Research Article A Clustering-Based Automatic Transfer ...downloads.hindawi.com/journals/mpe/2016/4547138.pdf · cation and visualization process is automatic and e cient. Experiments

Research ArticleA Clustering-Based Automatic Transfer Function Design forVolume Visualization

Tianjin Zhang12 Zongrui Yi1 Jinta Zheng23 Dong C Liu1 Wai-Mai Pang4

Qiong Wang2 and Jing Qin5

1College of Computer Science Sichuan University Chengdu 610065 China2Guangdong Provincial Key Laboratory of Computer Vision and Virtual Reality TechnologyShenzhen Institutes of Advanced Technology Chinese Academy of Sciences Shenzhen 518055 China3Department of Computer Science and Engineering The Chinese University of Hong Kong Sha Tin Hong Kong4Department of Computer Science Caritas Institute of Higher Education Tseung Kwan O Hong Kong5Centre for Smart Health School of Nursing Hong Kong Polytechnic University Hung Hom Hong Kong

Correspondence should be addressed to Qiong Wang wangqiongsiataccn

Received 12 April 2016 Revised 6 September 2016 Accepted 26 September 2016

Academic Editor Alberto Borboni

Copyright copy 2016 Tianjin Zhang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The two-dimensional transfer functions (TFs) designed based on intensity-gradient magnitude (IGM) histogram are effectivetools for the visualization and exploration of 3D volume data However traditional design methods usually depend on multipletimes of trial-and-error We propose a novel method for the automatic generation of transfer functions by performing the affinitypropagation (AP) clustering algorithm on the IGM histogram Compared with previous clustering algorithms that were employedin volume visualization the AP clustering algorithm has much faster convergence speed and can achieve more accurate clusteringresults In order to obtain meaningful clustering results we introduce two similarity measurements IGM similarity and spatialsimilarity These two similarity measurements can effectively bring the voxels of the same tissue together and differentiate thevoxels of different tissues so that the generated TFs can assign different optical properties to different tissues Before performingthe clustering algorithm on the IGM histogram we propose to remove noisy voxels based on the spatial information of voxels Ourmethod does not require users to input the number of clusters and the classification and visualization process is automatic andefficient Experiments on various datasets demonstrate the effectiveness of the proposed method

1 Introduction

Direct volume rendering [1] has been widely used in manyfields particularly for the visualization of medical data witha variety of modalities such as CT MRI and ultrasoundIt projects three-dimensional (3D) volumetric data to atwo-dimensional (2D) screen to facilitate observation andexploration By using appropriate transfer functions (TFs)which map the voxel properties (eg gray scale and gradientmagnitude) to optical properties (eg transparency andcolor) the major structures of volume data can be revealedDesigning effective TFs is a must for useful visualizationof volumetric medical data especially for clinical diagnosisand treatment However it remains a challenging task forradiologists and physicians as it usually requires them to

acquire technical knowledge on rendering techniques Unfor-tunately doctors do not have enough time to acquire therelated knowledge and skills In addition the complicatedinteractions in traditional direct volume rendering systemsprohibit their application in clinical practice In this regarddeveloping automatic methods for TFs generation is impor-tant for medical data visualization

TFs typically operate on the value domain of inputvolume and its derived feature domains (such as gradient)to separate different structures To assist usersrsquo explorationthe feature domains are always decomposed into severalmeaningful clusters and users can explore the data based onthese clusters The voxelsrsquo scale values and their gradientsare the most commonly used feature domains Kindlmannand Durkin [2] projected the scale values and gradients into

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 4547138 13 pageshttpdxdoiorg10115520164547138

2 Mathematical Problems in Engineering

a 2D space to make up the intensity-gradient magnitude(IGM) histogram and then designed TFs based on it by usingvarious manipulation widgets (such as rectangles trianglesand ellipses) Later IGM histogram was often used in directvolume rendering and has been implemented in several pop-ular visualization packages such as VisIt [3] Voreen [4] andImageVis3D [5] However most implementations involveda lot of manual operations to achieve satisfactory resultswhich made them not applicable for doctors without specialknowledge of IGM histogram In recent years Wang et al [6]proposed a volume exploration with the Gaussian mixturemodel and generated a set of suggestive elliptical transferfunctions semiautomatically Nonetheless this method is noteffective when the dataset is complicated as the widgetsare too regular to describe multidimensional features withcomplex shapes Ip et al [7] presented a semiautomaticapproach to detect embedded features and spatial structuresby visually segmenting the IGM histogram of a volumetricdataset However this method required too many operationsto annul undesired structures and achieve satisfactory resultsand hence was still not suitable for radiologists and physi-cians

To overcome these shortcomings we propose a novelautomatic TFs design method based on IGM histogram Ourmethod is based on the observation that voxels belonging tothe same structure usually have similar intensity and gradientmagnitude and are located together in the volume In thiscase we employ affinity propagation clustering algorithmto classify the scattering points in the IGM histogram intoclusters by defining two novel similarity measurements IGMsimilarity and spatial similarity These two similarity mea-surements can effectively bring the voxels of the same tissuetogether and differentiate the voxels of different tissues so thatthe generated TFs can assign different optical properties todifferent tissues making the visualization results unambigu-ous and easy to be used in diagnosis and treatment Note thatas we do not need to assign a cluster number to the affinitypropagation clustering algorithm in advance the clusteringprocess is fully automatic In addition before performingclustering we eliminate noisy voxels by leveraging the spatialinformation of the input volume data We also provide asimple interaction interface to allow users further polish theclustering results The TFs are automatically generated basedon the clustering results Experiments on various datasetsdemonstrate the effectiveness of the proposed method

The organization of this paper is as follows We presentsrelated work in Section 2 Section 3 describes our method indetail Section 4 provides experimental results Conclusionsare drawn in Section 6

2 Related Work

Although direct volume rendering technique has been widelyused transfer function design remains a challenging taskThe design of TFs is often classified into two categoriesimage-centric and data-centric [8] depending on whetherthey derive their parameters from the resulting images ororiginal data

Image-centric methods typically adjusted TFs by search-ing for the optimal rendered image in the rendering spaceThus the TFrsquos generation is considered as a process of param-eter optimization In early investigations genetic algorithmswere employed to select the most satisfactory renderingresult [9] However depth information cannot be sufficientlyconsidered in these algorithms To enhance the depth infor-mation of rendering images Chan et al [10] proposed a novelmetric based on psychological principles and Zheng et al[11] proposed an effective design by optimizing an energyfunction based on quantitative perception models Althoughimage-centric methods are intuitive and easy to implementthe final rendering results generated by thesemethods heavilydepend on the initial result In addition a lot of parametersneed to be tuned in these methods which is unacceptable formedical data visualization

In contrast with image-centric methods data-centric TFsdefine visual properties based on the information of voxels Inearly studies TFs were designed based only on the histogramof voxelsrsquo scalar values (these TFs are usually called 1D TFs)However 1D TFs have inherent difficulties in identifyingdifferent materials with similar intensities Then 2D TFsbased on intensity values and gradient magnitudes wereproposed which were more effective in detecting multiplematerials and their boundaries [2] Besides using derivativeproperties of scalar value other effective metrics were alsoincorporated into the feature space such as curvature [12]feature size [13] and ambient occlusion [14] In order to high-light important structures of volume some methods basedon visibility were introduced [15] and information divergencebetween visibility distribution and target distribution wasoptimized to generate automatic TFs [16ndash18] Recently Caiet al have proposed a rule-based transfer function based onthe local frequency distribution of a voxels neighborhood[19] The structures of userrsquos interest in a volume datasetcan be characterized by using the various feature spaceHowever higher dimensions of the feature space add morecomplication in transfer function design

Recently several semiautomatic transfer function designshave been proposed to reduce the number of degrees offreedom in transfer function design Tzeng andMa proposeda method based on material classes extracted from thecluster space using the ISODATA technique to generate TFs[20] Some other machine learning based methods have alsobeen employed to analyze the feature spaces For exampleMacIejewski et al proposed a nonparametric kernel densityestimation method onto the IGM feature space in order toextract feature patterns [21] FurthermoreWang et al appliedthe Morse theory to automatically decompose the IGMfeature space into a set of valley cells [22] In order to displaythe surface information of the volume data better Sereda et alproposed LH histogram which uses the lower FL and higherFH intensity values of the twomaterials that form a boundaryas the two axes to design the transfer function [23] Basedon LH histogram Nguyen et al proposed to oversegment LHhistogram usingmean shift clustering and then group similarvoxels by using hierarchical clustering [24] Roettger et alincorporated spatial information into the IGMhistogram and

Mathematical Problems in Engineering 3

Refined IGM histogram

Similarity

Similarityin IGM

Similarityin volume

AP clustering

TF design

Gradient magnitude

IGM histogram

IGMvalue

Spatialinformation

of bin

Threshold

Volume dataset

Visualization

Figure 1 The workflow of the proposed method

performed clustering based on the barycenter and varianceof each bin [25] Ip et al introduced a hierarchical volumeexploration scheme based on normalized-cut algorithm [7]

The affinity propagation (AP) clustering algorithm is aneffective clustering algorithm proposed by Frey and Dueck[26] It clusters based on similarity of data and has norequirement for the symmetry of the similaritymatrix of datagiving it a relatively wide application AP algorithm regardsall data points as potential cluster centers simultaneouslyand produces high-quality cluster centers by using messagetransfer between the data points Compared with 119870-meansclustering algorithms [27] AP algorithms do not need toprespecify the number of clusters119870 Although the hierarchalclustering algorithm [28] also does not require the numberof clusters its classification results [23 24] are sensitive tothe initial clustering result while AP algorithm clusteringresolves this issue by improving the reliability of clusteringresult As mentioned earlier the normalized cut algorithm[29] was also employed in volume visualization [7 17] whichovercomes the disadvantages of the 119870-means algorithm forits ability to identify nonconvex distribution clustering but itrequires the solution of matrix eigenvalues and eigenvectorswhich is quite computation-intensive and time-consumingIn contrast convergence speed of AP algorithm is muchfaster and its clustering results are more accurate [26]

In this paper we propose an automatic method to classifythe volume data not only considering the intensity value andgradient magnitude but also taking the spatial informationof voxels into consideration Then the TFs are automaticallygenerated based on the clustering results

3 Method

Figure 1 shows the workflow of the proposed method whichis composed of three steps First we calculate the gradient ateach voxel Together with the intensity the IGM histogramof the volume is constructed We call a set of voxels withthe same intensity and gradient value a bin We computethe mean and variance of position of the voxels withineach bin The bin with a larger variance of position than aspecified threshold is eliminated from the histogram as it

can be considered as noise or nonsignificant tissues locatedat the boundaries of the volume Second we cluster theremainder bins in the histogram using affinity propagationalgorithm according to both the intensity and gradient infor-mation and the spatial information of voxels In additionwe provide a simple interaction interface to allow usersto further polish the clustering results in order to achievemore desired visualization Finally the transfer functions areautomatically generated based on the clustering results andthe visualization of the volume can be obtained from thetransfer function

31 IGM Histogram The IGM histogram is a useful toolfor exploring the volume dataset [2 7 25] It is generallyrepresented as a 2D scatter plot To figure out the IGMhistogram from a given 3D intensity field we first computethe gradient magnitude at each voxel Then the intensity andgradient magnitude is set as the horizontal and vertical axisof the scatter plot Each point in the scatter plot representsa bin of voxels with the same intensity and gradient valueThe brightness value of each point is associated with thenumber of the corresponding voxels To avoid significantchanges of the magnitude of the brightness we set it as alogarithmic function of the number of voxels If all bins arefed into the following clustering algorithm the generatedtransfer function obtained from the clustering results aswell as the final visualization may be greatly affected by agreat deal of noise In this regard we should try our bestto maintain the bins of important tissues while eliminatingnonsignificant and noisy bins as much as possible to facilitatethe histogram clustering It is observed that the spatiallocation of important tissues and organs are concentratedBased on this observation we employ a simple and efficientscheme to refine the IGM histogram considering the spatialinformation of voxels We first calculate the mean position ofvoxels 120583119904 and the variance of voxel positions ]119904 by

120583119904 = 1119899 sumVisin119880119901 (V) (1)

]119904 = 1119899 sumVisin1198801003817100381710038171003817119901 (V) minus 1205831199041003817100381710038171003817 (2)


(a) (b)

(c)

Figure 2 The intensity-gradient histogram of abdomen dataset with different threshold values (a) the threshold is 0 (b) threshold is 025(c) the threshold is 041

where 119899 is the number of voxels in the bin 119880 is the voxelset and 119901(V) is the position of voxel V If ]119904 is larger thana threshold value we remove the bin from the histogramFigure 2 shows the intensity-gradient histogram of abdomendataset (see Figure 6) with different threshold values wheregray means the removed bins based on the prespecifiedthreshold while the red means the left bins for clusteringMore results can be found in Section 4

32 Clustering of IGM Histogram

321 Basics of Affinity Propagation Clustering For the com-pleteness of this paper we introduce the basics of the affinitypropagation (AP) clustering algorithm here and readers canrefer to [26] formore details AP clustering algorithm initiallyregards all the data points as potential cluster centers Mes-sages between data points then begin to transmit iteratively tomaximize a fitness function until an optimal set of exemplarsand relative clusters emerge The similarity between datapoint 119894 and data point 119896 notated as 119904119894119896 is generally calculatedby negative Euclidean distance between them The value ofsimilarity increases when the distance between the two pointsdecreases The elements 119904119896119896 on the diagonal of the similaritymatrix 119878 = [119904119894119896]119899times119899 are defined as ldquopreferencerdquo Initially thevalue of 119904119896119896 is set as the median of the similarities betweendata point 119896 and all other data points

The most important messages in AP algorithms areresponsibility and availability The responsibility 119903119894119896 is themessage sent from data point 119894 to candidate cluster center 119896

which reflects whether 119896 is suitable to be the cluster centerof data point 119894 taking into account other potential exemplarsfor 119894 The availability 119886119894119896 is the message sent from candidatecluster center 119896 to data point 119894 which reflects whether datapoint 119894 should choose 119896 as its cluster center The alternatingrenewal process of these two kinds of messages is the core ofthe AP algorithm

The initial values of responsibility and availability are setto be 0 For data point 119894 the responsibility and the availabilityare updated in an iterative way as

119903119894119896 larr997888 119904119894119896 minusmax1198961015840 =119896

(1198861198941198961015840 + 1199041198941198961015840) 119886119894119896 larr997888 min

0 119903119896119896 + sum1198941015840notin119894119896

max 0 1199031198941015840119896 (3)

The self-availability is updated in a slightly different way as

119886119896119896 larr997888 sum1198941015840 =119896

max 0 1199031198941015840119896 (4)

Upon convergence the exemplar 119896 for each data point 119894 willbe chosen by maximizing the following criterion

argmax119896

119886119894119896 + 119903119894119896 (5)

Finally we obtained a set of exemplars and correspondingclusters The similarity matrix plays an important role in AP


algorithm as the calculation of both the responsibility andavailability depends on the similarity matrix Traditionallythe similarity matrix in AP algorithm is calculated based onthe negative value of the Euclidean distance between twodata points However it is not sufficient for our applicationwhere the data points are bins on the IGM histogramand their Euclidean distance only contain the intensity andgradient magnitude information As mentioned the spatialinformation of voxels are also quite important to achievedesired visualization results To this end we design ourown similarity measurement which integrates both intensity-gradient information and spatial information of voxels

322 Similarity Measurement The similarity measurementemployed in our application should be designed to bring thevoxels of the same tissues together and differentiate the voxelsof different tissues so that the generated transfer function canassign different opacity and color to different tissues makingthe visualization results unambiguous and easy to be used indiagnosis As mentioned statistically the voxels of the sametissue have similar intensity and gradient magnitude and arelocated closely within the volume However most previousworks only use the information of IGMhistogram and ignorethe location information of voxels which may influence thevisualization result In order to take full advantage of bothIGM and spatial information we integrate IGM similaritymeasurement and spatial similaritymeasurement into the APclustering algorithm to achieve a more effective clusteringresults for more appealing visualization of the volumetricdata

IGMSimilarityUsually close bins in the IGMhistogramhavesimilar intensity and gradient magnitude In this regard weemploy Euclidean distance to measure the IGM similarity oftwo bins 119887119909 and 119887119910

1199041015840IGM (119887119909 119887119910) = 2radic10038161003816100381610038161003816119887119909119894 minus 119887119910119894100381610038161003816100381610038162 + 10038161003816100381610038161003816119887119909119892 minus 119887119910119892100381610038161003816100381610038162 (6)

where 119887119909119894 and 119887119910119894 are the intensity magnitude of the two binsand 119887119909119892 and 119887119910119892 are the gradient magnitude of the two binsWe normalize 1199041015840IGM and obtain the IGM similarity

119904IGM (119887119909 119887119910) = 1199041015840IGM minus 119904min119904max minus 119904min (7)

where 119904max and 119904min are themaximum andminimum value of1199041015840IGMSpatial Similarity The second similarity measurement isdesigned to group bins that are neighbored in the volumeWe call it spatial similarity We evaluate the spatial similaritybetween two bins using the number of direct neighborhoodrelations between the two bins we employed the methodintroduced in [23] to calculate the number Given two bins 119887119909and 119887119910 the number of direct neighborhood relations between119887119909 and 119887119910 can be calculated by

NR (119887119909 119887119910) = sumV119909isin119887119909

sumV119910isin119887119910

119873(V119909 V119910) 119887119909 = 119887119910 (8)

where V119909 represents voxels in 119887119909 V119910 represents voxels in 119887119910and119873(V119909 V119910) is a Boolean function and we set it as 1 if voxelV119910 is a neighbor of voxel V119909 and 0 otherwise Note that wedefine NR(119887119909 119887119910) = 0

The total number of neighborhood relations of bin 119887119909 isNR (119887119909) = sum

119887119894

NR (119887119909 119887119894) (9)

where 119887119894 represents all other bins in the IGM histogramThenthe spatial similarity between two bins (119904vol) is computed bytaking themaximumof the normalized sumof neighborhoodrelations

119904vol (119887119909 119887119910)

= maxNR (119887119909 119887119910)

NR (119887119909) NR (119887119910 119887119909)NR (119887119910) NR (119887119894) = 0

0 NR (119887119894) = 0(10)

where 119894 isin 119909 119910 and NR(119887119894) = 0 means that there no anyneighborhood voxels of 119887119894

By integrating the IGM similarity and the spatial similar-ity the similaritymeasurement employed in theAP clusteringalgorithm can be defined as

119904 (119887119909 119887119910) = minus1198961119904IGM (119887119909 119887119910) + 1198962119904vol (119887119909 119887119910) (11)

where the constant 1198961 and constant 1198962 are the weights of thetwo similarity measurements We set 1198961 = 065 and 1198962 = 035in our experiments

323 Affinity Propagation Clustering on IGM HistogramOnce the similaritymeasurement is defined we can figure outthe similarity matrix using (11) and then feed it into the APclustering algorithm In order to avoid numerical oscillationswe introduce a damping parameter 120582 into (3)ndash(4) and theseequations can be reformulated as

119903119905+1119894119896 larr997888 (1 minus 120582) (119904119894119896 minusmax1198961015840 =119896

119886119905119894119896 + 1199041198941198961015840) + 120582119903119905119894119896119894 = 119896

(12)

119886119905+1119894119896 larr997888 120582119886119905119894119896+ (1 minus 120582)(min

0 119903119905119896119896 + sum1198941015840notin119894119896

max 0 1199031198941015840119896) 119894 = 119896

(13)

119886119905+1119896119896 larr997888 (1 minus 120582)( sum1198941015840notin119894119896

max 0 119903119905+11198941015840119896 ) + 120582119886119905119894119896 (14)

where the superscripts 119905 and 119905 + 1 represent the number ofiterations The damping parameter 120582 is between 0 and 1and we set it as a default value 05 in our experiments Themessage-passing procedure may be terminated after changesin the messages fall below a threshold


The whole AP clustering can be summarized as follows

(1) Calculate the gradient magnitude of each voxel(2) Calculate the variance ]119904 of each bin(3) Remove the noisy bins according to the threshold(4) Calculate the IGM similarity and spatial similarity

and obtain the similarity matrix according to (11)(5) The availability and the responsibility matrixes are

initialized as 0(6) Do the following until changes in the messages fall

below a threshold

(i) Calculate the availability 119886119894119896 and responsibility119903119894119896(ii) Update 119886(119905+1)

119894119896and 119903(119905+1)

119894119896according to (12) and

(13)

(7) Output the AP clustering result

33 Generation of Transfer Function

331 Opacity Transfer Function After performing AP clus-tering on IGM histogram the voxels in the volume data areclassified into a set of clusters We first assign an opacityvalue to each cluster based on the average distance betweenthe voxels of the cluster and the center point of the volumeIn order to achieve more appealing visualization we furtherrefine the opacity value of each voxel in the cluster accordingto the voxelrsquos gradient value which usually indicates whetherthe voxel is near the boundary of the cluster or not

A cluster119862119894 ismore likely to occlude other clusterswhen itis located closed to the boundary of the volume and farther tothe center point of itThe average distance between the voxelsof the cluster 119862119894 and the center point of the volume can becalculated by

119889 (119862119894 1198810)= 1119873119894 sumVisin119862119894radic(V119909 minus V0119909)

2 + (V119910 minus V0119910)2 + (V119911 minus V0119911)2 (15)

where1198810(V0119909 V0119910 V0119911) is the volumersquos center119873119894 is the number ofvoxels in cluster 119862119894 and V represents voxels of the cluster 119862119894We employ the average distance to define the opacity value ofthe cluster

120572119894 = max (119889) minus 119889119894max (119889) minusmin (119889) (120572max minus 120572min) + 120572min (16)

where the [120572min 120572max] is predefined opacity range andmax (119889) andmin (119889) are themaximumandminimumaveragedistance between clusters and volumersquos center

For clear-cut visualization the boundaries of a cluster aremuch more important than the internal region of the clusteras boundaries determine the shape of a cluster (ie an organor tissue) In addition the depiction of boundaries can alsoenhance the perception of local occlusions which are quite

essential in diagnosis and surgical planning In this regardwe should make the boundaries of a cluster visually distinctby differentiating the voxels closer to the boundaries and thevoxels located in the internal region In order to achieve thiseffect we refine the opacity value at each voxel according tothe gradient magnitude of this voxel

120572V = 120572119894 ( 1003817100381710038171003817119892V10038171003817100381710038171003817100381710038171003817119892max1003817100381710038171003817)119896 (17)

where 120572V is the opacity value of the voxel 120572119894 is the opacityvalue of the cluster 119862119894 119892V is the gradient magnitude ofthe voxel 119892max is the maximum gradient magnitude in thecluster 119862119894 and 119896 is a parameter that determines how to mapthe gradient value to the opacity value (we set 119896 as 1 or 2 inour experiments)

332 Color Transfer Function Color is also an important toolfor distinguishing different organs or structures We assignvoxels in a cluster the same color In our implementationthe HSV model was applied for color mapping as it is aperception-enhanced color model We divided the space ofhue (119867) equally according to the number of the clusters firstThen the saturation (119878) and value (119881) are set to constants as1 and 067 Specifically for a voxel which is classified into thecluster 119895 the hue can be computed as

ℎ119895 = 119895 lowast 360119899 (18)

where 119899 is the number of clusters

333 Interaction Interface For flexibility we also providean interaction interface for our visualization system Intraditional visualization systems that employ IGM histogramto design transfer functions users are usually requestedto select voxels for visualization in the IGM histogram bymanipulating various widgets such as rectangle triangle andellipse In this case users are often required to have a goodunderstanding of the IGM histogram so that the widgets canbe put in appropriate places for meaningful visualizationIt is not an easy task for doctors In contrast in ourvisualization system as we segment the IGM histogram intoseveral clusters users can interact with the clusters insteadof scattered points in the IGM histogram which makesthe interaction more intuitive and effective For exampleusers can merge two or more clusters by setting the clusterswith the same color or modify the color of a cluster formore appealing visualization In addition users can changeparameter settings by dragging scroll bars provided in theinterface in order to obtain promising result quickly

4 Results

Our method was implemented on a PC with 350GHz IntelXeon E5-1620 8G memory and an NVIDIA Quadro K4200The GPU-accelerated ray-casting [30] volume algorithmwas developed to render the results using C++ and GLSLshading language Experiments were performed on various


(a) (b)

1 6

(c) (d)

Figure 3 Visualization of the Visible Human Male Head dataset using different TFs In every subfigure the top row shows the visualizationand the bottom row shows corresponding TFs design (a) the visualization using basic intensity based method (b) the visualization usingtraditional IGM basedmethod based on widgets (c) the visualization using hierarchical exploration [7] (this figure is obtained from [7]) and(d) the visualization using our method

datasets to demonstrate the effectiveness of the proposedmethod The datasets were obtained from the volume library(httpwww9informatikuni-erlangendeExternalvollib)

41 Comparison We first compared the proposed methodwith commonly used transfer function schemes includingbasic intensity based method traditional IGM based methodusing widgets and a state-of-the-art IGM based methodleveragingmultilevel segmentation [7]The experimentswereperformed on the Visible Human Male Head dataset Theresults are shown in Figure 3 Figure 3(a) shows the visual-ization using basic intensity basedmethod It is observed thatthe bone and other interior organs are occluded by the skinIn addition users have to extensively adjust TFs in a time-consuming trial-and-error way in order to achieve desiredresults Figure 3(b) shows the visualization using traditionalIGM based method which demonstrates the advantage ofusing intensity-gradient histogram For example the left ofFigure 3(b) is the initial result while the right of Figure 3(b)is the visualization by adjusting the sizes of the widgets andchanging the opacity of each widget This is a quite time-consuming task Figure 3(c) shows the result using [7] whichconsiders the intensity-gradient histogram as an image andsegments it using graph cut to explore the different organsin the volume data It is a hierarchical interaction processto remove the materials which users want to exclude in thevisualization result For example as shown in Figure 3(c) ifone wants to achieve the result in the right side he (she)needs to perform at least six interactions on the initial result

(left of Figure 3(c)) Figure 3(d) shows the visualization of ourmethod which can achieve satisfaction result automaticallyand users can further polish the result by performing verysimple interactions

42 Rendering Results

421 Visible Human Male Head Figure 4 shows more visu-alization results of the Visible Human Male Head dataset(128 times 256 times 256) using our method In order to more clearlyshow the effectiveness of our classification and interactivityapproaches we display each organ independently by settingthe transparency of other classes to 0 Figure 4(a) shows thedesign of TFs and Figure 4(b) shows the visualization resultof the whole volume The red green yellow light blue andmazarine represent the skin sinus bones in the outsidebones in the inside and teeth while the grey represents thenoise and nonsignificant tissues located at the boundaries ofthe volume As teeth and the chin bone are quite close inthe volume our method classifies them into one cluster Notethat as the skins distance to volume data center is larger thanother tissues and its gradientmagnitude is also quite large theopacity of the skin is set small in ourmethod and thus it lookstransparency in the visualization result to avoid occludingother inside organs

422 Knee Dataset Figure 5 shows the visualization of theknee dataset (379 times 229 times 305) The dataset is acquired from


(a) (b)

(c) (d) (e) (f) (g)

Figure 4 Visualization of the Visible Human Male Head dataset (a) the refined and clustered intensity-gradient histogram (b) thevisualization result using the left transfer function and (c) (d) (e) (f) and (g) the visualization results of the five clusters of the histogramrepresenting skin sinus outside bones inside bones and teeth respectively

(a) (b)

(c) (d) (e)

Figure 5 Visualization of knee dataset (a) the refined and segmented intensity-gradient histogram (b) the visualization result of ourmethodand (c) (d) and (e) the visualization results of the three clusters of the histogram representing skin tibia and fibula respectively

CT scan and is used for anterior tibia osteotomy The bluered and yellow in Figures 5(c) 5(d) and 5(e) represent theskin tibia and fibula respectively while the gray representsthe structures excluded by the threshold

423 Abdomen Dataset Figure 6 shows the visualizationof the abdomen dataset (512 times 512 times 147) The datasetwas acquired from CT scan of the abdomen and pelvisA doctor must identify the key structures from it before


(a) (b)

(c) (d) (e)

Figure 6 Visualization of the CT abdomen dataset (a) the refined and segmented intensity-gradient histogram (b) the visualization resultof the proposed method and (c) (d) and (e) the visualization results of the skin blood vessel and pelvis respectively

(a) (b)

Figure 7 Visualization of other datasets (a) visualization of a CTA dataset and (b) visualization of a MRI dataset

performing surgical planning based on it The blue red andgreen represent the skin blood vessel and pelvis respectivelyIt is observed that the proposed method can automaticallyproduce clear and meaningful visualization result for furtherdiagnosis and treatment

424 Other Datasets In addition to CT scan data ourmethod is general enough to be employed for the visual-ization of other imaging modalities Figure 7(a) shows thevisualization of a computed tomography angiography (CTA)dataset (512times512times120) Note that our method can visualizethe small aneurysm in the middle of the brain (labeled bythe green ellipse in the figure) which is quite important

for diagnosis and treatment planning Figure 7(b) shows thevisualization of a MRI data

The threshold of the spatial variance of a bin is a keyparameter in the proposed method It determines which binsshould be eliminated from histogram Figures 8 and 9 showthe results of the Visible HumanMaleHead dataset and a foot(256times256times256) dataset with different thresholdsThe smallerthe threshold is themore the bins will be eliminated from thehistogram Figures 8(a)ndash8(d) show the visualization resultsof the Visible Human Male Head dataset when we set thethreshold as 1 045 023 and 012 respectively Figures 9(a)ndash9(c) show the visualization results of the foot dataset whenweset the threshold as 075 033 and 021 respectively


(a) (b)

(c) (d)

Figure 8 Visualization of the Visible HumanMale Head dataset with different thresholds (a) the threshold is set as 1 (b) the threshold is setas 045 (c) the threshold is set as 023 and (d) the threshold is set as 012

43 Performance Six datasets have been tested using ourtransfer function and its effectiveness in improving renderingquality has been demonstrated in Figures 3ndash9 To evaluateour methodrsquos effectiveness of real time we do the timeperformance test on these datasets We also evaluate the timeperformance of the proposed method on these datasets Theresults are shown in Table 1 with some key parameters Thetime of computing ]119904 in (2) is also tested as the preprocessingtime

5 Discussion

As we mentioned in Section 2 some previous works havebeen proposed based on IGM histogram to help transferfunction design Figure 3 shows the comparison of ourmethod and most common approaches based on IGM his-togram From this comparison we can see the traditionalIGM based method needs to spend a lot of time to achievesatisfactory results while this is not convenient in clinicalpractice Particularly in Figure 3(b) it is difficult for radiol-ogists and physicians to find an appropriate location to put

the widget as they usually have no good understanding ofthe histogram Figure 3(c) also needs a lot of interactions toremove the tissues which users may not want to observe Inour approach it can remove noisy voxels by presetting thethreshold of the spatial variance of a bin and the classificationof voxels is automatically by usingAP clustering that is shownin Figure 3(d) In Figure 6 the blood vessel is so small inintensity-gradient histogram that the users cannot easily putwidget using traditional IGM based approaches while ourmethod can automatically figure it out using combined IGMand spatial information This is convenient for doctors tooperate to further improve the efficiency of diagnosis Ourmethod can be applied in many clinical applications such asmaxillofacial surgery [31] tumor detection [32] andwearabledevice design [33]

Except the convenient operation the other advantage ofour method is that the spatial information of each voxel isalso considered as the measurement of similarity and thiscan help distinguish different tissues especially for the tissueswhose intensity and gradient magnitude values are similarsuch as bone and teeth that are shown in Figures 4(f) and


(a) (b)

(c)

Figure 9 Visualization of foot dataset with different thresholds (a) the threshold is set as 075 (b) the threshold is set as 033 and (c) thethreshold is set as 021

Table 1 Time performance of the proposed method (FPS frame per second)

Volume Data size Figure Threshold Preprocessing (ms) FPSMale Head 128 times 256 times 256 Figures 3(c) 4 and 8(b) 045 400 198Knee 379 times 229 times 305 Figure 5 3 600 142Abdomen 512 times 512 times 147 Figure 6 08 890 123CTA Head 512 times 512 times 120 Figure 7(a) 055 860 135MRI Head 125 times 154 times 145 Figure 7(b) 040 280 210Foot 256 times 256 times 256 Figure 9(b) 033 480 187

4(g) In Figure 5 the segmentation of tibia and fibula is aquite challenging task as they belong to the same materialwhile in our method by taking both intensity and gradientinformation and spatial information into consideration thetibia and fibula can be successfully segmented and hence thedoctors and acquire more information from the visualizationresult for diagnosis and treatment planning

Although it needs some time to compute the spatialvariance of a bin our time performance test shows that thepreprocessing time cost is so little that does not affect the real

time rendering and the variance of each bin is stored in a filewhen they computed firstly to avoid repeated computing

6 Conclusion

In this paper we present a novel automatic transfer func-tion design scheme for medical data visualization based onthe IGM histogram Compared with previous studies alsobased on IGM histogram our method considers both theintensity-gradient information and the spatial information


of voxels making the clustering of the voxels more accurateand effective The AP clustering algorithm is employed toautomatically generate the TFs Compared with previousclustering algorithms employed in volume visualization theAP clustering algorithm has much faster convergence speedand can achieve more accurate clustering results To achievemeaningful clustering results two novel similarity measure-ments IGM similarity and spatial similarity are proposedand integrated into the AP clustering algorithm Experimentsdemonstrate the proposed method enable users to efficientlyexplore volumetric medical data withminimum interactionsFuture investigations include assessing our method on moreclinical data and attempting to automatically optimize theparameters based on the analysis of the input volume data

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The work described in this paper was supported by a grantfrom the Shenzhen-Hong Kong Innovation Circle FundingProgram (nos SGLH20131010151755080 andGHP00213SZ)a grant from the National Natural Science Foundationof China (Project nos 61233012 and 61305097) a grantfrom the Guangdong Natural Science Foundation (no2016A030313047) and a grant from Shenzhen Basic ResearchProgram (Project no JCYJ20150525092940988)

References

[1] M Levoy ldquoDisplay of surfaces from volume datardquo IEEE Com-puter Graphics and Applications vol 8 no 3 pp 29ndash37 1987

[2] G Kindlmann and JW Durkin ldquoSemi-automatic generation oftransfer functions for direct volume renderingrdquo in Proceedingsof the IEEE ACM Symposium on Volume Visualization (VVSrsquo98) pp 79ndash86 Research Triangle NC USA October 1998

[3] H Childs E Brugger K Bonnell et al ldquoA contract based systemfor large data visualizationrdquo in Proceedings of the VisualizationConference (VIS rsquo05) pp 191ndash198 October 2005

[4] J Meyer-Spradow T Ropinski J Mensmann and K HinrichsldquoVoreen a rapid-prototyping environment for ray-casting-based volume visualizationsrdquo IEEE Computer Graphics andApplications vol 29 no 6 pp 6ndash13 2009

[5] T Fogal and J Kruger ldquoTuvok an architecture for large scalevolume renderingrdquo in Proceedings of the 15th InternationalWorkshop on Vision Modeling and Visualization (VMV rsquo10) pp139ndash146 Berlin Germany November 2010

[6] YWangW Chen J Zhang T Dong G Shan and X Chi ldquoEffi-cient volume exploration using the gaussian mixture modelrdquoIEEE Transactions on Visualization and Computer Graphics vol17 no 11 pp 1560ndash1573 2011

[7] C Y Ip A Varshney and J JaJa ldquoHierarchical exploration ofvolumes usingmultilevel segmentation of the intensity-gradienthistogramsrdquo IEEE Transactions on Visualization and ComputerGraphics vol 18 no 12 pp 2355ndash2363 2012

[8] H Pfister B Lorensen C Bajaj et al ldquoThe transfer functionbake-offrdquo IEEE Computer Graphics and Applications vol 21 no3 pp 16ndash22 2001

[9] T He L Hong A Kaufman and H Pfister ldquoGenerationof transfer functions with stochastic search techniquesrdquo inProceedings of the 1996 IEEE Visualization Conference pp 227ndash234 IEEE San Francisco Calif USA November 1996

[10] M-Y Chan Y Wu W-H Mak W Chen and H QuldquoPerception-based transparency optimization for direct volumerenderingrdquo IEEE Transactions on Visualization and ComputerGraphics vol 15 no 6 pp 1283ndash1290 2009

[11] L Zheng Y Wu and K-L Ma ldquoPerceptually-based depth-ordering enhancement for direct volume renderingrdquo IEEETransactions on Visualization and Computer Graphics vol 19no 3 pp 446ndash459 2013

[12] G Kindlmann R Whitaker T Tasdizen and T MollerldquoCurvature-based transfer functions for direct volume render-ing methods and applicationsrdquo in Proceedings of the 14th IEEEVisualization (VIS rsquo03) pp 513ndash520 IEEE 2003

[13] C D Correa and K-L Ma ldquoSize-based transfer functionsa new volume exploration techniquerdquo IEEE Transactions onVisualization and Computer Graphics vol 14 no 6 pp 1380ndash1387 2008

[14] C D Correa and K-L Ma ldquoThe occlusion spectrum forvolume classification and visualizationrdquo IEEE Transactions onVisualization and Computer Graphics vol 15 no 6 pp 1465ndash1472 2009

[15] C D Correa and K-L Ma ldquoVisibility histograms and visibility-driven transfer functionsrdquo IEEE Transactions on Visualizationand Computer Graphics vol 17 no 2 pp 192ndash204 2011

[16] M Ruiz A Bardera I Boada I Viola M Feixas and MSbert ldquoAutomatic transfer functions based on informationaldivergencerdquo IEEE Transactions on Visualization and ComputerGraphics vol 17 no 12 pp 1932ndash1941 2011

[17] L Cai W-L Tay B P Nguyen C-K Chui and S-H OngldquoAutomatic transfer function design for medical visualizationusing visibility distributions and projective color mappingrdquoComputerizedMedical Imaging and Graphics vol 37 no 7-8 pp450ndash458 2013

[18] H Qin B Ye and R He ldquoThe voxel visibility model an efficientframework for transfer function designrdquo Computerized MedicalImaging and Graphics vol 40 pp 138ndash146 2015

[19] L-L Cai B P Nguyen C-K Chui and S-H Ong ldquoRule-enhanced transfer function generation for medical volumevisualizationrdquo Computer Graphics Forum vol 34 no 3 pp 121ndash130 2015

[20] F-Y Tzeng and K-L Ma ldquoA cluster-space visual interface forarbitrary dimensional classification of volume datardquo in Proceed-ings of the 6th Joint Eurographics-IEEE TCVG Conference onVisualization (VISSYM rsquo04) pp 17ndash24 2004

[21] R MacIejewski I Woo W Chen and D S Ebert ldquoStructuringfeature space a non-parametric method for volumetric transferfunction generationrdquo IEEE Transactions on Visualization andComputer Graphics vol 15 no 6 pp 1473ndash1480 2009

[22] Y Wang J Zhang D J Lehmann H Theisel and X ChildquoAutomating transfer function design with valley cell-basedclustering of 2D density plotsrdquo Computer Graphics Forum vol31 no 3 pp 1295ndash1304 2012

[23] P Sereda A V Bartrolı I W O Serlie and F A GerritsenldquoVisualization of boundaries in volumetric data sets using lhhistogramsrdquo IEEE Transactions on Visualization and ComputerGraphics vol 12 no 2 pp 208ndash217 2006

[24] B P Nguyen W-L Tay C-K Chui and S-H Ong ldquoAclustering-based system to automate transfer function design


for medical image visualizationrdquo The Visual Computer vol 28no 2 pp 181ndash191 2012

[25] S Roettger M Bauer andM Stamminger ldquoSpatialized transferfunctionsrdquo in Proceedings of the EuroVis pp 271ndash278 2005

[26] B J Frey and D Dueck ldquoClustering by passing messagesbetween data pointsrdquo Science vol 315 no 5814 pp 972ndash9762007

[27] J MacQueen ldquoSome methods for classification and analysis ofmultivariate observationsrdquo in Proceedings of the 5th BerkeleySymposium onMathematical Statistics and Probability vol 1 pp281ndash297 Oakland Calif USA 1967

[28] G Karypis E-H Han and V Kumar ldquoChameleon hierarchicalclustering using dynamic modelingrdquo Computer vol 32 no 8pp 68ndash75 1999

[29] J Shi and J Malik ldquoNormalized cuts and image segmentationrdquoIEEE Transactions on Pattern Analysis andMachine Intelligencevol 22 no 8 pp 888ndash905 2000

[30] M Strengert T Klein R Botchen S StegmaierM Chen andTErtl ldquoSpectral volume rendering using GPU-based raycastingrdquoThe Visual Computer vol 22 no 8 pp 550ndash561 2006

[31] M Tatullo M Marrelli and F Paduano ldquoThe regenerativemedicine in oral and maxillofacial surgery the most importantinnovations in the clinical application of mesenchymal stemcellsrdquo International Journal of Medical Sciences vol 12 no 1 pp72ndash77 2015

[32] P J Koo J J Kwak S Pokharel and P L Choyke ldquoNovelimaging of prostate cancer with MRI MRIUS and PETrdquoCurrent Oncology Reports vol 17 no 12 article 56 2015

[33] P N Kooren A G Dunning MM H P Janssen et al ldquoDesignand pilot validation of a-gear a novel wearable dynamic armsupportrdquo Journal of NeuroEngineering and Rehabilitation vol12 no 1 article 83 pp 1ndash12 2015

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


a 2D space to make up the intensity-gradient magnitude(IGM) histogram and then designed TFs based on it by usingvarious manipulation widgets (such as rectangles trianglesand ellipses) Later IGM histogram was often used in directvolume rendering and has been implemented in several pop-ular visualization packages such as VisIt [3] Voreen [4] andImageVis3D [5] However most implementations involveda lot of manual operations to achieve satisfactory resultswhich made them not applicable for doctors without specialknowledge of IGM histogram In recent years Wang et al [6]proposed a volume exploration with the Gaussian mixturemodel and generated a set of suggestive elliptical transferfunctions semiautomatically Nonetheless this method is noteffective when the dataset is complicated as the widgetsare too regular to describe multidimensional features withcomplex shapes Ip et al [7] presented a semiautomaticapproach to detect embedded features and spatial structuresby visually segmenting the IGM histogram of a volumetricdataset However this method required too many operationsto annul undesired structures and achieve satisfactory resultsand hence was still not suitable for radiologists and physi-cians

To overcome these shortcomings we propose a novelautomatic TFs design method based on IGM histogram Ourmethod is based on the observation that voxels belonging tothe same structure usually have similar intensity and gradientmagnitude and are located together in the volume In thiscase we employ affinity propagation clustering algorithmto classify the scattering points in the IGM histogram intoclusters by defining two novel similarity measurements IGMsimilarity and spatial similarity These two similarity mea-surements can effectively bring the voxels of the same tissuetogether and differentiate the voxels of different tissues so thatthe generated TFs can assign different optical properties todifferent tissues making the visualization results unambigu-ous and easy to be used in diagnosis and treatment Note thatas we do not need to assign a cluster number to the affinitypropagation clustering algorithm in advance the clusteringprocess is fully automatic In addition before performingclustering we eliminate noisy voxels by leveraging the spatialinformation of the input volume data We also provide asimple interaction interface to allow users further polish theclustering results The TFs are automatically generated basedon the clustering results Experiments on various datasetsdemonstrate the effectiveness of the proposed method

The organization of this paper is as follows We presentsrelated work in Section 2 Section 3 describes our method indetail Section 4 provides experimental results Conclusionsare drawn in Section 6

2 Related Work

Although direct volume rendering technique has been widelyused transfer function design remains a challenging taskThe design of TFs is often classified into two categoriesimage-centric and data-centric [8] depending on whetherthey derive their parameters from the resulting images ororiginal data

Image-centric methods typically adjusted TFs by search-ing for the optimal rendered image in the rendering spaceThus the TFrsquos generation is considered as a process of param-eter optimization In early investigations genetic algorithmswere employed to select the most satisfactory renderingresult [9] However depth information cannot be sufficientlyconsidered in these algorithms To enhance the depth infor-mation of rendering images Chan et al [10] proposed a novelmetric based on psychological principles and Zheng et al[11] proposed an effective design by optimizing an energyfunction based on quantitative perception models Althoughimage-centric methods are intuitive and easy to implementthe final rendering results generated by thesemethods heavilydepend on the initial result In addition a lot of parametersneed to be tuned in these methods which is unacceptable formedical data visualization

In contrast with image-centric methods data-centric TFsdefine visual properties based on the information of voxels Inearly studies TFs were designed based only on the histogramof voxelsrsquo scalar values (these TFs are usually called 1D TFs)However 1D TFs have inherent difficulties in identifyingdifferent materials with similar intensities Then 2D TFsbased on intensity values and gradient magnitudes wereproposed which were more effective in detecting multiplematerials and their boundaries [2] Besides using derivativeproperties of scalar value other effective metrics were alsoincorporated into the feature space such as curvature [12]feature size [13] and ambient occlusion [14] In order to high-light important structures of volume some methods basedon visibility were introduced [15] and information divergencebetween visibility distribution and target distribution wasoptimized to generate automatic TFs [16ndash18] Recently Caiet al have proposed a rule-based transfer function based onthe local frequency distribution of a voxels neighborhood[19] The structures of userrsquos interest in a volume datasetcan be characterized by using the various feature spaceHowever higher dimensions of the feature space add morecomplication in transfer function design

Recently several semiautomatic transfer function designshave been proposed to reduce the number of degrees offreedom in transfer function design Tzeng andMa proposeda method based on material classes extracted from thecluster space using the ISODATA technique to generate TFs[20] Some other machine learning based methods have alsobeen employed to analyze the feature spaces For exampleMacIejewski et al proposed a nonparametric kernel densityestimation method onto the IGM feature space in order toextract feature patterns [21] FurthermoreWang et al appliedthe Morse theory to automatically decompose the IGMfeature space into a set of valley cells [22] In order to displaythe surface information of the volume data better Sereda et alproposed LH histogram which uses the lower FL and higherFH intensity values of the twomaterials that form a boundaryas the two axes to design the transfer function [23] Basedon LH histogram Nguyen et al proposed to oversegment LHhistogram usingmean shift clustering and then group similarvoxels by using hierarchical clustering [24] Roettger et alincorporated spatial information into the IGMhistogram and



Similarity

Similarityin IGM

Similarityin volume

AP clustering

TF design

Gradient magnitude

IGM histogram

IGMvalue

Spatialinformation

of bin

Threshold

Volume dataset

Visualization





3 Method




120583119904 = 1119899 sumVisin119880119901 (V) (1)

]119904 = 1119899 sumVisin1198801003817100381710038171003817119901 (V) minus 1205831199041003817100381710038171003817 (2)


(a) (b)

(c)








119903119894119896 larr997888 119904119894119896 minusmax1198961015840 =119896

(1198861198941198961015840 + 1199041198941198961015840) 119886119894119896 larr997888 min

0 119903119896119896 + sum1198941015840notin119894119896

max 0 1199031198941015840119896 (3)


119886119896119896 larr997888 sum1198941015840 =119896

max 0 1199031198941015840119896 (4)


argmax119896

119886119894119896 + 119903119894119896 (5)










NR (119887119909 119887119910) = sumV119909isin119887119909

sumV119910isin119887119910

119873(V119909 V119910) 119887119909 = 119887119910 (8)



119887119894

NR (119887119909 119887119894) (9)


119904vol (119887119909 119887119910)

= maxNR (119887119909 119887119910)

NR (119887119909) NR (119887119910 119887119909)NR (119887119910) NR (119887119894) = 0

0 NR (119887119894) = 0(10)



119904 (119887119909 119887119910) = minus1198961119904IGM (119887119909 119887119910) + 1198962119904vol (119887119909 119887119910) (11)




119886119905119894119896 + 1199041198941198961015840) + 120582119903119905119894119896119894 = 119896

(12)

119886119905+1119894119896 larr997888 120582119886119905119894119896+ (1 minus 120582)(min

0 119903119905119896119896 + sum1198941015840notin119894119896

max 0 1199031198941015840119896) 119894 = 119896

(13)


max 0 119903119905+11198941015840119896 ) + 120582119886119905119894119896 (14)







below a threshold


119894119896and 119903(119905+1)


(13)






2 + (V119910 minus V0119910)2 + (V119911 minus V0119911)2 (15)






120572V = 120572119894 ( 1003817100381710038171003817119892V10038171003817100381710038171003817100381710038171003817119892max1003817100381710038171003817)119896 (17)



ℎ119895 = 119895 lowast 360119899 (18)



4 Results



(a) (b)

1 6

(c) (d)









(a) (b)

(c) (d) (e) (f) (g)


(a) (b)

(c) (d) (e)





(a) (b)

(c) (d) (e)


(a) (b)







(a) (b)

(c) (d)



5 Discussion





(a) (b)

(c)







6 Conclusion




Competing Interests


Acknowledgments


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Similarity

Similarityin IGM

Similarityin volume

AP clustering

TF design

Gradient magnitude

IGM histogram

IGMvalue

Spatialinformation

of bin

Threshold

Volume dataset

Visualization





3 Method




120583119904 = 1119899 sumVisin119880119901 (V) (1)

]119904 = 1119899 sumVisin1198801003817100381710038171003817119901 (V) minus 1205831199041003817100381710038171003817 (2)


(a) (b)

(c)








119903119894119896 larr997888 119904119894119896 minusmax1198961015840 =119896

(1198861198941198961015840 + 1199041198941198961015840) 119886119894119896 larr997888 min

0 119903119896119896 + sum1198941015840notin119894119896

max 0 1199031198941015840119896 (3)


119886119896119896 larr997888 sum1198941015840 =119896

max 0 1199031198941015840119896 (4)


argmax119896

119886119894119896 + 119903119894119896 (5)










NR (119887119909 119887119910) = sumV119909isin119887119909

sumV119910isin119887119910

119873(V119909 V119910) 119887119909 = 119887119910 (8)



119887119894

NR (119887119909 119887119894) (9)


119904vol (119887119909 119887119910)

= maxNR (119887119909 119887119910)

NR (119887119909) NR (119887119910 119887119909)NR (119887119910) NR (119887119894) = 0

0 NR (119887119894) = 0(10)



119904 (119887119909 119887119910) = minus1198961119904IGM (119887119909 119887119910) + 1198962119904vol (119887119909 119887119910) (11)




119886119905119894119896 + 1199041198941198961015840) + 120582119903119905119894119896119894 = 119896

(12)

119886119905+1119894119896 larr997888 120582119886119905119894119896+ (1 minus 120582)(min

0 119903119905119896119896 + sum1198941015840notin119894119896

max 0 1199031198941015840119896) 119894 = 119896

(13)


max 0 119903119905+11198941015840119896 ) + 120582119886119905119894119896 (14)







below a threshold


119894119896and 119903(119905+1)


(13)






2 + (V119910 minus V0119910)2 + (V119911 minus V0119911)2 (15)






120572V = 120572119894 ( 1003817100381710038171003817119892V10038171003817100381710038171003817100381710038171003817119892max1003817100381710038171003817)119896 (17)



ℎ119895 = 119895 lowast 360119899 (18)



4 Results



(a) (b)

1 6

(c) (d)









(a) (b)

(c) (d) (e) (f) (g)


(a) (b)

(c) (d) (e)





(a) (b)

(c) (d) (e)


(a) (b)







(a) (b)

(c) (d)



5 Discussion





(a) (b)

(c)







6 Conclusion




Competing Interests


Acknowledgments


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) (b)

(c)








119903119894119896 larr997888 119904119894119896 minusmax1198961015840 =119896

(1198861198941198961015840 + 1199041198941198961015840) 119886119894119896 larr997888 min

0 119903119896119896 + sum1198941015840notin119894119896

max 0 1199031198941015840119896 (3)


119886119896119896 larr997888 sum1198941015840 =119896

max 0 1199031198941015840119896 (4)


argmax119896

119886119894119896 + 119903119894119896 (5)










NR (119887119909 119887119910) = sumV119909isin119887119909

sumV119910isin119887119910

119873(V119909 V119910) 119887119909 = 119887119910 (8)



119887119894

NR (119887119909 119887119894) (9)


119904vol (119887119909 119887119910)

= maxNR (119887119909 119887119910)

NR (119887119909) NR (119887119910 119887119909)NR (119887119910) NR (119887119894) = 0

0 NR (119887119894) = 0(10)



119904 (119887119909 119887119910) = minus1198961119904IGM (119887119909 119887119910) + 1198962119904vol (119887119909 119887119910) (11)




119886119905119894119896 + 1199041198941198961015840) + 120582119903119905119894119896119894 = 119896

(12)

119886119905+1119894119896 larr997888 120582119886119905119894119896+ (1 minus 120582)(min

0 119903119905119896119896 + sum1198941015840notin119894119896

max 0 1199031198941015840119896) 119894 = 119896

(13)


max 0 119903119905+11198941015840119896 ) + 120582119886119905119894119896 (14)







below a threshold


119894119896and 119903(119905+1)


(13)






2 + (V119910 minus V0119910)2 + (V119911 minus V0119911)2 (15)






120572V = 120572119894 ( 1003817100381710038171003817119892V10038171003817100381710038171003817100381710038171003817119892max1003817100381710038171003817)119896 (17)



ℎ119895 = 119895 lowast 360119899 (18)



4 Results



(a) (b)

1 6

(c) (d)









(a) (b)

(c) (d) (e) (f) (g)


(a) (b)

(c) (d) (e)





(a) (b)

(c) (d) (e)


(a) (b)







(a) (b)

(c) (d)



5 Discussion





(a) (b)

(c)







6 Conclusion




Competing Interests


Acknowledgments


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

















NR (119887119909 119887119910) = sumV119909isin119887119909

sumV119910isin119887119910

119873(V119909 V119910) 119887119909 = 119887119910 (8)



119887119894

NR (119887119909 119887119894) (9)


119904vol (119887119909 119887119910)

= maxNR (119887119909 119887119910)

NR (119887119909) NR (119887119910 119887119909)NR (119887119910) NR (119887119894) = 0

0 NR (119887119894) = 0(10)



119904 (119887119909 119887119910) = minus1198961119904IGM (119887119909 119887119910) + 1198962119904vol (119887119909 119887119910) (11)




119886119905119894119896 + 1199041198941198961015840) + 120582119903119905119894119896119894 = 119896

(12)

119886119905+1119894119896 larr997888 120582119886119905119894119896+ (1 minus 120582)(min

0 119903119905119896119896 + sum1198941015840notin119894119896

max 0 1199031198941015840119896) 119894 = 119896

(13)


max 0 119903119905+11198941015840119896 ) + 120582119886119905119894119896 (14)







below a threshold


119894119896and 119903(119905+1)


(13)






2 + (V119910 minus V0119910)2 + (V119911 minus V0119911)2 (15)






120572V = 120572119894 ( 1003817100381710038171003817119892V10038171003817100381710038171003817100381710038171003817119892max1003817100381710038171003817)119896 (17)



ℎ119895 = 119895 lowast 360119899 (18)



4 Results



(a) (b)

1 6

(c) (d)









(a) (b)

(c) (d) (e) (f) (g)


(a) (b)

(c) (d) (e)





(a) (b)

(c) (d) (e)


(a) (b)







(a) (b)

(c) (d)



5 Discussion





(a) (b)

(c)







6 Conclusion




Competing Interests


Acknowledgments


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














below a threshold


119894119896and 119903(119905+1)


(13)






2 + (V119910 minus V0119910)2 + (V119911 minus V0119911)2 (15)






120572V = 120572119894 ( 1003817100381710038171003817119892V10038171003817100381710038171003817100381710038171003817119892max1003817100381710038171003817)119896 (17)



ℎ119895 = 119895 lowast 360119899 (18)



4 Results



(a) (b)

1 6

(c) (d)









(a) (b)

(c) (d) (e) (f) (g)


(a) (b)

(c) (d) (e)





(a) (b)

(c) (d) (e)


(a) (b)







(a) (b)

(c) (d)



5 Discussion





(a) (b)

(c)







6 Conclusion




Competing Interests


Acknowledgments


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) (b)

1 6

(c) (d)









(a) (b)

(c) (d) (e) (f) (g)


(a) (b)

(c) (d) (e)





(a) (b)

(c) (d) (e)


(a) (b)







(a) (b)

(c) (d)



5 Discussion





(a) (b)

(c)







6 Conclusion




Competing Interests


Acknowledgments


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) (b)

(c) (d) (e) (f) (g)


(a) (b)

(c) (d) (e)





(a) (b)

(c) (d) (e)


(a) (b)







(a) (b)

(c) (d)



5 Discussion





(a) (b)

(c)







6 Conclusion




Competing Interests


Acknowledgments


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) (b)

(c) (d) (e)


(a) (b)







(a) (b)

(c) (d)



5 Discussion





(a) (b)

(c)







6 Conclusion




Competing Interests


Acknowledgments


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) (b)

(c) (d)



5 Discussion





(a) (b)

(c)







6 Conclusion




Competing Interests


Acknowledgments


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) (b)

(c)







6 Conclusion




Competing Interests


Acknowledgments


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Competing Interests


Acknowledgments


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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Research Article A Clustering-Based Automatic Transfer ...downloads.hindawi.com/journals/mpe/2016/4547138.pdf · cation and visualization process is automatic and e cient. Experiments