Segmental Wall Motion Classification in Echocardiograms Using Compact Shape Descriptors

Segmental Wall Motion Classification inEchocardiograms Using Compact

Shape Descriptors1

K. Y. Esther Leung, MSc, Johan G. Bosch, PhD

Rationale and Objectives. Parametric shape representations of endocardial contours, obtained with principal compo-nent analysis (PCA) and the orthomax criterion, provide compact descriptors for classifying segmental left ventricu-lar wall motion.

Materials and Methods. Endocardial contours were delineated in the left ventricular echocardiograms of 129 patients.Parametric models of these shapes were built with PCA and subsequently rotated using the orthomax criterion, producingmodels with local variations. Shape parameters of this localized model were used to predict the presence of wall motionabnormalities, as determined by expert visual wall motion scoring.

Results. Best results were obtained using the varimax criterion and full variance models. Although traditional PCAmodels needed 8.0 � 3.0 parameters to classify segmental wall motion, only 5.1 � 3.2 parameters were needed us-ing the orthomax rotated models (P � .05) to achieve similar classification accuracy. The classification space wasalso better behaved.

Conclusions. Orthomax rotation generates more local parameters, which are successful in reducing the complexity of wallmotion classification. Because pathologies are typically spatially localized, many medical applications involving local clas-sification should benefit from orthomax parameterizations.

Key Words. Cardiac ultrasound; computer-aided diagnosis; orthomax rotations; principal component analysis; wall motionclassification.© AUR, 2008

Coronary artery diseases are a major cause of death in theWestern world. Detection of wall motion abnormalities ofthe left ventricle, widely accepted as predictors for thesediseases, is therefore of great clinical importance. Echo-cardiographic examination is often used for diagnosing

Acad Radiol 2008; 15:1416–1424

1 From Biomedical Engineering, Thoraxcenter, Office Ee 2302, P.O. Box2040, 3000 CA, Rotterdam, The Netherlands (K.Y.E.L., J.G.B.). ReceivedApril 4, 2008; accepted July 1, 2008. This research is supported by theDutch Technology Foundation STW (grant 06666), applied science divisionof NWO and the Technology Program of the Ministry of Economic Affairs,The Netherlands. Address correspondence to: K.Y.E.L. e-mail: [email protected]

©
AUR, 2008doi:10.1016/j.acra.2008.07.005
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these wall motion abnormalities, because of the speed ofacquisition, the high spatial and temporal resolution, andthe relatively low cost of the technique. A well-estab-lished method is stress echocardiography, which comparesthe left ventricular (LV) wall motion in images acquiredat different stages of stress (1). However, because theimages are often evaluated visually, a quantitative andobjective measure of wall motion is still lacking. To ob-tain such quantitative measures, automated analysis of LVwall motion may be preferred to currently visual, andtherefore subjective, assessments.

Various quantitative measures have been proposed toassess wall motion in the literature (2). Methods such as
acoustic quantification and color kinesis (3) observe the
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mailto:[email protected]

Academic Radiology, Vol 15, No 11, November 2008 SEGMENTAL WALL MOTION CLASSIFICATION

backscatter of the ultrasonic signal to measure the endo-cardial motion. Tissue Doppler imaging (4) generatesmeasurements of velocity, displacement, and strain/strain–rate values. These values can also be obtained using im-age-based speckle-tracking methods (5), which are be-coming increasingly popular. Other image-based methodsuse manual or automated delineations of the endocardialborder to assess, for example, regional volumes, whichmay be of help in quantifying wall motion (6).

Methods have also been proposed to automaticallydetect wall motion abnormalities on echocardiograms.Assmann et al proposed to measure wall motion usingregional ejection fraction, area reduction, and fractionalshortening, based on endocardial contours (7). Frouin etal proposed factor analysis to separate the image-intensitytime curves into a “constant” and a “contraction-relax-ation” factor, to distinguish between different levels ofwall motion (8). Aoued et al (9) applied principal compo-nent analysis (PCA) to strain and strain–rate time curves.A distance measure was proposed, which expresses thedistance between a patient’s time curve and a model ofhealthy subjects. A special classifier was developed byFung et al to distinguish between normal and abnormalhearts (10). Previously, we have suggested using point-distribution models of contour sequences in two-dimen-sional echocardiograms for automated classification ofwall motion abnormalities (11).

The goal of this study is to evaluate a new automatedclassification approach for detecting local wall motion abnor-malities. Building on our previous work, the wall motionparameters are derived from contour sequences of the leftventricle (11). Point-distribution models with localized varia-tions are obtained with orthomax rotations. The parametersare used to classify segmental wall motion.

MATERIALS AND METHODS

Stress Echo and Visual Wall Motion ScoringStress echo is a commonly used diagnostic technique

for assessing LV dysfunction and underlying coronarydisease (1). This technique consists of acquiring echocar-diographic images of the left ventricle when the patient isat rest and at stress (maximum workload for the cardiacmuscle). By studying the regional motion patterns of theLV wall, myocardial tissue functionality can be diag-nosed. Deteriorating contractility in one or more segmentsin stress is a sign of local ischemia, associated with a
stenosis in the corresponding coronary artery.
The wall motion is assessed visually by assigning aqualitative score to each segment. Different scoring sys-tems are in use; the data in this study use a four-pointsystem (0: normokinesia, 1: hypokinesia; 2: akinesia; 3:dyskinesia) and 13 segments (12). Because each qualita-tive score is associated with a numeric value, semiquanti-tative results can be calculated, such as the total score ofall segments. However, because the scoring is performedvisually, the work is time-consuming and the scoring maysuffer from intra- and interobserver variabilities (13). De-velopment of an automatic method for classifying wallmotion, which emulates visual wall motion scoring, istherefore highly desirable.

Analysis of Endocardial Contours via ShapeModels

Previously, we have proposed using shape models forclassifying LV wall motion automatically (11). Shapemodels, or point-distribution models, are parametric repre-sentations of a set of shapes (14). These models are usu-ally built using PCA. This technique generates a statisti-cal representation of the global shape variations encoun-tered in the input data (eg, LV endocardial contours of aset of patients). This allows a shape, represented by spa-tial point coordinates concatenated in a vector x, to beaccurately approximated by a limited number of shapeparameters, or modes, concatenated in a vector b:

x � x � �b, (1)

where x is the average shape, and � is the eigenvectormatrix. Typically, the number of shape modes is similarto the number of input training samples. Any new shapecan be projected to this model using the pseudoinverse(��1) of the eigenvector matrix: b � ��1�x � x�.

The modeling technique was extended to time sequencesof echocardiograms (15). In this way, the model comprisedvariations in LV wall motion patterns across a group of pa-tients. The complex motion pattern of the left ventricle couldthen be accurately described with these shape modes.

These shape modes were then used to classify globalclinical parameters (eg, LV volume) and local parameters(eg, visual wall motion scores) (11). Although clear corre-lations were found, relatively many shape modes wereneeded to classify these local wall motion scores accu-rately. This is because the PCA technique inherently gen-erates models of global variations, whereas for classifying
local parameters, local variations are desired. Therefore,
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LEUNG AND BOSCH Academic Radiology, Vol 15, No 11, November 2008

we hypothesized that models with local variations aremore compact representations of local wall motion. Fewermodes should be needed to classify the wall motion withthe same degree of accuracy.

Orthomax RotationsRecently, a method called orthomax rotations has been

suggested to obtain local models from the global PCAmodel (16). The main advantages of this technique are:(1) computational feasibility in high-dimensional spaces,as often is the case for shape models; (2) automatic local-ization with little fine-tuning of method necessary; and (3)availability of implementation in a variety of statistical pack-ages (SPSS, SPSS Inc, Chicago, IL; SAS/STAT, SAS Insti-tute, Cary, North Carolina; Matlab, The MathWorks Inc.,Natick, MA).

Orthomax rotations are basically reparameterizations ofthe PCA model space (17). The PCA eigenvectors, whichact as orthogonal axes in a high-dimensional space, arerotated so that the axes are more “sparse.” This can bestbe understood from a visual example (Fig 1): rotation ofthe PCA axes results in a matrix with a minimal numberof nonzero elements. This implies that the variation in thepositions of certain shape points (part of x) will dependmore on the variation of only one model parameter (oneelement of b). In other words, varying one model parame-ter leads to a localized variation.

In mathematical terms, the orthomax rotation seeks tofind a rotation matrix R, so that the following criterion ismaximized:

� ��j�1

k

�i�1

n

Gij4 �

�

n�j�1

k ��i�1

n

Gij2�2� ⁄ n, (2)

where Gij denotes the scalar element in the ith row and jth

column in the rotated eigenvector matrix G � �R, and �

is the orthomax type. The shape coefficients after rotationbR can be found with bR � R�1b.

The orthogonal orthomax criterion is equivalent to theCrawford-Ferguson criterion, which is a weighted sum ofrow and column complexity of the eigenvector matrix(18). Therefore, orthomax rotations can be interpreted asa redistribution of the elements of the eigenvector matrixso that each row or column has a minimal number ofnonzero elements, as can be observed in Figure 1. Thetwo extremes are quartimax (� � 0), favoring row spar-sity, and factor parsimony (� � n), which favors column
sparsity (17). Varimax (� � 1), a commonly used type,
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resides somewhere in between (19). In practice, completerow or column sparsity cannot be achieved because theshape model is restricted to the observed, physically al-lowed variations in the training samples.

Because PCA orders the shape modes automatically ac-cording to variance, modes with low variation generally con-tain noise. Eliminating some of these modes may lead tomore representative local variations in the rotated shapemodel. However, if too many modes are removed, subtlevariations in wall motion may be lost; thus, the accuracy ofclassification may be reduced. Therefore, we investigate howdifferent proportions f of the total variance V (sum over all peigenvalues) affect the classification:

�i�1

k

�i � fV, (3)

where k denotes the number of eigenvectors with thelargest eigenvalues �i. Modes with low eigenvalues,corresponding with the rightmost columns of the eigen-vector matrix �, are removed before the orthomax ro-tation.

Orthomax rotation was applied to the four- and two-chamber shape models, using an iterative method basedon singular value decomposition (16), as implemented in

Figure 1. Top row, Eigenvector matrixes of principal componentanalysis (PCA) and orthomax rotated shape model of the four-chamber. Bottom row, Shape variations of the fifth mode, show-ing localization of variation for the rotated model.

Matlab (20).


Clinical Data and Contour DelineationThe effect of orthomax rotations on wall motion classifi-

cation was demonstrated on low-dose dobutamine stressecho data from 129 unselected infarct patients (11,12). Fromall patients, the two-dimensional transthoracic apical four-and two-chamber sequences from the resting stage wereavailable for shape modeling (Fig 2). Each time sequencewas normalized to 16 frames, of which the first and lastframe corresponded to end-diastole and the ninth frame toend-systole. Endocardial borders were delineated using asemi-automated tracing program (ECHO-CMS system,MEDIS Medical Imaging Systems, Leiden, The Netherlands)(21), independent of the visual wall motion scoring. Thecontour in each frame was modeled by 37 points.

For training of the PCA model, the x and y coordinatesof these contour points of all frames were concatenated inx. This was carried out for all training patient data sets.These vectors were the input to the PCA.

Wall Motion ClassificationTwo classification experiments were performed, denoted

as the “TRN L-1-O” and the “TST” situation. The total dataset was split randomly into a training set (TRN) of 65 pa-tients and a testing set (TST) of 64 patients. Shape modelsof the four- and two-chamber were built with the trainingset, as in our previous work (11). Shape parameters b and bR

were calculated for all data sets. In the “TRN L-1-O” case, aleave-one-out approach was used, where the classifier wastrained on the TRN set except for one sample and tested withthat sample. This process was then repeated for all TRN sam-ples. In the “TST” situation, the classifier was trained on thewhole TRN set and then tested on all TST cases. This resem-bled classification in the real world: both shape model and clas-sifier were trained with a limited training set and tested on com-

Figure 2. Left ventricular (LV) segments in four- and two-cham-ber views. LAD, left anterior descending artery; LCX, left circum-flex artery; RCA, right coronary artery.

pletely “new” shapes.

For single segments and combinations of two to fivesegments, a distinction was made between normal(summed score � 0) and abnormal (summed score � 0)motion. For combinations of more than five segments, inwhich many scores were summed, this distinction wouldresult in very biased classes (because the patient set con-tained only infarct patients and no normals). Therefore, adistinction was made between mild (summed score �3)and severe (summed score �3) wall motion abnormali-ties. This distinction in scores was used as the response(dependent) variables in the classification.

To classify the wall motion abnormalities, linear dis-criminant analysis was used. This classifier searches for alinear combination of the shape parameters that providesthe best discrimination between classes (22). In the pa-rameter space spanned by the shape parameters, this dis-criminant can be observed as a hyperplane, separating thenormal motion and abnormal motion classes: �i�1

qibi �

0 � 0, where q is the number of shape parameters in theclassification space. Linear discriminant analysis was per-formed using the statistical package SPSS (v. 11.0.1,2001). Shape parameters were added automatically by theclassifier using the “stepwise” option and the “unex-plained variance” criterion, so that an optimal subset q ofthe k shape parameters were selected that best discrimi-nate between normal and abnormal motion.

To investigate whether the normal and abnormalclasses were better separated in the orthomax parameterspace than the original PCA space, cluster measures werecomputed. After the classifier has selected a subset q ofthe k shape parameters that best predicts normal or abnor-mal wall motion, these q parameters of a particular pa-tient can be seen as a point in the q-dimensional classifi-cation space. Ideally, for a linear classifier such as lineardiscriminant analysis, points of the normal class shouldform a compact cloud (or cluster), completely separatedfrom the point cluster of the abnormal class. A commonmeasure of cluster compactness is the within-class scattermatrix SW, whereas the between-class scatter matrix SB isoften used to describe cluster separation. A measure ofoverall cluster quality is the ratio J of the trace of the twoscatter matrixes: J � tr(SB)/tr(SW). (22)

RESULTS

Orthomax RotationsOrthomax rotations were applied to four- and two-

chamber shape models. Although PCA shape modes are

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s loc

ape p


ordered according to variance, thus exhibiting global vari-ations in the first modes, orthomax modes show localvariations in most modes (Fig 3). Because the whole car-diac cycle was modeled, the variations were localizedalong the spatial as well as the temporal extent.

Orthomax CriteriaShape parameters were used to predict the presence of

Figure 3. Global principal componentmodes of the four-chamber model. Absparameter variation are shown on the atained variance (f ), the variations are les

Table 1Classification Accuracy of Individual SegmenUsed (mean � SD) for Different Orthomax Cr

fClassificat

TRN

PCA 99.9% 88.9Quartimax 99.9% 90.1Factor parsimony 99.9% 89.4Varimax 99.9% 91.1

99% 88.998% 87.795% 86.3

PCA, principal component analysis; TRN, traif denotes different proportions of retained var*Significantly (P � .05, paired t-test) fewer sh

wall motion abnormalities for each individual LV seg-

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ment. Significantly fewer orthomax modes were neededthan PCA modes, without compromising classificationaccuracy (ie, the proportion of segments correctly classi-fied as normal or abnormal) (Table 1). The varimax crite-rion needed the least number of modes.

Proportion of Retained VarianceWe investigated the effect of using different propor-

sis (PCA) and localized orthomaxdisplacements resulting from �3 SDe shapes. For lower proportion of re-alized.

rsus the Number of Shape Parameters, Averaged over Nine Segments

ccuracyParametersO TST

% 74.0 � 9.4% 8.0 � 3.0% 75.4 � 9.8% 5.6 � 3.9*% 76.3 � 10.3% 5.4 � 3.2*% 76.5 � 10.5% 5.1 � 3.2*% 76.0 � 8.7% 5.7 � 3.3*% 75.8 � 9.5% 6.4 � 3.5% 76.3 � 9.0% 6.6 � 3.7

set; TST, testing set.in the shape models.

arameters than PCA.

analyoluteverag

ts veiteria

ion AL-1-

� 5.9� 5.2� 5.7� 4.5� 5.9� 7.0� 6.1

ningiance

tions of retained variance in the shape model (Eqn. 3).

eters


Figure 3 shows the motion patterns of models with differ-ent proportions. Interestingly, a lower f results in less lo-calized shape variations, because each mode must capturemore variation.

Results for classification accuracy are shown in Ta-ble 1. Proportions of f � 95%, 98%, 99%, and 99.9%were investigated, corresponding to k � 27, 40, 48,and 63 modes in the four-chamber model and to k �

Table 2Classification Accuracy in Multiple Segments versus the Numb

View Segment (n) Normal

P

Classification

TRN L-1-O

4C�2C All (9) 25.6% 89.2%4C Total 4C (5) 22.5% 87.7%2C Total 2C (5) 6.2% 100%4C Septal (2) 29.5% 100%4C Lateral (2) 49.6% 81.5%2C Anterior (2) 61.2% 93.8%2C Inferior (2) 30.2% 93.8%2-segment combinations:

Mean 42.6% 92.3%SD 15.5% 7.8%

2C, two-chamber; 4C, four-chamber; PCA, principal componentn denotes the number of segments combined.The percentage of normal motion of all datasets is also listed.

Table 3Classification Accuracy in Individual Segments versus the Num

View Segment Normal

PCA f �

Classification Accur

TRN L-1-O TST

4C�2C Apical 34.9% 93.8% 81.3%4C Septal Basal 59.7% 86.6% 70.3%4C Septal Mid 42.6% 92.3% 73.4%4C Lateral Basal 84.5% 86.2% 64.1%4C Lateral Mid 51.2% 76.9% 69.8%2C Anterior Basal 97.7% 95.4% 95.3%2C Anterior Mid 61.2% 93.8% 74.6%2C Inferior Basal 42.6% 84.6% 71.9%2C Inferior Mid 39.5% 90.8% 65.6%

Mean 57.1% 88.9% 74.0%SD 21.4% 5.9% 9.4%

2C, two-chamber; 4C, four-chamber; PCA, principal componentThe percentage of normal motion of all datasets is also listed.*Denotes significantly (P � .05, paired t-test) fewer shape param

25, 38, 46, and 62 modes in the two-chamber model.

Models with higher f needed fewer shape modes duringclassification.

Segmental ClassificationThe classification results for combinations of seg-

ments and for each individual segment are given inTables 2 and 3 for the varimax criterion. For combina-tions of two segments, the reduction in the number of

PCA and Varimax Shape Parameters Used

� 99.9% Varimax f � 99.9%

curacy # Parameters Classification Accuracy # Parameters

ST TRN L-1-O TST

.3% 11 100% 70.9% 47

.1% 13 96.9% 81.3% 18

.9% 21 100% 96.9% 35

.2% 28 92.3% 64.1% 24

.3% 10 90.8% 70.3% 8

.4% 11 93.8% 75.0% 14

.0% 14 98.5% 76.6% 14

.5% 15.8 93.9% 71.5% 13.8

.5% 8.3 3.3% 5.6% 7.3

sis; TRN, training set; TST, testing set.

of PCA and Varimax Shape Parameters Used

Varimax f � 99.9%

# Parameters Classification Accuracy # Parameters

TRN L-1-O TST

7 95.4% 85.9% 610 90.8% 64.1% 512 92.3% 73.4% 118 90.8% 82.8% 26 81.5% 71.4% 23 96.9% 96.9% 2

11 93.8% 76.2% 95 90.8% 71.9% 5

10 87.7% 65.6% 48.0 91.1% 76.5% 5.1*3.0 4.5% 10.5% 3.2

sis; TRN, training set; TST, testing set.

than PCA.

er of

CA f

Ac

T

70789667707375

713

analy

ber

99.9%

acy

analy

classification parameters was not statistically signifi-

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ent in

r the


cant. However, for the single segments, significantlyfewer modes were needed.

Figure 4 shows the cluster quality J in PCA and vari-max space. Clearly, varimax rotation resulted in betterdefinition of the classification space, as shown by highervalues J for most segments.

The actual modes used for classifying the individualsegments are depicted in Figure 5. As expected, many

Figure 4. Cluster quality J is higher for(PCA) classification space, meaning bettetained variance f � 99.9%, the improvem

Figure 5. Histogram of the modes useMany principal component analysis (PCvalues; no clear relation can be seen fo

PCA modes used for classification had large eigenval-

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ues, which corresponded with large global variation.For the orthomax rotated modes, no clear relationcould be seen.

DISCUSSION

Localized shape models of the left ventricle, generated

ax than principal component analysister separation. *For proportion of re-J is statistically significant (P � .05).

classification of individual segments.odes corresponded with large eigen-varimax rotated modes.

varimr clus

d forA) m

using orthomax rotations, were more effective in classify-


ing local wall motion. Significantly fewer parameterswere needed for classifying segmental wall motion intwo-dimensional echocardiographic sequences while pre-serving classification accuracy.

Orthomax CriteriaSimilar results were obtained for the quartimax, factor

parsimony, and varimax criteria, probably because thereparameterizations were restricted by the allowable varia-tions in the training set, rather than the actual criterion.Similar modes can be found for each criterion. This sug-gests that the criterion used has only a minor influence onthe localization. Varimax is a commonly used criterionand often implemented in statistical packages, making itdesirable for use.

Proportion of Retained VarianceBest results were obtained using all PCA modes before

the orthomax rotation. More modes in the original PCAmodel means more degrees of freedom for the rotation.As a result, the rotated basis is sparser, so the variationsare more localized (Fig 3). Thus, fewer parameters areneeded to be combined to classify an individual segment.

Segmental ClassificationFor classification of five or more segments, more or-

thomax parameters are used than PCA parameters. This isas expected, because the more localized modes must becombined to explain global variations. However, whenclassifying combinations of two segments, a slight de-crease in the number of parameters can already be ob-served. Localization of model modes results in muchmore subtle motion patterns, which are more meaningfulfor local wall motion classification.

It is important to keep in mind that the orthomaxmethod automatically generates sparse representations,given the variations in the training set. The relativelylarge improvement in the number of used modes in thelateral region may be due to the combination of the ortho-max rotation and these training variations, which bychance produced a sparser parameter representation inthose segments. The same holds for segments in whichthe improvement was smaller (eg, the septal mid-seg-ment).

The fact that the used orthomax modes did not corre-late with variance suggests that better methods may beneeded to categorize these modes (Fig 5). Recently,
Suinesiaputra et al suggested that modes may be ordered
locally, which is of particular interest for local classifica-tion (23). This is a subject of further investigation.

Visual Wall Motion Scoring and AlternativesThe absence of a true gold standard for diagnosing

cardiac disease is a limitation of this study. Stress echo isnoted for its variability in visual scoring (13). Regretta-bly, this variability could not be determined for this dataset because the scoring was performed by consensus.

More objective measures as gold standard would behelpful in evaluating the orthomax method. For example,quantitative deformation parameters, derived from tissueDoppler imaging (4), or strain/strain–rate of speckletracking techniques (5), can be used. A step further wouldbe to relate quantitative coronary angiography data withthe orthomax shape modes. This is more challenging, be-cause shape modes are related to the myocardial contrac-tility, and not directly to coronary artery disease.

Limitations of Study SetupAlthough the goal of this study is to show the com-

pactness of the orthomax representation, it will be inter-esting to see if the classification accuracy actually im-proves with the sparse approach. To draw solid conclu-sions, many more patients need to be analyzed. Also, thechoice of classifier may impact the performance. Thecluster quality J (Fig 4) revealed that there is still a rea-sonable amount of overlap of the classes, suggesting thatbetter classification may be achieved with a nonlineardiscriminant function. More advanced feature selectionmethods and nonlinear classifiers, such as support vectormachines, may lead to better classification accuracy. Thisis a subject of further investigation.

The proposed method could be directly applied to do-butamine stress images, rather than the rest images alone.Also, the parasternal short- and long-axis images were notavailable for modeling. Combined models for rest andstress, or models of differences between these two, arealso an option to explore (24). Finally, with the recentinterest in three-dimensional stress echocardiography(25,26), orthomax rotations should be useful in that ter-rain.

Orthomax Extensions and AlternativesAlthough orthomax is particularly interesting in terms

of computational efficiency, it would be interesting tocompare the proposed orthomax method with other local-ization methods, such as independent component analysis
(27) and sparse PCA methods (28,29). Alternatives that
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make use of the intensity patterns, instead of shapes,might also be interesting (8).

Although not explored in this study, sparse texturemodels can be constructed in a very similar manner (16).The direct cardiac application would be to examine localmyocardial thickening, which might also be a predictor ofcoronary disease. Such texture models might be moresuitable for image modalities such as magnetic resonanceimaging rather than echocardiography, which suffers fromspeckle noise.

CONCLUSIONS

In this study, we showed that classification of local leftventricular wall motion is feasible using local shape mod-els, obtained with orthomax rotations. Because of the lo-calization of shape variation, the classification needsfewer parameters to obtain similar accuracy with respectto normal global models.

Because pathologies are typically spatially localized,we anticipate many medical applications in which sparserepresentations are preferred to the conventional PCAapproach. The orthomax criterion is shown to be suitablefor building these sparse representations with relativeease. Researchers interested in local modeling are encour-aged to explore the technique and use it for their ownapplication.

ACKNOWLEDGMENTS

We thank Otto Kamp and Francisca Nijland for pro-viding the patient data, Gerard van Burken for the dataanalysis support, and Marijn van Stralen, Marco M. Voor-molen, Milan Sonka, and Mikkel B. Stegmann for theuseful discussions.

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