Clustering-based robust three-dimensional phase unwrapping algorithm

Clustering-based robust three-dimensionalphase unwrapping algorithm

Miguel Arevalillo-Herráez,1,* David R. Burton,2 and Michael J. Lalor2

1Department of Computer Science, University of Valencia, Avenida Vicente AndrésEstellés s/n, 46100 Burjassot, Valencia, Spain

2General Engineering Research Institute (GERI), Liverpool John Moores University,James Parsons Building, Room 114, Byrom Street, Liverpool L3 3AF, UK

*Corresponding author: [email protected]

Received 6 July 2009; revised 15 January 2010; accepted 5 February 2010;posted 25 February 2010 (Doc. ID 113685); published 25 March 2010

Relatively recent techniques that produce phase volumes have motivated the study of three-dimensional(3D) unwrapping algorithms that inherently incorporate the third dimension into the process. We pro-pose a novel 3D unwrapping algorithm that can be considered to be a generalization of the minimumspanning tree (MST) approach. The technique combines characteristics of some of the most robust ex-isting methods: it uses a quality map to guide the unwrapping process, a region growing mechanism toprogressively unwrap the signal, and also cut surfaces to avoid error propagation. The approach has beenevaluated in the context of noncontact measurement of dynamic objects, suggesting a better performancethan MST-based approaches. © 2010 Optical Society of America

OCIS codes: 100.5088, 350.5030.

1. Introduction

Phase unwrapping refers to a set of techniques thatare used to reconstruct signals in a large number ofapplication areas, such as optical interferometry,synthetic aperture radar (SAR), and magnetic reso-nance imaging (MRI). In these application areasthere exist certain analysis techniques that producedata in the form of a phase distribution, with valuesdefined in the range ½−π;þπ�. This causes 2π discon-tinuities to appear in the result, making the dataunusable until they are resolved. The removal ofthese discontinuities is known as phase unwrapping.In principle, phase unwrapping seems to be a sim-

ple operation that consists of detecting phase jumpsand adding (or subtracting) the most appropriatemultiple of 2π to each signal value. However thepresence of noise, processing errors, undersampling,and spurious artifacts makes this a cumbersome pro-cess. For this reason, phase unwrapping has been a

major topic of research during the past two decades,to such an extent that a book has been published thatis dedicated solely to this topic [1].

Since in the past most relevant techniques pro-duced two-dimensional (2D) data, most of the workhas focused on the implementation of 2D algorithms.In this context, most existing 2D techniques canbe classified into one of the following four groups:global error minimization algorithms, branch-cutmethods, quality guided techniques, and region-growing approaches.

Global error minimization algorithms attempt tominimize some measure of the difference betweenthe gradients of the wrapped and unwrapped signals.Both the Lp-norm [2] and least squares [3] algo-rithms fall within this category. Although theseare generally robust, they are also computationallyintensive.

Branch-cut methods (e.g., [4]) are based on findingsingularities in the phase map (a set of four pixelslaying on the vertices of a square such that it isnot possible to define a consistent phase map byadding integer multiples of 2π to the wrapped signal

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values). Once these are identified, a series of branch-cut lines joining singularities of different sign areplaced. Then, the original 2D signal is converted intoa one-dimensional (1D) signal, by devising a paththat does not cross the cut lines. Finally the resultingsignal is unwrapped by using a simple threshold-based 1D algorithm, which consists of adding or sub-tracting multiples of 2π to all successive pixels after aphase discontinuity is detected. Although these algo-rithms are usually computationally efficient, theyare not very robust [5].Quality guided techniques [6–8] also convert the

2D signal into a 1D one, according to a quality mea-sure that guides the construction of the path. In anattempt to avoid error propagation, the best pixelsare the first to be processed. In essence, the 2D signalis considered as a graph, with a vertex per value. Ver-tices that are adjacent in either the x or y dimensionsare connected by an edge. A reliability value is thenassigned to every existing edge, and a minimumspanning tree (MST) is built according to edge coststhat are related to the inverse of their reliabilityvalues. Finally, the signal is unwrapped using thesimple threshold-based 1D algorithm that was men-tioned above, applied on the path determined by theMST. The main advantage of these algorithms istheir processing speed, but their robustness dependson the mechanism used to produce the quality map.Region-growing approaches (e.g., [9]) are based on

dividing the image into smaller areas, unwrappingthese areas independently, and then bringing themtogether using a joining strategy. Although these al-gorithms are normally very fast, they usually fail atdetecting large discontinuities in the wraps.Relatively recent techniques that produce phase

volumes have motivated the study of 3D algorithms.This is the case for noncontact measurement of dy-namic objects, multitemporal SAR interferometricmeasurements [10], andMRI [11].Althoughphasevo-lumes can easily be processed as a set of independent2Dmaps, such an approach would not consider usefulinformation contained in the third dimension, whichcan be used to increase the quality of the result. Forthis reason, there is an interest in 3D unwrappingalgorithms that inherently incorporate the third di-mension into the process. In this direction, a numberof techniques have already been proposed, most ofthem extensions of their 2D counterparts.In [11], a quality guided approach has been pro-

posed. As a first step, a noise-level map is builtand a voxel is set as the starting seed. Thereafter,the technique operates in an iterative fashion. Ateach iteration, only the voxels with a noise levelbelow a threshold are unwrapped. The unwrappingof the remaining voxels is deferred to subsequentiterations, during which the threshold value isgradually increased until the volume has been fullyunwrapped.In [5], the technique above is improved by avoiding

the iterative procedure and the selection of a startingseed. In particular, the technique presented in [8] is

adapted to 3D, by using voxels instead of pixels andconsidering the edges in each of the three dimen-sions. In this context, several mechanisms to produce3D quality maps are evaluated, and the approach iscompared to other existing 3D unwrapping methods,showing a higher robustness.

A multidimensional algorithm which uses a best-pair-first region merging approach has also beenpresented in [12]. In particular, the n-dimensionalvolume is divided into a number of regions, each ofwhich contains no phase wraps, in a similar wayas in the 2D algorithm in [13]. These regions are thentreated as single entities, and a cost function is usedto determine which regions are merged first. Thisprocess continues until a single region remains.

Another interesting extension of a 2D algorithm ispresented in [14]. In this case, the concept of a branchcut has been replaced by that of a 3D branch-cut sur-face. In 2D, path independence is achieved by placingbranch-cut lines between singularity loops of oppo-site sign. However, the location of the cut lines fora given wrapped phase map is not unique, and theymay be placed in many different arrangements andorientations. In 3D, phase singularities appear in adisposition such that they form closed loops in space.The authors claim that this makes it possible to de-termine the location of the cuts uniquely, in contrastwith the 2D case for which there are many possibleways in which singularities can be paired up. Otherapproaches that build on this idea have also beenpresented in [15–17].

In this paper, a different 3D phase unwrappingmethod is proposed, by combining characteristic ofseveral algorithms mentioned above. In particular,the approach presented in [18] is extended to 3Dand evaluated in the context of noncontact measure-ment of dynamic objects. As in other quality guidedapproaches, it makes use of a quality map to achievea high immunity to noise. At the same time, it uses aregion growing mechanism to progressively unwrapthe signal. Besides, the signal is dynamically exam-ined to detect inconsistent areas and cut surfaces arestrategically positioned to avoid further error propa-gation. The combination of these features yields arobust technique that can be applied to multidimen-sional signals.

The remainder of the paper is organized as follows.Section 2 presents some basic definitions that arerequired for the rest of the paper. Then, Section 3 in-troduces the classical MST unwrapping algorithm.In Section 4, some disadvantages of the MSTapproach are analyzed and the proposed method isdescribed. In Section 5, the performance of the tech-nique is compared to that of an MST-based imple-mentation. Finally, Section 6 discusses the mostrelevant aspects of the approach and describes somefuture work.

2. Definitions

A graph G ¼ ðV ;EÞ is an abstract representationof a set of objects where some pairs of the objects

1 April 2010 / Vol. 49, No. 10 / APPLIED OPTICS 1781

are connected by links. The interconnected objectsare represented by mathematical abstractions calledvertices vi ∈ V, and the links that connect some pairsof vertices are called edges ðvi; vjÞ ∈ E. In a weightedgraph, a weightwðvi; vjÞ is also assigned to each edge.When edges have no orientation, the graph is calledundirected.A tree is a connected graph without cycles, and a

spanning tree for an undirected graphG is defined asa subgraph that is a tree and connects all verticestogether. By considering the weight of a spanningtree as the sum of the weights of all edges in thatspanning tree, a minimum spanning tree (MST) orminimum weight spanning tree is then a spanningtree with weights less than or equal to other span-ning trees in the same graph.One way to compute the MST for a graph G ¼

ðV;EÞ composed of n vertices vi ∈ V and edgesðvi; vjÞ ∈ E is by using Kruskal’s algorithm, pre-sented in Algorithm 1.

3. Implementation of an MST-based PhaseUnwrapping Algorithm

A phase unwrapping path following an MST usuallyavoids going through dipole structures [19] and thusyields reliable phase unwrapping results. A simpleMST approach to phase unwrapping can be imple-mented by considering a graphG ¼ ðV ;EÞwith a ver-tex vi ∈ V for each signal value, an edge ðvi; vjÞ ∈ Efor each pair of neighboring vertices, and a weightwððvi; vjÞÞ for each edge ðvi; vjÞ ∈ E (a measure ofthe noise level of the edge).

Algorithm 1: T← Kruskal (G)

Input:G: The GraphOutput:T: The Minimum Spanning Treebegin

Create a sorted list L with all edges in G ordered by costwðvi; vjÞ in increasing order)

Assign each vertex vi in G to a different elementary clusterCðviÞ

Define a tree T←Ø (Twill ultimately contain the edges of theMST)

while T contains fewer than n − 1 edges doðu; vÞ←L:getEdgeðÞif CðvÞ ≠ CðuÞ then

Add edge ðu; vÞ to TMerge CðuÞ and CðvÞ into a single cluster

endend

end

This is shown in Algorithm 2, where the localunwrapping of CðuÞwith respect to CðvÞ is performedby adding the following integer multiple of 2π to allvertices in CðuÞ:

2π⌊v:value − u:value2π þ 0:5⌋: ð1Þ

Indeed, the selection of the most appropriate noiselevel estimator is a major factor that affects the

performance of an MST based phase unwrappingalgorithm. To this end, several methods have beenproposed in the literature (see [5] for a brief descrip-tion of the most relevant ones).

4. The Algorithm

The formulation of the MST-based phase unwrap-ping method presented in Algorithm 2 allows oneto easily observe three major weaknesses of thisapproach:

• The first one is that clusters may be forced to-gether due to certain single elements being close toeach other, even thoughmany of the elements in eachcluster may be very distant from each other. This isknown as the chaining phenomenon, and it is causedby the use of the single-linkage approach.

• The second weakness is that, when two clu-sters are locally unwrapped with respect to eachother, a single signal value from each cluster isconsidered.

• The third issue is related to the determinationof the unwrapping path. In an MST-based approach,this is usually decided before the unwrapping opera-tion starts, and other important information thatmay arise during the process is disregarded.

Algorithm 2: unwrap (G)

Input:G: The Graphbegin

Create a sorted list Lwith all edges inG sorted by noise level(in increasing order)


while L:sizeðÞ > 0 doðu; vÞ←L:getEdgeðÞif CðvÞ ≠ CðuÞ then

Locally unwrap CðuÞ with respect to CðvÞMerge CðuÞ and CðvÞ into a single cluster

endend

end

By using a more dynamic approach, some informa-tion discovered during the unwrapping process canbe employed to detect inconsistencies and improvethe robustness of themethod. To this end, several im-provements have been made to Algorithm 2:

1. The selection of themost reliable nonprocessededge has been replaced by a new function thatobtains the closest clusters, not necessarily basedon a single edge. The determination of the two closestregions can be determined, e.g., according to thereliability of all signal values within the regions,or restricted to the value pairs that define theirboundary.

2. The local unwrapping function has been mod-ified so that the constant added to the values in one ofthe groups considers all pairs of vertices that definethe border between the two regions.


3. Cluster similarities are recalculated each timethat a merging operation takes place, and assesseddynamically. Additionally, a new mechanism hasbeen added to impose cut surfaces on the unwrap-ping path as soon as an inconsistency is detected.

The resulting algorithm is shown in Algorithm 3.Dynamic recalculation of cluster similarities is a

computationally intensive operation. At any parti-cular instant in time, the current state can be re-presented in the form of a graph (each clusterrepresents a vertex and borders are representedas edges). When two clusters C1 and C2 are mergedinto a new cluster CNew, a search for other clustersis required that simultaneously maintains edgeswith both C1 and C2. For each such cluster, the pre-vious edges are removed, and a new single edgewith CNew is created, after computing the newreliability value for it. Besides, a check for consis-tency is performed and an edge is banned if, afterthe merging operation, all pairs of vertices that de-fine the border between the two regions do notagree on the constant required to unwrap the clus-ter with respect to CNew. Banned edges are consid-ered only when no other edges remain. This is, infact, equivalent to establishing a branch cut thatthe unwrapping path cannot cross, in a similarway as in [4,14]. All other borders are simply redir-ected from C1 and C2 to CNew, and the reliabilityvalues are maintained.

Algorithm 3: unwrap (G)

Input:G: The Graphbegin


while more than one cluster remains doðu; vÞ←L:getEdgeðÞðC1;C2Þ←getClosestClustersðÞLocally unwrap C1 with respect to C2

Merge C1 and C2 into a single clusterUpdate similarity measures between clusters and ban

inconsistent edgesend

end

As in the case of the MST-based approach, the ro-bustness and performance of the algorithm dependson the definition of a distance measure. In this MSTmethod, this was used as a noise level estimator tocalculate the reliability of the edges. In the case ofthe approach presented in this paper, the distancemeasure is used to determine the closest clusters.Although this dependence provides the method witha higher flexibility and allows one to tune it tospecific types of signals, it also converts it into aparametric technique.In our particular implementation, the reliability

of each vertex has been calculated according to thefollowing expression:

SDi;j;k¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiH2ði;j;kÞþD2ði;j;kÞþN2ði;j;kÞþ

X10n¼1

D2nði;j;kÞ

vuut ;

ð2Þ

whereHði; j; kÞ ¼ ωðφði − 1; j; kÞ − φði; j; kÞÞ

− ωðφði; j; kÞ − φðiþ 1; j; kÞÞ; ð3Þ

Vði; j; kÞ ¼ ωðφði; j − 1; kÞ − φði; j; kÞÞ− ωðφði; j; kÞ − φði; jþ 1; kÞÞ; ð4Þ

Nði; j; kÞ ¼ ωðφði; j; k − 1Þ − φði; j; kÞÞ− ωðφði; j; kÞ − φði; j; kþ 1ÞÞ; ð5Þ

D1ði; j; kÞ ¼ ωðφði − 1; j − 1; kÞ − φði; j; kÞÞ− ωðφði; j; kÞ − φðiþ 1; jþ 1; kÞÞ; ð6Þ

D2ði; j; kÞ ¼ ωðφðiþ 1; j − 1; kÞ − φði; j; kÞÞ− ωðφði; j; kÞ − φði − 1; jþ 1; kÞÞ; ð7Þ

D3ði; j; kÞ ¼ ωðφði − 1; j − 1; k − 1Þ − φði; j; kÞÞ− ωðφði; j; kÞ − φðiþ 1; jþ 1; kþ 1ÞÞ; ð8Þ

D4ði; j; kÞ ¼ ωðφði; j − 1; k − 1Þ − φði; j; kÞÞ− ωðφði; j; kÞ − φði; jþ 1; kþ 1ÞÞ; ð9Þ

D5ði; j; kÞ ¼ ωðφðiþ 1; j − 1; k − 1Þ − φði; j; kÞÞ− ωðφði; j; kÞ − φði − 1; jþ 1; kþ 1ÞÞ; ð10Þ

D6ði; j; kÞ ¼ ωðφði − 1; j; k − 1Þ − φði; j; kÞÞ− ωðφði; j; kÞ − φðiþ 1; j; kþ 1ÞÞ; ð11Þ

D7ði; j; kÞ ¼ ωðφði − 1; jþ 1; k − 1Þ − φði; j; kÞÞ− ωðφði; j; kÞ − φðiþ 1; j − 1; kþ 1ÞÞ; ð12Þ

D8ði; j; kÞ ¼ ωðφðiþ 1; j; k − 1Þ − φði; j; kÞÞ− ωðφði; j; kÞ − φði − 1; j; kþ 1ÞÞ; ð13Þ

D9ði; j; kÞ ¼ ωðφði; jþ 1; k − 1Þ − φði; j; kÞÞ− ωðφði; j; kÞ − φði; j − 1; kþ 1ÞÞ; ð14Þ


D10ði; j; kÞ ¼ ωðφðiþ 1; jþ 1; k − 1Þ − φði; j; kÞÞ− ωðφði; j; kÞ − φði − 1; j − 1; kþ 1ÞÞ:

ð15Þ

Then, the reliability of an edge between any twoneighboring vertices u and v has been defined asthe inverse of SDðuÞ2 þ SDðvÞ2. To compute the relia-bility of a border between two regions, the averagereliability of the value pairs that define the boundarybetween them has been used.Another important issue is the high computational

cost associated with the current implementation ofthe algorithm. Some applications that use unwrap-ping algorithms need execution speeds that matchtypical video capturing rates. At the present time,

our algorithm would be unable to meet such real-time requirements and would be more suitable forpractical applications that require offline processing.

5. Evaluation

A. Experimental Setting

For evaluation purposes, we have tested theproposed algorithm by using simulated and realwrapped phase volumes, and have compared the re-sults to those obtained with two representative stateof the art unwrapping algorithms:

• The latest best-path 3D unwrapping algo-rithm, as presented in [5]. To allow for a fair compar-

Fig. 1. (Color online) First 3D phase map contains a noisy regionlocated at its center. A transversal cut of the signal is shown forillustrative purposes.

Fig. 2. Results of unwrapping the phase map in Fig. 1 using (a) asimple flood fill algorithm; (b) best path; (c) noise immune; and(d) the proposed approach.

Fig. 3. 3D plot of synthetic surface used in the second experiment.

Fig. 4. Results of unwrapping the phase map generated by simu-lating fringe projection on the growing surface shown in Fig. 3by using (a) a simple flood fill algorithm, (b) best path, (c) noiseimmune, and (d) the proposed approach.


ison, the same distance measure that is used in ourproposed algorithm has also been used here.• Huntley’s noise-immune method [14].

Furthermore, to evidence the effects of noise andother effects in all phase maps, the results obtainedwhen applying a flood fill algorithm are also given.To show the results, a typical range image repre-

sentation has been employed, using gray levels to re-present phase values, darkening as the phase valuesdecrease.

B. Noisy Areas

A first simple experiment aims to illustrate to whatdegree the algorithm is able to isolate noisy regionswithin a phase map. For this purpose, a synthetic 3Dphase distribution has been generated. In this map,the phase grows slowly along the x, y, and z axes andcontains a box of random noise located at the centerof the signal (Fig. 1).

This signal has been unwrapped by using the tech-nique presented in this paper and the other techni-ques mentioned above. The results for the middleframe are shown in Fig. 2. Except for the simple floodfill algorithm, the other three algorithms evaluatedare sufficiently robust to isolate the cube of noise in-troduced at the center of signal.

C. Genuine Phase Wraps

A second experiment has the objective to examinethe capability of the algorithms to respond to genuinephase wraps in the 3D map. To this end, we have cre-ated a synthetic phase distribution composed of 32frames of size 128 × 128. Each of these frames corre-spond to the measurement of the shape shown inFig. 3, in which the width of the ring shape placedat the center of the image grows with time. Figure 4

Fig. 5. Middle frame of simulated growing sphere.

Fig. 6. Results of unwrapping the phase map in Fig. 5 using different number of frames, for the different algorithms being compared.

Fig. 7. Frame 13th in the wrapped phase volume.


shows the results for the middle frame for such aphase volume.In this case, only the best pathmethod and the pro-

posed method obtain satisfactory results, unwrap-ping the two surfaces correctly.

D. Response to Low Resolution and Undersampling

The following experiment aims to compare the re-sponse of the three algorithms when the resolutionis decreased.For a first test, a computer generated phase volume

was used. In particular, 50 frames of size 128 × 128that correspond to a single growing sphere whose ra-dius increases with time was simulated. Then, Gaus-siannoisewasadded, andunwrappingwasperformedat three different resolutions, first using all 50 framesin the phase volume, second using every other frame(25 frames in total) and thirdusing every fourth frame(13 frames in total). Figure 5 shows the middle framefor such a phase volume, and the results for this sameframe after unwrapping are shown in Fig. 6.As a second experiment to test the response of the

algorithm to low resolution and undersampling, avideo sequence has been built by projecting a fringepattern on a synthetic human head and torso that is

used in radiotherapy calibration (undergoing manu-ally induced pseudorespiratorymotion), capturing 25frames of size 512 × 512. These were analyzed usingFourier fringe analysis, producing a 3D phase signal.This phase volume was downsampled to different re-solutions and unwrapped with the three algorithmsthat were being compared. The middle frame of thewrapped signal is shown in Fig. 7, and some illustra-tive results after phase unwrapping are shownin Fig. 8.

E. Execution Times and Memory Requirements

Despite the higher robustness obtained by the pro-posed algorithm in the cases tested, it also presentsslightly larger memory requirements and signifi-cantly higher execution times. In some sense, the al-gorithm belongs to a different class of techniques,more suitable for offline processing. Table 1 showsexecution times for all the experiments above, mea-sured on a PC with 4 Gbytes of memory and an IntelDual Core 6300 processor running at 1:86GHz. It canbe observed that these are several orders of magni-tude higher in the case of the proposed algorithm.Table 2 depicts the memory consumed in each case.In this context, memory requirements are about 15%

Fig. 8. Results of unwrapping the phase map in Fig. 7 for different resolutions using the algorithms being compared.

Table 1. Execution Times (in Seconds) for Experiments

Figure Size Floodfill Best Path Noise Immune Proposed

Fig. 2 128 × 128 × 64 0.250 4.984 2.484 10064Fig. 4 128 × 128 × 32 0.109 2.437 1.250 5006Fig. 6 top 128 × 128 × 50 0.188 5.078 2.203 21388Fig. 6 middle 128 × 128 × 25 0.078 2.281 1.093 8442Fig. 6 bottom 128 × 128 × 13 0.047 1.031 0.547 1549Fig. 8 top 128 × 128 × 25 0.079 2.266 0.969 7150Fig. 8 middle 128 × 128 × 13 0.047 1.031 0.500 2380Fig. 8 bottom 128 × 128 × 7 0.032 0.469 0.266 867


higher for the proposed algorithm than for the bestpath technique and 65% higher than for Huntley’snoise immune method.

6. Discussion and Conclusions

In this paper we have presented a novel algorithm forunwrapping 3D phase signals, by combining someof the characteristics of the most popular existingtechniques. As with quality guided algorithms, ituses a quality measure to guide the unwrapping pro-cess. However, a region growing approach has alsobeen adopted, with several regions growing simulta-neously and merging to form larger regions. Thismakes it possible to replace quality evaluationsbased on a single edge by a more reliable approachthat considers all voxels in the boundary. Addition-ally, dynamically built cut surfaces have been usedto avoid further growth of the regions in directionswhere inconsistencies occur.The technique has been tested in both simulated

and real phase volumes, and the results have beenshown to outperform the classical best path methodbased on the construction of a MST [5] and the noiseimmune algorithm presented in [14].As is the case with any quality-guided technique,

the algorithm is sensitive to the quality measureused. In general, this can be defined in terms of all sig-nal values contained within the regions, or restrictedto the value pairs that define the boundary betweenthem. If this measure is fixed to the minimum dis-tance between all valued pairs that define the bound-ary, theMST-based algorithm is obtained. This provesthat the technique presented in thiswork is a general-ization of theMST,which endows the approachwith ahigher flexibility. Determining the most suitablemeasure for each particular type of phase volume isan issue that is still under investigation.Finally, the application of classical clustering the-

ory and algorithms to this particular case may yieldfaster implementations and allow the use of the al-gorithm in real-time applications, but this is some-thing that has not yet been achieved.

We thank Hussein S. Abdul-Rahman and MuntherA. Gdeisat for their useful advice and providing theimages used for the evaluation of the unwrappingtechnique presented in this paper. This work hasbeen funded by the Ministry of Education and Eur-opean Regional Development Fund (ERDF), through

project Consolider Ingenio 2010 CSD2007-00018 andby Engineering and Physical Sciences ResearchCouncil (EPSRC) through grants “MEGURATH”

EP/D0077702/1 and “ECSON” EP/F013698/1.

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Table 2. Memory Consumption (in Megabytes) for Experiments

Figure Size Floodfill Best Path Noise Immune Proposed

Fig. 2 128 × 128 × 64 4.2 298.1 202.7 335.4Fig. 4 128 × 128 × 32 2.1 148.1 100.6 166.6Fig. 6 top 128 × 128 × 50 3.3 232.4 158.0 261.6Fig. 6 middle 128 × 128 × 25 1.6 115.2 78.3 129.7Fig. 6 bottom 128 × 128 × 13 0.9 59.0 40.0 66.4Fig. 8 top 128 × 128 × 25 1.6 115.2 78.3 129.7Fig. 8 middle 128 × 128 × 13 0.9 59.0 40.0 66.4Fig. 8 bottom 128 × 128 × 7 0.5 30.9 20.9 34.8


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Clustering-based robust three-dimensional phase unwrapping algorithm