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Eurographics/ IEEE-VGTC Symposium on Visualization 2010G. Melançon, T. Munzner, and D. Weiskopf(Guest Editors)

Volume 29 (2010), Number 3

Isosurface Similarity Maps

Stefan Bruckner and Torsten Möller

Graphics, Usability and Visualization (GrUVi) LabSchool of Computing Science, Simon Fraser University, Canada

Abstract

In this paper, we introduce the concept of isosurface similarity maps for the visualization of volume data. Iso-surface similarity maps present structural information of a volume data set by depicting similarities betweenindividual isosurfaces quantified by a robust information-theoretic measure. Unlike conventional histograms, theyare not based on the frequency of isovalues and/or derivatives and therefore provide complementary information.We demonstrate that this new representation can be used to guide transfer function design and visualization pa-rameter specification. Furthermore, we use isosurface similarity to develop an automatic parameter-free methodfor identifying representative isovalues. Using real-world data sets, we show that isosurface similarity maps canbe a useful addition to conventional classification techniques.

Categories and Subject Descriptors (according to ACM CCS): I.3.3 [Computer Graphics]: Picture/ImageGeneration—Display algorithms

1. Introduction

The field of volume visualization aims to provide insightfuldepictions of three-dimensional data. Modalities such as CT,MRI, or laser-scanning confocal microscopy allow physi-cians, scientists, and engineers to investigate the interior ofcomplex objects. However, providing clear visualizations ofthe structures contained in a volume data set is a major chal-lenge. One of the issues is the lack of explicit geometric in-formation and limited semantics. A volume data set containsa large number of isosurfaces at different target scalar fieldvalues, while its structure is typically characterized by a fi-nite number of feature isosurfaces that segment the data setinto several important components. The data may be visu-alized directly by mapping the scalar values and/or derivedattributes to optical properties, or a geometric surface repre-sentation may be extracted using techniques like the popularMarching Cubes algorithm [LC87]. Irrespective of the cho-sen visualization method, providing guidance in the identi-fication of salient isovalues plays an important role in im-proving the exploration process. In this paper, we present anew approach for the visualization and detection of relevantisovalues which provides additional information comparedto conventional histograms.

Histogram-based methods typically infer similarity from

the frequency at which isovalues occur. While there are caseswhere this assumption holds, in general only limited infor-mation can be deduced using this approach. Moreover, thepresence of large homogenous regions, acquisition artifacts,and noise introduces additional problems. Instead of the fre-quency of individual data values and/or derived attributes,our method is based on the global notion of isosurface sim-ilarity. Using the information-theoretic measure of mutualinformation, we compare all combinations of isosurfaces todetermine their degree of dependency. This process resultsin an isosurface similarity map which provides a compactoverview of the similarities.

The main contribution of this paper is a new approachfor quantifying and visualizing the similarity between iso-surfaces in a scalar field. We demonstrate its applicabil-ity for simplifying isovalue selection and enhancing scalar-field visualization. However, we want to stress that we donot directly compete with the plethora of techniques whichemploy multi-dimensional transfer functions based on localvoxel properties. We recognize that it is useful and, for manytypes of data, unavoidable to employ such classifiers in orderto separate features which share the same value ranges. In-deed, as the isovalue is one axis in many multi-dimensional

c© 2010 The Author(s)Journal compilation c© 2010 The Eurographics Association and Blackwell Publishing Ltd.Published by Blackwell Publishing, 9600 Garsington Road, Oxford OX4 2DQ, UK and350 Main Street, Malden, MA 02148, USA.

S. Bruckner & T. Möller / Isosurface Similarity Maps

transfer function domains, our approach complements thesetechniques.

2. Related Work

Due to the complex nature of volumetric data sets, tech-niques for providing a simplified view of the data are fre-quently used. Their purpose is to guide the user to inter-esting regions which can subsequently be investigated ina three-dimensional view. Histograms are one of the mostcommon representations employed for this purpose. Theyvisualize the data set by depicting the number of voxels foreach data value. Carr et al. [CDD06], in a result later refinedby Scheidegger et al. [SSD∗08], demonstrated that this ac-tually converges to the distribution of isosurface areas butis formally equivalent to the nearest neighbor interpolant.They proposed several practical measures which show bet-ter convergence behavior. Isosurface area, however, still con-veys limited information about the nature and structure of adata set. For this reason, Bajaj et al. [BPS97] displayed a va-riety of additional isosurface statistics in their contour spec-trum. Pekar et al. [PWH01] suggested the use of a Laplacian-weighted histogram to assist in the detection of significantisovalues.

Several approaches apply topological analysis to volumedata. The contour tree [CSVDP10] is an abstraction of ascalar field that encodes the nesting relationships of isosur-faces. Takahashi et al. [TTFN06] employed a volume skele-ton tree to identify isosurface embeddings in order to provideadditional structural information. Hyper Reeb graphs, pro-posed by Fujishiro et al. [FTAT00], capture the topologicalskeleton of a volumetric data set and can serve as a refer-ence structure for designing comprehensible volume visual-izations. One problem of these methods is that they rely ongeometric extraction processes which suffer from noise inreal-world data.

Much work has focused on the collection of local prop-erties such as first and second derivatives. Plotting these at-tributes against isovalues provides guidance for feature se-lection. Kindlmann and Durkin [KD98] demonstrated that atwo-dimensional histogram of data value and gradient mag-nitude enables the identification of boundary regions whichmanifest themselves as arches. Kniss et al. [KKH02] ex-tended this idea to the common notion of two-dimensionaltransfer functions and developed a direct manipulation in-terface for their specification. Lum et al. [LM04] used ad-ditional gradient-aligned samples depicted in a line-basedhistogram. Šereda et al. [ŠVSG06] extended this idea bysearching for high and low values in paths that followthe gradient. Tenginakai et al. [TLM01] employed multi-dimensional histograms based on local higher-order mo-ments to detect important data values. To better charac-terize the shape of local features, Sato et al. [SWB∗00]used the matrix of second derivatives. The role of curva-ture was investigated by Kindlmann et al. [KWTM03] and

Hladuvka et al. [HKG00]. Roettger et al. [RBS05] extendedhistograms by incorporating spatial information. Lundströmet al. [LLY06] proposed the use of local histograms tobetter represent the distribution of intensity values in agiven neighborhood which improves tissue separation forthe case of overlapping intensity ranges. Correa and Ma pro-posed classification approaches based on size [CM08], oc-clusion [CM09a], and visibility [CM09b].

Due to this large number of different classification cri-teria, several approaches proposed the use of dimensional-ity reduction techniques to identify regions based on high-dimensional voxel signatures. Tzeng et al. [TM04] used ma-chine learning methods to generate classifications based ona simple painting interface. They also presented a cluster-space approach for interacting with multiple classificationcriteria [TLM05]. Šereda et al. [ŠVG06] employed hierar-chical clustering of material boundaries. Pinto et al. [PF07]utilized self-organizing maps to reduce the dimensionality ofthe classification space.

Previous methods attempt to characterize volume data setsby analyzing global isosurface statistics, extracting topo-logical relationships, or collecting local voxel signatures.Our approach is fundamentally different in that we mea-sure similarities of isosurfaces as a whole based on a robustinformation-theoretic measure. We show that this notion canyield additional insights into the structure of the data and canserve as the basis for enhanced visualization.

3. Isosurface Similarity

Isosurfaces play a crucial role in visualizing and interpret-ing volumetric data. In many cases they represent importantobject and/or material boundaries. However, in practice itis difficult to identify salient isovalues. Traditionally, one-and two-dimensional histograms have been employed to as-sist the user in this process. These approaches depict the fre-quency of isovalues and other attributes (e.g, gradient mag-nitude). Peaks or clusters in these plots then guide the explo-ration and visualization process. Frequency, however, can bea problematic measure as large regions such as the back-ground intensity tend to dominate.

We propose isosurface similarity as an alternative mea-sure for identifying features and guiding visualization pa-rameter specification. Instead of collecting statistics of in-dividual isosurfaces and using their variation to obtain in-formation about their significance, we are interested in in-vestigating similarities between individual isosurfaces di-rectly, i.e., how much does knowledge about one surfacetell us about the others. In this section, we first introduceour new measure for isosurface similarity. We then applythis measure to the isosurfaces of a scalar field to providean overview of the mutual similarities which complementstraditional frequency-based depictions.

c© 2010 The Author(s)Journal compilation c© 2010 The Eurographics Association and Blackwell Publishing Ltd.


3.1. Similarity Measure

An isosurface, or level set, of a volumetric scalar-valuedfunction f : R3 → R is the locus of all points at which fattains the isovalue h:

Lh ={

x ∈ R3 : f (x) = h}

(1)

A popular information-theoretic measure of similarity whichhas been applied in many areas including shape registra-tion [HPM06], multi-modality fusion [HBKG08], and view-point selection [VFSG06] is mutual information. The mutualinformation of two discrete random variables X and Y can bedefined as [Yao03]:

I(X ,Y ) = ∑x∈X

∑y∈Y

pX ,Y (x,y) log(

pX ,Y (x,y)pX (x)pY (y)

)(2)

where pX ,Y is the joint probability distribution function of Xand Y , and pX and pY are the marginal probability distribu-tion functions of, respectively, X and Y . Mutual informationquantifies the information that X and Y share by measur-ing how much knowing one of these variables reduces theuncertainty about the other. In other words, mutual informa-tion measures the dependence between the joint distributionof X and Y and what the joint distribution would be if Xand Y were independent. The mutual information of X andY is zero if and only if they are statistically independent.In the case of identity of X and Y , the mutual informationis equal to the uncertainty associated with the random vari-able, i.e., its entropy. Mutual information can be equivalentlyexpressed in terms of entropy as:

I(X ,Y ) = H(X)+H(Y )−H(X ,Y ) (3)

with

H(X) = − ∑x∈X

pX (x) log(pX (x)) (4)

H(Y ) = − ∑y∈Y

pY (y) log(pY (y)) (5)

H(X ,Y ) = − ∑x∈X

∑y∈Y

pX ,Y (x,y) log(

pX ,Y (x,y))

(6)

where H(X) and H(Y ) denote the marginal entropies andH(X ,Y ) is the joint entropy of X and Y . As it is convenientto work with values in [0,1], a normalized measure can beobtained by [Kvå87]:

I(X ,Y ) =2I(X ,Y )

H(X)+H(Y )(7)

Volume data frequently exhibit an onion-peel-like structureand contain material inhomogeneities, partial volume ef-fects, and noise. This results in several redundant isosur-faces, i.e., they do not represent substantial additional infor-mation. We would like to obtain a measure which classifiesthese kind of isosurfaces as similar. In the registration liter-ature, shape representations based on implicit distance func-tions are commonly used as they have proven to be stableand robust to shape perturbations and noise [HPM06]. For

these reasons, we choose to represent individual isosurfacesusing their distance transform. The distance transform Dh ofan isosurface with isovalue h gives the minimum distance ofa point x to the surface [JBŠ06]:

Dh(x) = min∀y∈Lh

d(x,y) (8)

where d is a distance measure (for the remainder of this pa-per we will assume the Euclidean distance). We can nowconsider the distances from any point to a pair of isosurfacesLp and Lq as random variables X and Y . In order to com-pute the mutual information between the two isosurfaces, weneed to estimate the joint distribution of X and Y . This canbe accomplished using the joint histogram of Dp and Dq: forevery voxel position x, we record the distances Dp(x) andDq(x) in a two-dimensional histogram where each bin rep-resents a certain range of distances. The marginal probabilitydistributions of X and Y can be estimated by summing overthe columns and rows, respectively, of the joint histogram.This allows us to directly compute H(X ,Y ), H(X), and H(Y )to evaluate the normalized mutual information, as defined inEquation 7, of the two isosurfaces as a measure of their sim-ilarity.

3.2. Similarity Maps

In order to obtain information about the similarity relation-ships in a data set, we introduce the isosurface similaritymap. Given the discrete set of N isovalues V = {h1, . . . ,hN}in a data set, we generate a N ×N matrix SMV (i, j) con-taining the isosurface similarity, computed as described inthe previous section, for each combination of isovalues hiand h j. We will also use the notation SMV (x,y) to denotethe matrix element SMV (i, j) with x = hi and y = h j whenconvenient. Due to the properties of mutual information, themap is symmetric and one along the main diagonal. As itrecords the similarity between every pair of isosurfaces, itprovides an overview of the similarity relationships in thedata set. In contrast to histograms, which visualize the fre-quency of individual values, it instead depicts the similarityof isosurfaces measured by the mutual information of theirdistance fields.

By summing over the rows (or columns) of the isosur-face similarity map and normalizing the result by the numberof isovalues, we obtain the isosurface similarity distributionSDV :

SDV (i) =1|V |

|V |

∑j=1

SMV (i, j) (9)

The isosurface similarity distribution describes the averagesimilarity for each individual isosurface. Peaks in the simi-larity distribution correspond to those isosurfaces which aremost similar to others while valleys indicate regions of rapidchange. As will be shown in Section 4, it is frequently usefulto investigate the similarity distribution for a specific subset



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Figure 1: Example of our isosurface similarity measure. Row (1): column (a) shows an image of three concentric circles, incolumn (b) the innermost sphere is replaced by a square of approximately equal area, and columns (c) and (d) show the sametwo images with added noise. In row (2) the respective histograms are shown. Row (3) depicts the corresponding isosurfacesimilarity maps. Row (4) shows the isosurface similarity distributions.

of isovalues. In practice, a summed-area table of the similar-ity map enables the efficient evaluation of similarity distri-bution values for arbitrary continuous subranges.

Figure 1 shows a simple example. The image in row (1),column (a) contains three concentric circles with differentisovalues. In column (b) the innermost circle is replaced bya square of approximately equal area. Expectedly, the corre-sponding histograms shown in row (2) are essentially identi-cal. The isosurface similarity maps depicted in row (3), how-ever, show considerable differences. Similarity is linearlymapped to grayscale intensity where white means a simi-larity of zero and black corresponds to a similarity of one.While row (3), column (a) shows a high degree of mutualsimilarity between the three spheres, the square’s presenceis clearly indicated in row(3), column (b). This is also re-

flected in the corresponding similarity distributions depictedin row (4). Columns (c) and (d) of the figure demonstrate thatthe basic structure of similarity map and distribution remainsthe same even though noise has been added to the images.

4. Applications

In this section, we present applications of isosurface simi-larity maps and distributions for the visualization of volumedata. We do not advocate replacement of well-proven meth-ods such as histograms which are clearly useful for manypurposes nor do we propose similarity as a sole classifica-tion criterion. Instead, we want to demonstrate that isosur-face similarity provides additional information which can beexploited to build improved tools for volume visualization.



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Figure 2: Result images for similarity-enhanced isosurface visualization are shown in (a) and (b), (c) depicts the isosurfacesimilarity map of the data set, and (d) shows the isosurface similarity distribution for the highlighted region in the similaritymap. The isovalues depicted in (a) and (b) are marked with corresponding colors in (c) and (d).

4.1. Similarity-Enhanced Isosurface Visualization

A common problem in volume visualization is that even mi-nor changes in the selected isovalues can have dramatic im-pact on the depicted features. The importance of providingthe user with information on this kind of uncertainty wasdemonstrated by Lundström al al. [LLPY07] in the contextof stenosis assessment. In a similar spirit, isosurface simi-larity allows us to indicate the stability of an isosurface byvisually encoding the similarity of a sample point with re-spect to a specified isovalue.

We want to depict the isosurface at an user-selected iso-value hu in a focus+context manner using direct volume ren-dering [HMBG01]. In addition to the isosurface itself (fo-cus), information about similar regions (context) should beconveyed to the user. To prevent visual overload, the con-textual regions should be depicted in a sparse way. We usethe following importance function γ(x) which determines thedegree-of-interest in a sample at location x:

γ(x) = ∏y∈C(x)

SMV (hu, f (y)) (10)

where f is the scalar field and C(x) denotes a local neighbor-hood of samples around x. In practice, we reuse the samplevalues required for gradient estimation using central differ-ences as the sample neighborhood. The effect of this func-tion is that, due to the product of similarities of neighbor-ing voxels, only contextual regions with high local similar-ity will tend to exhibit high importance. For visualizing theisosurface hu we specify the opacity of a sample α(x) as:

α(x) =

{1 if f (x)≥ hu

γ(x) otherwise(11)

To clearly distinguish between focus and context, γ is alsoused to control the color and the degree of surface shad-ing. Additionally, the directional occlusion model of Schottet al. [SPH∗09] is used uniformly for all samples.

An example is shown in Figure 2. The isovalue selected

in Figure 2 (a) is very unstable – the extent of the cloud sur-rounding the surface indicates that it only captures part ofthe structure of interest. Figure 2 (b) shows more stability.Note that the similarity cloud has the same shape in both im-ages – this means that both isosurfaces are part of a clusterof high mutual similarity. This is confirmed by locating theisovalues in the similarity map shown in Figure 2 (c) – bothlie within one large cluster. Figure 2 (d) shows the similaritydistribution for the indicated square region. It can be seenthat the isovalue used in Figure 2 (a) has low average simi-larity, while the value from Figure 2 (b) is located at a peak,i.e., it represents the region well.

4.2. Similarity-Based Isovalue Remapping

Isovalues are typically selected and modified using userinterface widgets such as sliders or by linearly mappingthem to mouse movement. This, however, can be quite non-intuitive: if a subrange of isovalues corresponds to very sim-ilar isosurfaces, large changes of the value will have almostno effect on the depicted structures. Conversely, in regions ofhigh dissimilarity even a minor modification can completelyalter the appearance. Ideally, the effects in the visualizationshould correspond to the magnitude of change in the corre-sponding user interface component. Thus, instead of directlytranslating changes of the controlling element to changes inthe isovalue, we use a nonlinear mapping based on isosur-face similarity. Let MLV be the conventional direct mappingfunction which maps values [0,1] linearly to the range ofisovalues [hmin,hmax] in the set V :

MLV (x) = hmin + x(hmax�hmin) (12)

The idea is to use a monotonous function MSV (x) withMSV (0) = MLV (0) and MSV (1) = MLV (1) whose deriva-tive is controlled by the similarity of neighboring isovalues.For this purpose, we define the cumulative similarity of an



ML V

(x)

MS V

(x)

x = 0.22 x = 0.24 x = 0.26 x = 0.28 x = 0.30 x = 0.32

Figure 3: Transition using linear mapping MLV (x) (top row) and similarity-based mapping MSV (x) (bottom row) for x ∈[0.22,0.32].

ordered set of isovalues V :

SCV (i) =i

∑j=1

SMV ( j�1, j) (13)

where SMV (0,1) = 0. Our similarity-based mapping func-tion can then be written as:

MSV (x) = MLV

(SCV (x(|V |�1)+1)

SCV (|V |)

)(14)

The fact that this function is piecewise constant does notmatter in practice, since we are typically only interested indiscrete isovalues. However, one can simply use any inter-polant for SCV to remedy this.

Figure 3 depicts a transition using MLV (x) (top row) andMSV (x) (bottom row) with x varying from 0.22 to 0.32 in in-crements of 0.02. Even though the differences are subtle, theimages generated using the similarity-based mapping func-tion show a more uniform progression. The isosurface sim-ilarity map for the data set is shown an Figure 4 (a) – thehighlighted area indicates the range of isovalues of the tran-sition. Note that the dissimilar nature of the isosurfaces in theinterval is clearly visible. The graph in Figure 4 (b) depictsthe function MLV (x)�MSV (x) for the chosen interval, i.e.,the difference in isovalue resulting from using the similarity-based mapping function instead of the linear one.

4.3. Representative Isovalue Selection

An important problem in volume visualization has been theidentification of relevant isovalues. Many approaches com-bine different isosurface statistics to infer salient isovalueswhich characterize the function well. The isosurface simi-larity map can be used to provide guidance in finding rep-resentative isovalues. By selecting rectangular regions in themap and investigating their similarity distribution, insight on

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the relationships of isovalues and the corresponding struc-tures can be gained. While manual analysis is useful and un-avoidable for many exploratory tasks as the intent of the useris not known, the highly structured nature of the isosurfacesimilarity map also provides us with means to automaticallyidentify relevant isovalues. Regions of high similarity man-ifest themselves as distinct squares in the isosurface simi-larity map. We developed a simple algorithm which allowsus to automatically identify these regions and, based on theirsimilarity distributions, select the most representative isoval-ues for each of them:

Step 1 – Our aim is to identify representative isovalues, i.e.,values with high similarity to many other values. Initially,the value with the highest average similarity to all others isidentified. Then, the maxima of the similarity distributionfor only the values below and above this value are cho-



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Figure 5: Representative isovalue selection algorithm applied to a CT data set. The isosurface similarity map is shown on theleft and the six most representative isovalues are marked. The corresponding isosurfaces are depicted in the middle sectionnumbered from one to six with decreasing relevance. The image on the right shows a cutaway view of the data set classifiedaccording to maximum similarity with the six isovalues.

sen, one so on. Thus, our strategy recursively partitionsthe set of isovalues V by selecting the maximum of thesimilarity distribution for the current subset. The chosenisovalue m is inserted into a priority queue Q based on itssimilarity distribution value weighted by the number ofisovalues in the current subset. The following procedureprioritize(Q,V ) summarizes these operations:

m ← argmaxhi∈V

SDV (i)

p ← |V |SDV (m)

enqueue(Q,m, p)

V1 ← {hi ∈V : 1≤ i < m} (15)

prioritize(Q,V1)

V2 ← {hi ∈V : m < i≤ |V |}prioritize(Q,V2)

Step 2 – Next, we remove the isovalue hm with maximumpriority from the queue. To prevent similar values frombeing chosen, all remaining entries are penalized basedon their similarity with the selected value in the followingmanner:

pi←pi

1+SMV (hm,hi)(16)

where pi is the priority of the i-th item in the queue and hmis the selected value with the highest priority. This processrepeats until no more items remain in the queue.

This simple algorithm results in a reordering of the isoval-ues. Early values in the resulting order have high similaritywith many other isovalues, i.e., they represent a certain rangeof isovalues well, but low mutual similarity meaning thatthey are likely to correspond to distinct structures. One ma-jor advantage of this approach is that it does not require anykind of threshold or parameter. The user can simply exam-ine the isosurfaces in the order generated by the algorithm.After the first few isovalues corresponding to distinct fea-tures of the data set, subsequent values will typically be less

Figure 6: Automatically classified CT data set using theeight most representative isovalues.

representative values for the same structures as no furtherdissimilar values can be found.

An example is shown in Figure 5. The six most repre-sentative isovalues of a CT data set determined using thedescribed algorithm are marked in the isosurface similaritymap and the corresponding isosurfaces are shown numberedfrom 1 to 6 with decreasing relevance. While the first threevalues correspond to distinct structures, the remaining onesonly partially segment these features. The righthand side ofthe figure depicts a cutaway view where each voxel is clas-sified according to its maximum similarity with any of thesix isovalues – as the first three isovalues exhibit more sim-ilarity, the three less relevant isosurfaces are not visible. Afurther result for the classification of CT data using the mostrepresentative isovalues is shown in Figure 6. In CT datasets, there is a clear correspondence between isovalues anddifferent tissue types. Even though other types of volumedata do not exhibit the same kind of relationships and are



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Figure 7: Representative isovalue selection algorithm applied to an MRI angiography data set. The isosurface similarity mapis shown on the left and the eight most representative isovalues are marked. The corresponding isosurfaces are depicted in themiddle section numbered from one to eight with decreasing relevance. The image on the right shows a cutaway view of the dataset classified according to maximum similarity with the eight isovalues.

therefore difficult to classify based on isovalues alone, ourapproach can still identify salient structures in these cases.Figure 7 shows an MRI data set and its eight most represen-tative isovalues as identified by our method. In all depictedexamples the only manual interventions were specificationof the viewpoint and clipping.

5. Implementation

Our tool for the generation of isosurface similarity maps wasimplemented in C++. The computation process involves twosteps. First, the distance transforms for all isovalues are com-puted. As this can, depending on the resolution, require asubstantial amount of space they are immediately written todisk in compressed form. In the second step, we computethe mutual information of the distance transforms for eachpair of isovalues. Since mutual information is a symmetricmeasure, only half of the combinations need to be evaluated.The computation is performed by generating the joint his-togram of the two distance transforms which allows estima-tion of the joint and marginal entropies as described in Sec-tion 3.1 using the CUDA-based implementation of Shamsand Barnes [SB07].

6. Discussion

In our experiments we found that isosurface similarity mapsprovide a concise overview of a data set. Distinct featuresmanifest themselves as squares and their size informs theuser about the corresponding value range. Transitional re-gions can be detected through the nesting structure of thesesquares. In contrast to histograms, large regions do not tendto dominate the depiction. Due to the choice of mutual infor-mation as a similarity measure, uncorrelated noise has littleinfluence. These properties indicate that the presented con-cept has the potential to improve the visualization and anal-ysis of volume data even beyond the examples shown in thispaper.

One obvious disadvantage of our approach is the consid-erable cost of generating the isosurface similarity map. Ourimplementation can require several hours of processing time.Even though this is a one-time preprocess, we consider thisfact a serious limitation of our current method. The mostperformance-critical component is the construction of thejoint histogram of two distance transforms since it has to beperformed for each pair of isovalues. In order to reduce thecomputation time, we performed experiments with down-sampled distance transforms. Interestingly, it seems that theresolution can be considerably reduced without substantialchanges in the resulting similarity map. Figure 8 comparessimilarity maps computed from distance transforms at sev-eral resolutions. Note that the distance transforms are gener-ated at the original resolution of the data set and then down-sampled as opposed to computing them from a downsampledversion of the volume. We believe that the reason for thisstability is that the distance transform captures the uniquecharacteristics of an isosurface even at reduced resolutions.Since it is not used to perform accurate spatial comparisonsbut rather as a shape descriptor, we consider downsamplinga viable strategy. Table 1 lists the distance transform resolu-tions and similarity map computation times for all data setsused in the paper. On a typical notebook, generation of theisosurface similarity map using a distance transform resolu-tion of approximately 64× 64× 64 takes about 25 minutesfor standard data sets. Throughout the paper, we used a fixednumber of 128× 128 bins in the computation of the jointhistogram.

Even though lowering the resolution of the distance trans-form dramatically reduces the computation time to a level weconsider acceptable, our generation method is still a brute-force approach. For a more fundamental improvement, onepotential direction is the use of a different method for jointprobability density estimation. While joint histograms arewidely used, other methods are gaining increasing recog-nition. It may even be possible to use an alternative ap-



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(c) 64×64×57 (d) 32×32×28

Figure 8: Comparison of isosurface similarity maps com-puted with different distance transform resolutions for thedata set shown in Figure 5. The total compuation times (inminutes) were (a) 569.1, (b) 35.8, (c) 22.1, and (d) 20.6.

proach which does not require explicit computation of thedistance transform. Interesting recent work by Rajwade etal [RBR09] points in this direction and remains to be ex-plored. Additionally, an adaptive strategy for approximat-ing the full similarity map could also be employed. Whileour current method for generating isosurface similarity mapscan be considered a proof-of-concept, we have shown thatthey have useful applications for the visualization of volumedata. There are several other areas, however, where the pro-posed concept may be of interest. In closing, we would liketo briefly list some avenues which could be promising direc-tions for further exploration:

Volume quantization and compression. The notion ofsimilarity may be useful in developing new approachesfor quantizing and/or compressing volume data withhigher fidelity. Isosurfaces which exhibit a high degreeof redundancy could be grouped together while stillpreserving essential features in the data set.

Volume segmentation. Isosurface similarity could alsobe employed as an alternate metric for segmentationalgorithms such as region growing. These methodstypically use similarity criteria based on the differencebetween isovalues, so our measure may help to increaserobustness.

Multi-dimensional classification. While there is noth-

Figure(s) Orig. Resolution DT Resolution Time2 512×512×361 64×64×45 23.9

3, 4 512×512×333 64×64×41 21.85 256×256×230 64×64×57 22.16 512×512×361 64×64×45 23.77 512×512×125 64×64×15 21.6

Table 1: Depicting figure, original data set resolution, res-olution of the downsampled distance transform, and totalcomputation time (in minutes) for the isosurface similaritymaps used in the paper are listed. System configuration: In-tel Core 2 Duo 2.53 GHz CPU, 4 GB RAM, NVidia GeForce9600M GT GPU.

ing in our approach that prevents combination withmulti-dimensional classification approaches using gradi-ent magnitude [KKH02], curvature [KWTM03], occlu-sion [CM09a], or other measures proposed in the litera-ture, we did not investigate this area. Indeed, as the iso-value typically represents one axis in multi-dimensionaltransfer function schemes, similarity could help to betterseparate features and to decrease the influence of noise.

7. Conclusions

In this paper, we introduced the notion of isosurface similar-ity for the visualization of volume data. This new measurequantifies the similarity of two isosurfaces as the normalizedmutual information of their respective distance transforms.The resulting isosurface similarity map provides a visual-ization of these similarities and gives an overview of a dataset which complements traditional depictions. Additionally,the similarity map can be used to improve rendering and pa-rameter specification. Its structured nature enables automaticdetection of representative isovalues to assist the explorationprocess. The presented concept opens up several interestingdirections for future investigation.

Acknowledgements

The work presented in the paper was funded by the Natural Sciencesand Engineering Research Council of Canada (NSERC). The datasets used in Figures 2, 5, and 6 are courtesy of the OsiriX Foundation(http://www.osirix-viewer.com).

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