
712
.015
.121
.953
.7-
-D
F10
.17.
37.
28.
510
.313
.721
.449
.4
AM
AM
WA
8.2
5.6
7.2
9.9
12.1
14.8
20.0
46.0
SR2.
85.
06.
58.
811
.014
.221
.151
.9
SS
WA
12.6
5.4
6.9
9.4
11.8
14.6
21.3
48.2
SR3.
25.
06.
58.
911
.114
.621
.852
.7
AA
WA
5.4
5.2
6.7
9.2
11.7
14.9
21.7
45.0
SR3.
34.
96.
48.
811
.114
.721
.953
.3
RR
WA
10.2
5.3
6.8
9.2
11.8
15.1
22.1
50.8
SR3.
25.
16.
48.
911
.214
.822
.253
.5
S/A
S/A
WA
6.1
5.5
7.0
9.4
11.8
14.5
21.2
43.4
SR3.
15.
16.
58.
911
.214
.621
.752
.8
S/R
S/R
WA
11.3
5.7
7.0
9.4
11.9
15.3
2251
.4SR
3.3
5.5
6.7
9.0
11.3
14.8
22.0
53.2
S/A
/RS/
A/R
WA
5.0
5.4
6.8
9.2
11.3
14.1
21.0
44.9
SR3.
35.
16.
58.
810
.814
.421
.552
.9
88
6.2.7 Example 6: ERS – SPOTA grid was generated for the input DEMs at 25 m intervals according to the spatial resolutionof the ERS DEM. After resampling, the input DEMs have a grid size of 480 × 700. Theprocessing time required for the fusion was using SR was 2.1 minutes while using WA was0.6 seconds. The weights for the input DEMs for the performed tests are calculated using thegiven accuracy map of the ERS DEM, geomorphological characteristics and edginess (Table6.11). The othoimage is first resampled at 25 m. On points where edge elements exist we givemaximum weight value. Fig. 6.14 shows a detail of the Z difference images of the three DEMs.The errors of the SPOT and ERS DEMs are reduced significantly in the final fused DEM whichsupports the choice of the weights.
The DEM fusion method which performs best is the data fitting using as interpolation theregularized spline with tension method. In Table 6.11, we see that compared to the ERS DEM,the fusion achieved up to 4% improvement in RMSE and similarly, as compared to the SPOTDEM, the fusion improved the RMSE by 20%.
89
(a) ALOS residual map.
(b) IKONOS residual map.
(c) PRISM residual map.
Figure 6.2: The coloured absolute Z residuals (m) between the LiDAR DEM and the trans-formed ALOS, IKONOS and PRISM DEMs after co-registration.
90
(a) ERS residual map.
(b) SPOT residual map.
Figure 6.3: The coloured absolute Z residuals (m) between the LiDAR DEM and the trans-formed ERS, and SPOT DEMs after co-registration.
91
(a) ASTER residual map.
(b) SRTM-C residual map.
(c) SRTM-X residual map.
Figure 6.4: The coloured absolute Z residuals (m) between the LiDAR DEM and the trans-formed ASTER, SRTM-C and SRTM-X DEMs after co-registration.
92
(a)
(b)
Figure 6.5: Profiles of fused DEM, and reference LiDAR DEM along with that of (a) ALOSDEM and (b) IKONOS DEM.
93
(a) ALOS DEM (b) IKONOS DEM. (c) Fusion DEM.
(d) ALOS residual map. (e) IKONOS residual map. (f) Fusion residual map.
Figure 6.6: ALOS–IKONOS fusion example. (a)-(c) Input and output DEMs, (d) Residualsbetween the ALOS and the LiDAR DEMs, (e) Residuals between the IKONOS andthe LiDAR DEMs, (f) Residuals between the fusion and the LiDAR DEMs. Theresiduals bar unit is in meters.
94
(a)
(b)
Figure 6.7: Profiles of fused DEM, and reference LiDAR DEM along with that of (a) IKONOSDEM and (b) PRISM DEM.
95
(a) IKONOS DEM. (b) PRISM DEM. (c) Fusion DEM.
(d) IKONOS residual map. (e) PRISM residual map. (f) Fusion residual map.
Figure 6.8: IKONOS–PRISM fusion example. (a)-(c) Input and output DEMs, (d) Residualsbetween the IKONOS and the LiDAR DEMs, (e) Residuals between the PRISMand the LiDAR DEMs, (f) Residuals between the fusion and the LiDAR DEMs.The residuals bar unit is in meters.
96
(a)
(b)
Figure 6.9: Profiles of fused DEM, and reference LiDAR DEM along with that of (a) ALOSDEM and (b) SPOT DEM.
97
(a) ALOS DEM. (b) SPOT DEM. (c) Fusion DEM.
(d) ALOS residual map. (e) SPOT residual map. (f) Fusion residual map.
Figure 6.10: ALOS–SPOT fusion example. (a)-(c) Input and output DEMs, (d) Residuals be-tween the ALOS and the LiDAR DEMs, (e) Residuals between the SPOT andthe LiDAR DEMs, (f) Residuals between the fusion and the LiDAR DEMs. Theresiduals bar unit is in meters.
98
(a)
(b)
Figure 6.11: Profiles of fused DEM, and reference LiDAR DEM along with that of (a) ALOSDEM and (b) ERS DEM.
99
(a) ALOS DEM. (b) ERS DEM. (c) Fusion DEM.
(d) ALOS residual map. (e) ERS residual map. (f) Fusion residual map.
Figure 6.12: ALOS–ERS fusion example. (a)-(c) Input and output DEMs, (d) Residuals be-tween the ALOS and the LiDAR DEMs, (e) Residuals between the ERS and theLiDAR DEMs, (f) Residuals between the fusion and the LiDAR DEMs. Theresiduals bar unit is in meters.
100
(a)
(b)
Figure 6.13: Profiles of fused DEM, and reference LiDAR DEM along with that of (a) ERSDEM and (b) SPOT DEM.
101
(a) ERS DEM. (b) SPOT DEM. (c) Fusion DEM.
(d) ERS residual map. (e) SPOT residual map. (f) Fusion residual map.
Figure 6.14: ERS–SPOT fusion example. (a)-(c) Input and output DEMs, (d) Residuals be-tween the ERS and the LiDAR DEMs, (e) Residuals between the SPOT and theLiDAR DEMs, (f) Residuals between the fusion and the LiDAR DEMs. Theresiduals bar unit is in meters.
102
6.2.8 Example 7: ERS – ASTER
A grid was generated for the input DEMs at 25 m intervals according to the spatial resolution ofthe ERS DEM. After resampling, the input DEMs have a grid size of 480×700. The processingtime required for the fusion was using SR was 2 minutes while using WA was 0.6 seconds. Theweights for the input DEMs for the performed tests are calculated using the given accuracy mapof the ERS DEM, geomorphological characteristics and edginess (Table 6.13). The othoimageis first resampled at 25 m. On points where edge elements exist we give maximum weightvalue. Fig. 6.16 shows a detail of the Z difference images of the three DEMs. The errors of theASTER and ERS DEMs are reduced significantly in the final fused DEM which supports thechoice of the weights.
The DEM fusion method which performs best is the data fitting using as interpolation theregularized spline with tension method. In Table 6.13, we see that compared to the ERS DEM,the fusion achieved up to 6% improvement in RMSE and similarly, as compared to the ASTERDEM, the fusion improved the RMSE by 19%.
6.2.9 Example 8: SPOT – SRTM band-C
A grid was generated for the input DEMs at 30 m intervals according to the spatial resolutionof the SPOT DEM. After resampling, the input DEMs have a grid size of 400 × 583. Theprocessing time required for the fusion was using SR was 1.4 minutes while using WA was0.5 seconds. The weights for the input DEMs for the performed tests are calculated usinggeomorphological characteristics and edginess (Table 6.15). The othoimage is first resampledat 30 m. On points where edge elements exist we give maximum weight value. Fig. 6.18 showsa detail of the Z difference images of the three DEMs. The big blunders of the SRTM-C DEMare introduced but their area is significantly reduced.
The DEM fusion method which performs best is the data fitting using as interpolation theregularized spline with tension method. In Table 6.15, we see that compared to the SRTM-CDEM, the fusion achieved up to 46% improvement in RMSE while the SPOT DEM was notimproved, so DEM fusion failed.
6.2.10 Example 9: SRTM band-C – ASTER
A grid was generated for the input DEMs at 30 m intervals according to the spatial resolutionof the ASTER DEM. After resampling, the input DEMs have a grid size of 400 × 583. Theprocessing time required for the fusion was using SR was 1.4 minutes while using WA was0.4 seconds. The weights for the input DEMs for the performed tests are calculated usinggeomorphological characteristics and edginess (Table 6.17). The othoimage is first resampledat 30 m. On points where edge elements exist we give maximum weight value. Fig. 6.20 showsa detail of the Z difference images of the three DEMs. The big blunders of the SRTM-C DEMare introduced but their area is significantly reduced.
The DEM fusion method which performs best is the data fitting using as interpolation theregularized spline with tension method. In Table 6.13, we see that compared to the SRTM-C
103
Tabl
e6.
11:S
tatis
tical
resu
ltsof
the
fusi
onof
the
ER
San
dSP
OT
DE
Ms
fort
heco
mpl
ete
area
.All
units
are
inm
eter
s.
Wei
ghts
ER
SW
eigh
tsSP
OT
Met
hod
Min
Max
Mea
nM
edia
nST
DR
MSE
MA
D0
MA
D1
ER
S-
--
-111
.221
8.8
1.0
-1.1
12.4
12.5
7.8
3.9
SPO
T-
--
-275
.721
7.1
0.3
-0.1
15.0
15.0
8.6
4.2
Fusi
onD
EM
--
AV-1
70.1
157.
60.
6-0
.312
.512
.57.
63.
9-
-D
F-2
44.8
242.
30.
9-0
.712
.012
.07.
33.
7
SS
WA
-156
.515
6.9
0.8
-0.3
12.3
12.3
7.6
3.9
SR-1
60.5
159.
70.
7-0
.312
.112
.17.
43.
8
AA
WA
-164
.915
8.4
0.7
-0.3
12.4
12.4
7.6
3.9
SR-1
62.4
161.
80.
6-0
.312
.212
.37.
43.
8
RR
WA
-167
.515
8.4
0.6
-0.3
12.4
12.5
7.6
3.9
SR-1
65.8
165.
20.
6-0
.312
.312
.37.
53.
8
SS
&E
dges
WA
-156
.515
6.9
0.7
-0.2
12.3
12.3
7.6
3.9
SR-1
58.9
158.
40.
7-0
.312
.112
.27.
53.
9
SA
MW
A-1
47.7
153.
10.
8-0
.212
.212
.27.
64.
0SR
-146
.115
8.8
0.8
-0.2
12.0
12.1
7.4
3.9
S/A
/RS/
A/R
WA
-174
.015
5.1
0.6
-0.3
12.2
12.3
7.4
3.8
SR-1
72.3
158.
40.
6-0
.312
.212
.27.
43.
8
104
Tabl
e6.
12:F
used
DE
MR
MSE
(m)i
nre
latio
nto
slop
e.Fu
sion
ofth
eE
RS
and
SPO
TD
EM
s.
Mea
nR
MSE
per
Slop
eC
lass
DE
MW
eigh
tsE
RS
Wei
ghts
SPO
TM
etho
dSl
ope
Cla
sses
in%
0-0.
50.
5-2
2-5
5-10
10-1
515
-30
30-6
0>6
0E
RS
--
-2.
33.
85.
27.
08.
611
.217
.438
.8SP
OT
--
-9.
25.
25.
87.
49.
311
.718
.742
.2
Fusi
onD
EM
--
AV2.
63.
85.
06.
78.
611
.417
.734
.5-
-D
F6.
13.
84.
65.
86.
89.
515
.432
.7
SS
WA
2.8
3.7
5.0
6.7
8.6
11.2
17.5
34.0
SR2.
93.
44.
56.
28.
210
.817
.233
.1
AA
WA
3.3
3.4
4.6
6.6
8.7
11.3
17.5
34.4
SR3.
53.
14.
46.
38.
210
.817
.333
.6
RR
WA
2.8
3.8
4.9
6.7
8.7
11.3
17.6
34.5
SR2.
73.
44.
66.
38.
110
.817
.333
.9
SS
&E
dges
WA
2.9
3.7
5.0
6.7
8.7
11.3
17.4
33.5
SR3.
03.
34.
56.
38.
010
.917
.232
.9
SA
MW
A2.
83.
94.
96.
78.
611
.217
.333
.4SR
2.9
3.3
4.5
6.3
8.0
10.8
17.1
32.5
S/A
/RS/
A/R
WA
3.6
3.7
4.9
6.7
8.6
11.2
17.5
33.1
SR2.
93.
44.
56.
38.
110
.817
.233
.8
105
Tabl
e6.
13:S
tatis
tical
resu
ltsof
the
fusi
onof
the
ER
San
dA
STE
RD
EM
sfo
rthe
com
plet
ear
ea.A
llun
itsar
ein
met
ers.
Wei
ghts
ER
SW
eigh
tsA
STE
RM
etho
dM
inM
axM
ean
Med
ian
STD
RM
SEM
AD
0M
AD
1
ER
S-
--
-111
.221
8.8
1.0
-1.1
12.4
12.5
7.8
3.9
AST
ER
--
--2
37.7
177.
40.
6-0
.314
.514
.58.
95.
1
Fusi
onD
EM
--
AV-1
69.9
168.
50.
8-0
.712
.412
.47.
74.
2-
-D
F-1
05.4
239.
71.
0-0
.911
.811
.87.
43.
7
AM
S/A
/RW
A-1
61.7
176
0.9
-0.5
12.2
12.2
7.7
4.2
SR-1
59.1
171.
50.
9-0
.612
.012
.17.
64.
2
SS
WA
-169
.916
8.5
0.8
-0.7
12.4
12.4
7.7
4.2
SR-1
61.7
167.
20.
9-0
.712
.212
.27.
64.
1
AA
WA
-171
.316
3.0
0.8
-0.6
12.4
12.4
7.7
4.2
SR-1
70.3
160.
90.
8-0
.712
.312
.37.
74.
1
RR
WA
-173
.116
6.6
0.8
-0.7
12.4
12.4
7.7
4.1
SR-1
73.4
181.
10.
8-0
.712
.312
.37.
64.
1
S/A
/RS/
A/R
WA
-176
.816
8.6
0.8
-0.6
12.2
12.2
7.5
4.0
SR-1
82.2
177.
40.
8-0
.712
.212
.27.
54.
0
S/A
/RS/
A/R
&E
dges
WA
-176
.816
8.6
0.8
-0.6
12.3
12.3
7.6
4.1
SR-1
71.3
169.
70.
8-0
.612
.212
.27.
54.
0
106
Tabl
e6.
14:F
used
DE
MR
MSE
(m)i
nre
latio
nto
slop
e.Fu
sion
ofth
eE
RS
and
AST
ER
DE
Ms.
Mea
nR
MSE
per
Slop
eC
lass
DE
MW
eigh
tsE
RS
Wei
ghts
AST
ER
Met
hod
Slop
eC
lass
esin
%
0-0.
50.
5-2
2-5
5-10
10-1
515
-30
30-6
0>6
0E
RS
--
-2.
33.
85.
27.
08.
611
.217
.438
.8A
STE
R-
--
9.4
6.4
6.9
8.0
9.6
12.1
18.5
37.2
Fusi
onD
EM
--
AV2.
84.
15.
26.
88.
811
.417
.533
.6-
-D
F3.
14.
04.
86.
47.
910
.115
.632
.6
SS
WA
2.9
4.2
5.2
6.9
8.8
11.3
17.3
33.7
SR3.
23.
84.
96.
58.
511
.016
.931
.6
AA
WA
3.2
3.8
5.1
6.7
8.8
11.4
17.6
33.6
SR3.
33.
84.
86.
58.
411
.117
.031
.8
RR
WA
2.9
4.0
5.2
6.8
8.9
11.4
17.5
33.6
SR3.
13.
84.
86.
58.
411
.117
.031
.8
S/A
/RS/
A/R
&E
dges
WA
3.8
4.2
5.1
6.7
8.8
11.3
17.2
32.6
SR3.
43.
64.
86.
38.
310
.916
.833
.3
AM
S/A
/RW
A3.
14.
05.
26.
68.
611
.117
.233
.7SR
3.1
3.6
4.9
6.2
8.1
10.8
16.7
33.4
S/A
/RS/
A/R
WA
3.4
3.9
5.0
6.7
8.7
11.3
17.2
32.7
SR3.
13.
84.
76.
58.
410
.916
.931
.6
107
Tabl
e6.
15:S
tatis
tical
resu
ltsof
the
fusi
onof
the
SPO
Tan
dSR
TM
-CD
EM
sfo
rthe
com
plet
ear
ea.A
llun
itsar
ein
met
ers.
Wei
ghts
SPO
TW
eigh
tsSR
TM
-C
Met
hod
Min
Max
Mea
nM
edia
nST
DR
MSE
MA
D0
MA
D1
SPO
T-
--
-267
.920
9.9
0.3
-0.2
14.8
14.8
8.4
4.2
SRT
M-C
--
--2
15.7
361.
83.
9-0
.929
.029
.317
.88.
2
Fusi
onD
EM
--
AV-2
02.2
216.
72.
10.
419
.519
.612
.05.
9-
-D
F-2
51.3
262.
51.
30.
815
.615
.79.
44.
9
SS
WA
-202
.823
8.3
2.4
0.4
19.7
19.8
12.1
5.8
SR-2
09.9
250.
82.
30.
419
.219
.311
.75.
6
AA
WA
-202
.122
8.9
2.2
0.4
19.6
19.7
12.1
5.9
SR-1
91.5
227.
42.
10.
319
.219
.311
.75.
7
RR
WA
-202
.422
5.7
2.1
0.4
19.5
19.6
11.9
5.7
SR-1
88.0
227.
52.
10.
419
.019
.111
.55.
5
S/A
/RS/
A/R
WA
-199
.920
7.9
1.8
0.3
18.2
18.2
10.9
5.2
SR-2
01.5
205.
41.
70.
317
.918
.010
.75.
1S/
A/R
&E
dges
S/A
/RW
A-1
99.9
206.
41.
70.
317
.717
.810
.65.
1SR
-208
.019
9.9
1.6
0.2
17.4
17.5
10.4
5.1
108
Tabl
e6.
16:F
used
DE
MR
MSE
(m)i
nre
latio
nto
slop
e.Fu
sion
ofth
eSR
TM
-Can
dSP
OT
DE
Ms.
Mea
nR
MSE
per
Slop
eC
lass
DE
MW
eigh
tsSP
OT
Wei
ghts
SRT
M-
C
Met
hod
Slop
eC
lass
esin
%
0-0.
50.
5-2
2-5
5-10
10-1
515
-30
30-6
0>6
0SP
OT
--
-4.
74.
44.
75.
97.
510
.317
.640
.4SR
TM
-C-
--
28.9
9.7
13.9
16.8
20.4
24.4
34.4
48.0
Fusi
onD
EM
--
AV4.
04.
76.
49.
813
.820
.030
.837
.9-
-D
F8.
05.
24.
65.
46.
910
.117
.937
.7
SS
WA
7.0
5.1
6.8
10.7
15.2
21.7
29.6
37.5
SR9.
24.
16.
09.
312
.617
.727
.940
.3
AA
WA
8.5
4.7
6.2
10.2
14.3
19.2
29.7
40.0
SR9.
53.
95.
78.
812
.117
.227
.840
.6
RR
WA
5.6
4.5
6.0
9.4
13.3
19.2
30.5
39.3
SR7.
74.
15.
88.
511
.616
.527
.941
.2S/
A/R
&E
dges
S/A
/RW
A5.
54.
65.
78.
611
.816
.325
.838
.2SR
6.6
3.9
5.3
7.6
10.5
14.6
24.4
39.2
S/A
/RS/
A/R
WA
5.0
4.4
5.9
8.9
12.4
17.2
27.4
38.4
SR6.
73.
75.
68.
110
.815
.425
.540
.0
109
(a)
(b)
Figure 6.15: Profiles of fused DEM, and reference LiDAR DEM along with that of (a) ERSDEM and (b) ASTER DEM.
DEM, the fusion achieved up to 53% improvement in RMSE and similarly, as compared to theASTER DEM, the fusion improved the RMSE by 4%.
6.2.11 Example 10: SRTM band-X – SRTM band-C
A grid was generated for the input DEMs at 30 m intervals according to the spatial resolutionof the SRTM band-X DEM. After resampling, the input DEMs have a grid size of 400 × 583.
110
Tabl
e6.
17:S
tatis
tical
resu
ltsof
the
fusi
onof
the
SRT
M-C
and
AST
ER
DE
Ms
fort
heco
mpl
ete
area
.All
units
are
inm
eter
s.
Wei
ghts
AST
ER
Wei
ghts
SRT
M-
C
Met
hod
Min
Max
Mea
nM
edia
nST
DR
MSE
MA
D0
MA
D1
AST
ER
--
--2
30.7
168.
00.
5-0
.314
.314
.38.
75.
0SR
TM
-C-
--
-215
.736
1.8
3.9
-0.9
29.0
29.3
17.8
8.2
Fusi
onD
EM
--
AV-1
68.9
251.
32.
2-0
.419
.619
.712
.26.
2-
-D
F-2
30.3
169.
20.
7-0
.213
.813
.88.
55.
1
SS
WA
-170
.127
0.0
2.5
-0.3
19.9
20.1
12.4
6.1
SR-1
72.8
278.
52.
5-0
.419
.419
.512
.05.
9
AA
WA
-165
.726
5.0
2.4
-0.3
19.5
19.7
12.2
6.1
SR-1
69.1
262.
72.
3-0
.419
.019
.211
.85.
9
RR
WA
-176
.623
9.5
2.0
-0.3
18.3
18.4
11.5
5.9
SR-1
82.4
248.
41.
9-0
.417
.918
11.1
5.7
S/A
/RS/
A/R
WA
-177
.924
1.5
1.9
-0.2
18.2
18.3
11.2
5.6
SR-1
83.2
243.
21.
9-0
.317
.817
.910
.95.
5S/
A/R
&E
dges
S/A
/RW
A-1
77.9
241.
51.
8-0
.217
.817
.910
.95.
6SR
-178
.824
4.8
1.8
-0.3
17.4
17.5
10.7
5.5
111
Tabl
e6.
18:F
used
DE
MR
MSE
(m)i
nre
latio
nto
slop
e.Fu
sion
ofth
eSR
TM
-Can
dA
STE
RD
EM
s.
Mea
nR
MSE
per
Slop
eC
lass
DE
MW
eigh
tsA
STE
RW
eigh
tsSR
TM
-C
Met
hod
Slop
eC
lass
esin
%
0-0.
50.
5-2
2-5
5-10
10-1
515
-30
30-6
0>6
0A
STE
R-
--
3.9
5.1
6.0
6.9
8.4
10.9
17.2
36.2
SRT
M-C
--
-28
.99.
713
.916
.820
.424
.434
.448
.0
Fusi
onD
EM
--
AV4.
15.
26.
79.
713
.719
.731
.038
.0-
-D
F7.
44.
05.
86.
57.
49.
815
.731
.6
SS
WA
6.8
5.8
7.0
10.5
15.5
21.9
30.1
37.3
SR10
.25.
06.
69.
412
.417
.827
.841
.1
AA
WA
7.1
5.2
6.9
9.8
13.6
18.9
29.4
39.9
SR8.
74.
56.
58.
711
.816
.827
.241
.2
RR
WA
4.8
5.1
6.3
8.7
12.3
17.0
27.6
38.7
SR6.
84.
35.
98.
110
.715
.325
.539
.8S/
A/R
&E
dges
S/A
/RW
A5.
95.
06.
28.
411
.616
.225
.738
.6SR
6.8
4.0
6.0
7.7
10.2
14.7
24.0
39.4
S/A
/RS/
A/R
WA
5.2
4.8
6.1
8.8
12.2
17.1
27.2
39.0
SR6.
54.
25.
98.
010
.915
.325
.239
.8
112
(a) ERS DEM. (b) ASTER DEM. (c) Fusion DEM.
(d) ERS residual map. (e) ASTER residual map. (f) Fusion residual map.
Figure 6.16: ERS–ASTER fusion example. (a)-(c) Input and output DEMs, (d) Residuals be-tween the ERS and the LiDAR DEMs, (e) Residuals between the ASTER andthe LiDAR DEMs, (f) Residuals between the fusion and the LiDAR DEMs. Theresiduals bar unit is in meters.
The processing time required for the fusion was using SR was 1.3 minutes while using WAwas 0.4 seconds. The weights for the input DEMs for the performed tests are calculated usinggeomorphological characteristics (Table 6.19). Fig. 6.21 shows a detail of the Z differenceimages of the three DEMs. Most of the blunders of the two input DEMs are significantlycorrected.
The DEM fusion method which performs best is the weighted average using as weights forboth DEMs the composite measure of slope, aspect and roughness. In Table 6.19, we see thatcompared to the SRTM-C DEM, the fusion achieved up to 29% improvement in RMSE andsimilarly, as compared to the SRTM-X DEM, the fusion improved the RMSE by 35%.
113
Tabl
e6.
19:S
tatis
tical
resu
ltsof
the
fusi
onof
the
SRT
M-X
and
SRT
M-C
DE
Ms
fort
heco
mpl
ete
area
.All
units
are
inm
eter
s.
Wei
ghts
SRT
M-
C
Wei
ghts
SRT
M-
X
Met
hod
Min
Max
Mea
nM
edia
nST
DR
MSE
MA
D0
MA
D1
SRT
M-C
--
--2
15.7
295.
011
.09.
550
.651
.739
.432
.4SR
TM
-X-
--
-289
.192
2.1
3.2
1.9
56.2
56.2
28.9
15.8
Fusi
onD
EM
--
AV-1
89.8
608.
67.
14.
843
.444
.030
.723
.7-
-D
F-9
83.6
549.
8-4
.01.
549
.049
.227
.315
.1
SS
WA
-194
.654
5.3
7.4
4.9
42.0
42.6
29.6
22.5
SR-2
07.2
480.
94.
63.
437
.037
.327
.822
.2
AA
WA
-191
.459
7.0
6.1
4.0
39.9
40.4
29.2
22.8
SR-1
74.2
539.
94.
62.
837
.737
.928
.522
.8
RR
WA
-190
.860
8.6
7.1
4.8
43.2
43.7
30.5
23.5
SR-1
70.9
528.
54.
23.
137
.938
.128
.723
.1
S/A
S/A
WA
-193
.058
9.9
6.5
4.6
39.4
39.9
29.1
23.0
SR-1
75.1
512.
45.
13.
637
.437
.828
.423
.0
S/R
S/R
WA
-190
.560
8.5
6.1
4.6
40.4
40.9
30.0
23.9
SR-1
75.0
521.
14.
73.
538
.338
.629
.323
.8
A/R
A/R
WA
-189
.559
6.0
7.4
5.1
43.1
43.7
31.1
24.3
SR-1
65.3
511.
64.
83.
438
.538
.829
.524
.1
S/A
/RS/
A/R
WA
-194
.353
4.3
4.9
3.5
39.2
39.5
27.5
20.8
SR-1
86.0
523.
03.
22.
436
.636
.726
.820
.9
114
Tabl
e6.
20:F
used
DE
MR
MSE
(m)i
nre
latio
nto
slop
e.Fu
sion
ofth
eSR
TM
-Can
dSR
TM
-XD
EM
s.
Mea
nR
MSE
per
Slop
eC
lass
DE
MW
eigh
tsSR
TM
-C
Wei
ghts
SRT
M-
X
Met
hod
Slop
eC
lass
esin
%
0-0.
50.
5-2
2-5
5-10
10-1
515
-30
30-6
0>6
0SR
TM
-C-
--
53.8
42.6
46.6
42.2
46.1
45.0
47.6
54.2
SRT
M-X
--
-25
354
.644
.034
.543
.733
.832
.978
.7
Fusi
onD
EM
--
AV14
2.9
39.7
33.2
38.3
32.3
32.4
38.8
56.9
--
DF
72.3
50.7
39.7
37.8
24.5
22.6
22.7
60.2
SS
WA
132.
847
.242
.141
.131
.131
.236
.155
.1SR
57.3
27.4
34.6
31.3
30.6
29.9
33.3
46.7
AA
WA
86.4
37.6
42.4
35.5
28.9
29.9
36.1
52.4
SR56
.634
.133
.829
.431
.230
.434
.247
.1
RR
WA
131.
732
.538
.435
.930
.431
.538
.356
.9SR
58.6
29.8
35.0
29.8
29.1
30.0
34.4
47.8
S/A
S/A
WA
92.7
39.0
46.9
38.1
32.2
31.9
36.8
49.5
SR57
.235
.034
.234
.331
.031
.534
.445
.9
S/R
S/R
WA
105.
250
.341
.435
.932
.232
.039
.055
.5SR
56.0
37.1
33.5
31.4
30.6
31.4
35.4
47.7
A/R
A/R
WA
79.0
55.9
37.1
34.8
30.4
30.1
38.0
51.7
SR56
.432
.632
.430
.629
.431
.635
.347
.5
S/A
/RS/
A/R
WA
112.
256
.640
.636
.830
.129
.034
.252
.3SR
63.5
39.7
29.9
30.7
30.6
27.7
31.5
47.4
115
(a)
(b)
Figure 6.17: Profiles of fused DEM, and reference LiDAR DEM along with that of (a) SPOTDEM and (b) SRTM-C DEM.
116
(a) SPOT DEM. (b) SRTM-C DEM. (c) Fusion DEM.
(d) SPOT residual map. (e) SRTM-C residual map. (f) Fusion residual map.
Figure 6.18: SRTM-C–SPOT fusion example. (a)-(c) Input and output DEMs, (d) Residualsbetween the SPOT and the LiDAR DEMs, (e) Residuals between the SRTM-Cand the LiDAR DEMs, (f) Residuals between the fusion and the LiDAR DEMs.The residuals bar unit is in meters.
117
(a)
(b)
Figure 6.19: Profiles of fused DEM, and reference LiDAR DEM along with that of (a) SRTM-CDEM and (b) ASTER DEM.
118
(a) SRTM-C DEM. (b) ASTER DEM. (c) Fusion DEM.
(d) SRTM-C residual map. (e) ASTER residual map. (f) Fusion residual map.
Figure 6.20: SRTM-C–ASTER fusion example. (a)-(c) Input and output DEMs, (d) Residualsbetween the SRTM-C and the LiDAR DEMs, (e) Residuals between the ASTERand the LiDAR DEMs, (f) Residuals between the fusion and the LiDAR DEMs.The residuals bar unit is in meters.
119
(a) SRTM-C DEM. (b) SRTM-X DEM. (c) Fusion DEM.
(d) SRTM-C residual map. (e) SRTM-X residual map. (f) Fusion residual map.
Figure 6.21: SRTM-X–SRTM-C fusion example. (a)-(c) Input and output DEMs, (d) Residualsbetween the SRTM-X and the LiDAR DEMs, (e) Residuals between the SRTM-Cand the LiDAR DEMs, (f) Residuals between the fusion and the LiDAR DEMs.The residuals bar unit is in meters.
120
6.3 Summary and Discussion
In the previous sections the developed methodology has been applied to several datasets withdifferent characteristics. In this section experimental results obtained from the tests are dis-cussed.
In assessing the height discrepancies between the input DEMs for the different fusion ex-periments and the corresponding points from the LiDAR DEM, it should be kept in mind the0.5 − 1.5 m accuracy of the laser ranging component. Additional errors are introduced to theorginal DEM grids because the input DEMs were interpolated twice. The first interpolationwas made for the transformation to the Swiss coordinate system LV03 LN02 and the secondinterpolation was made after the co-registration.
Bearing in mind the initial assumptions (Sec.3.1) the results show that fusion can be suc-cessfully performed between two DEMs acquired at different resolutions and with differenttechniques. Tables Tables 6.3, 6.5, 6.7, 6.9, 6.11, 6.13, 6.15, 6.17 and 6.19 show the statisticsfor the comparison of the input and output fusion DEMs to the reference LiDAR DEM. Theyshow several effects generated by the DEM fusion processes. Improvements can be seen fromthe standard deviations. The minimum/maximum values, which are the highest differences tothe reference DEM are reduced by fusion. This implies that the fusion DEM is more reliableeven if the standard deviation does not change a lot. Similar results are received for both DEMfusion methods (WA and SR) with slightly better values for the SR DEM fusion.
We notice that when we fuse DEMs produced by different technology (e.g. RADAR andoptical stereoscopy) the reduction of the RMSE on the fused DEM are more significant sincewe exploit the complementary advantages of two different technologies like in the examples inSec. 6.2.3 to 6.2.5 and 6.2.7 to 6.2.10.
We notice that, in practice, the DEM fusion method of the Weighted Average works sur-prisingly well, which confirms the intuition that the correct choice of weights has a far greaterinfluence on the results than the mathematical recipe for fusing the inputs. Its main weaknessesfrom an algorithmic point of view are on one hand that it looks at each grid point independently,disregarding the resulting surface shape; and on the other hand that the up-sampling of the in-put DEMs could potentially introduce further artefacts – which however is a largely theoreticalconcern with little practical relevance.
Composite measures that incorporates multiple individual quality measures to provide asingle score perform better than individual quality measures.
The area investigated represents different terrain classes from almost no slopes to steepslopes. It is well established that the performance of DEM generation technologies is generallydegraded as a function of increasing terrain slope. Steep terrain adversely impacts especiallyupon image matching in stereo photogrammetric techniques and upon radar interferometry. Thecurrent experiments offered a favourable opportunity to examine how different DEM fusiontechniques behaved in areas of differing topography. The results of the slope assessment onTables 6.4, 6.6, 6.8, 6.10, 6.12, 6.14, 6.16, 6.18 and 6.20 show that the DEM fusion methodof the Weighted Average works better on steeper slopes (>30%) while the DEM fusion methodof sparse representations performs better on flatter terrain (<30%) . We notice that the fusionleads to an improvement especially for medium and high slopes and less for low slopes.
For a more detailed evaluation of the DEM fusion, profiles of the various DEMs along a
121
given line crossing a slope are presented in Figs. 6.5, 6.7, 6.9, 6.11, 6.13, 6.15, 6.17 and 6.19.Qualitatively, the terrain accuracy was improved from the DEM fusion process as can be seenfrom the closer resemblance of the fused terrain profiles to the reference profiles.
122
Chapter 7
Conclusions and Outlook
It has been the aim of this study to develop a new set of tools that fuse two DEMs. In particular,the thesis argument is that this can only be achieved effectively by (i) evaluating the qualityof the input DEMs, and (ii) fusing using such information. This concluding chapter considersthe success of the approach be re-evaluating the research objectives stated in the introduction,identifying weaknesses of the approach, and avenues for future research.
7.1 Summary
This thesis has dealt with the fusion of DEMs. Existing DEMs are 2.5-dimensional non-parametric surfaces represented by samples on regular grids, and the goal is to obtain a newDEM in the form of new height values for the denser one of the two grids. The crucial techni-cal difficulty of DEM fusion is that it requires weights to quantify the influence of the inputs atevery surface location. These weights are a function of the height accuracy, and typically varysignificantly across each DEM, due to the sensor technology, scene characteristics and methodused to generate it. These weights, sometimes referred to as “height error maps”, should be anatural by-product of DEM generation, but are nevertheless often not available, in which caseone can try to estimate them statistically from local surface or scene properties.
We reviewed several methods (data fitting, averaging, weighted average and sparse repre-sentations) for DEM fusion, and presented an empirical investigation of how to derive fusionweights from a given DEM. To mitigate the weakness of weighted averaging, it has been pro-posed to exploit the framework of sparse representations for DEM fusion (Papasaika et al.,2011b). In a nutshell, the rationale is the following: to be more robust to blunders it would bedesirable to include prior knowledge about plausible surface shapes, such that improbable localsurface geometries caused by blunders are suppressed. Such a fusion scheme can be designedby representing a local DEM patch as a sparse combination of basis patches, such that the ba-sis can (theoretically) represent all local terrain shapes. The use of patches instead of singleheight values, together with the sparsity constraint (i.e. only very few basis patches shall becombined to generate the output) regularises the output to probable surface shapes; generatingthe generic patch basis by down-sampling patches from a DEM with much higher grid spacingavoids sampling artefacts. The fused DEM is estimated by minimising the deviations from the
123
two inputs while enforcing that both be represented by the same sparse combination of basispatches. It is well known that sparsity of a linear combination can be achieved by minimisingthe L1-norm of the coefficient vector. Overall, the estimation thus amounts to solving a L1-regularised least-squares system, for which efficient solvers exist. We notice that, in practice,the DEM fusion method of the Weighted Average works almost as well as the Sparse Represen-tation, which confirms the intuition that the correct choice of weights has a far greater influenceon the results than the mathematical recipe for fusing the inputs.
For the described methods weights are required, which govern the relative influence of thetwo input DEMs at a given raster location. These weights are critical for proper DEM fusion.If uncertainties σi are available for every single raster point computing the weights is straight-forward. Unfortunately, this is often not the case. In many cases only a global uncertaintyfor the entire DEM is reported, while local variations of the errors are lost. Simply using thisglobal number everywhere is unsatisfactory, because the physical characteristics of differentsensors, data acquisition conditions, processing methods and scene characteristics (especiallygeomorphology and land-cover) imply strongly varying accuracy. If adequate error measuresare not provided, it is natural to ask whether the missing accuracies can be estimated directlyfrom the DEM. To investigate this question we have analysed the residuals of different DEMsw.r.t. LiDAR ground truth. It turns out that in fact clear correlations exist between height errorsand certain surface properties.
We then experimentally evaluated DEM fusion for a test site in central Switzerland, havebeen performed on DEMs with different resolution and production technology and discussedthe findings of that study.
7.2 Open Issues and Further Improvements
The experiments confirm that significant improvements are possible by fusion of existing DEMs.Most of the gain is due to the reduction of gross errors, of which there are quite many in large-area DEMs. The combination of complementary DEM generation technologies, such as forexample interferometric SAR and optical image matching, has the biggest potential, becausethe different recording and processing principles introduce different blunders, which can be atleast partially remedied with more correct data from the other technique.
Although different fusion algorithms are available, the choice which mathematical toolboxto employ makes little difference. More important is to use the correct influence weights forthe inputs. Fusion requires fine-grained knowledge of the local uncertainty to determine theappropriate fusion weights. If these are not supplied together with a DEM, it is neverthelessfeasible to determine them from local geomorphological characteristics and land-cover.
Models generated with similar technology and/or very different accuracy are less suitablefor fusion, since they tend to contain blunders in the same areas, and the weaker DEM will notbe able to contribute any important information. Furthermore, one needs to be aware that evenin the case of successful fusion, the errors are increased for some points.
Regarding the data-driven estimation of fusion weights, the method requires access to train-ing data consisting of both estimated heights and residuals, thus it can only be done if DEMdata as well as sufficiently accurate ground truth is available, such that the dependency between
124
surface properties and height uncertainties can be estimated for the specific DEM generationtechnology.
Since only data in the overlapping DEM zone are used the possibility of area extension wasnot investigated.
Here, we have only investigated the fusion of two input DEMs. While the described meth-ods can trivially be extended to more inputs, an empirical investigation is still missing. The useof more than two DEMs can also support blunder detection. A further interesting domain isthe fusion of airborne photogrammetry and airborne LiDAR, which could potentially be usefulfor high-accuracy applications. The methods we have investigated are generic, but again theapplication to high-density airborne data remains to be verified experimentally.
So far we have only looked at the statistics of height errors with regard to ground truth, anerror measure that is independent of any particular application. For certain problems, DEMquality may not be primarily the per-point uncertainty, e.g. for hydrological applications it maybe more important to correctly delineate watersheds, even at the cost of higher overall errors.We have not investigated this issue yet, and at least in principle it is quite possible that alteringheights to reduce their individual residuals significantly changes gradient directions and thusrun-off patterns, especially in flat regions.
The accuracy of fusion could improve if blunder detection of the input DEMs would be apre-processing step. Once located, the blunder could be edited within the fusion process.
The sparse representation method involves the choice of a dictionary which is the set ofelementary terrain forms used to decompose the input DEMs. A possible improvement of thesparse representation DEM fusion method would be the use of a better dictionary. An optimaldictionary is constructed from DEMs with different characteristics: i.e. resolution, productiontechnology, multi-temporal, land use, geomorphological characteristics.
Synthetic data with known errors could be used to test further the proposed fusion DEMmethods and check their behaviour towards different types of errors.
At the moment the proposed DEM fusion methods are restricted to regular grids, theirextension to the use of TINs is an important research open issue.
The fusion results could improve by integrating breaklines. Breaklines are linear elementsthat describe changes in smoothness or continuity of the surface and their preservation is mostessential to obtaining a reliable DEM.
When the element of time is added to the fusion of multi-temporal DEMs, the the effects ofchanges to the physical shape of the land surface may be studied. Such is the case with analysisof historical (pre-mining) and recent (post-mining) topographic and other geospatial data sets,including land cover maps derived from remote sensing.
125
126
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