Posture-Invariant Statistical Shape Analysis Using …people.scs.carleton.ca/~c_shu/pdf/shape_variation_Laplace_SMI12.pdf · Posture-Invariant Statistical Shape Analysis Using Laplace

Posture-Invariant Statistical Shape Analysis Using Laplace Operator

Stefanie Wuhrerb,c, Chang Shua, Pengcheng Xia

aNational Research Council of CanadabSaarland University, Germany

cMax Plank Institute Informatik, Germany

Abstract

Statistical shape analysis is a tool that allows to quantify the shape variability of a population of shapes. Tradi-tional tools to perform statistical shape analysis compute variations that reflect both shape and posture changessimultaneously. In many applications, such as ergonomic design applications, we are only interested in shape vari-ations. With traditional tools, it is not straightforward to separate shape and posture variations. To overcome thisproblem, we propose an approach to perform statistical shape analysis in a posture-invariant way. The approachis based on a local representation that is obtained using the Laplace operator.

1. Introduction

Statistical shape analysis is a tool that allows toquantify the shape variability of a population ofshapes. This tool helps to gain a thorough understand-ing of a population of shapes, and it has various ap-plications, such as designing products for a specifictarget population, computing shape priors of a popu-lation to aid in solving statistical inference problems,and comparing different populations.

For applications that require a user to understandcertain properties of a population, it is helpful to visu-alize the shape variations of the population. An exam-ple of such an application is the design of garments orgear for a target population of human subjects.

In most ergonomic design applications, we areinterested in computing the shape variations of adatabase of shapes in a standard posture. Anthropo-metric surveys are conducted to aid the study of shapevariations of a population sample. However, evenwhen subjects are instructed to maintain a standardposture, there are slight posture variations, especiallyin the position of the arms. For example, this occursin the Civilian American and European Surface An-thropometry Resource (CAESAR) database [1].

In some ergonomic design applications, it is impor-tant for the product to fit a typical subject of the pop-ulation even when the subject moves. Hence, for thisapplication, we are interested in computing the shapevariations of a database of shapes captured in varyingpostures.

A popular approach to perform shape analysis is torepresent the shapes using their vertex coordinates andto analyze these shapes using approaches such as prin-cipal component analysis. This approach has the ad-vantage that reconstructing a shape based on its vertex

coordinate representation is straightforward, which al-lows an intuitive visualization of the shape variabil-ity. However, for a population of shapes in differentpostures, approaches operating on vertex coordinatescompute variations that reflect both shape and pos-ture changes of the models simultaneously. It is notstraightforward to extract only shape variations usingthese approaches.

To overcome this problem, we propose a methodto perform statistical shape analysis of a populationof shapes in varying postures in a posture-invariantway, and to visualize the resulting shape variationsin a standard posture. To achieve this, we developa posture-invariant shape representation based on theLaplace operator, and we perform statistical shapeanalysis using this representation. Since we aim to vi-sualize the shape variability of a population, this shaperepresentation needs to allow for the reconstruction ofa shape based on its representation. We will show thatcomputing the representation from a given model isstraightforward. Furthermore, we will present a solu-tion to the more challenging problem of computing amodel from the representation.

2. Related Work

To perform statistical analysis on a population ofshapes, we require intrinsic correspondence informa-tion between the shapes. Much work has been con-ducted to compute correspondences between pairs andpopulations of shapes, see [2] for a recent survey onthe topic.

Once the database of models is in correspondence,corresponding vertices on different models have thesame index. This correspondence information is

Preprint submitted to Computer & Graphics March 12, 2012

then commonly used to find the best rigid align-ment of the shapes using generalized Procrustes anal-ysis [3]. After this step, a common representation isto model each shape using a vector of vertex coor-dinates

[x1, y1, z1, . . . , xn, yn, zn

]T , where xi (yi, zi, re-spectively) is the x- (y−, z−, respectively) coordinateof the vertex with index i. In the following, we reviewa few statistical shape analysis tools that are com-monly used in combination with this shape represen-tation.

Principal Component Analysis (PCA) is a standardtool to perform linear shape analysis of a set of shapesrepresented by vectors in a Euclidean vector space.An average object is computed as the linear average ofthe data set. PCA then finds the linear d-dimensionalsubspace of the vector space that maximizes the vari-ability of the model. Here, d is a positive integersmaller than the input dimension. PCA has been ap-plied to compute the shape variation of a populationof human shapes in a standard posture [4]. While thisapproach yields satisfactory results when computingthe shape variation of models in a standard posture,it is not suitable for computing the shape variation ofmodels in varying postures. The reason is that in thiscase, some of the principal components reflect posturechanges of the models.

To allow for posture changes, Fletcher et al. [5] pro-pose an approach called Principal geodesic analysis.This approach aims to simulate PCA in a Riemanniansymmetric space. By using this space rather than Eu-clidean space, posture variations are less strongly em-phasized. While the approach is general, one needsto be able to efficiently compute geodesics to use thismethod. The original method by Fletcher et al. there-fore focuses on shapes represented by a medial axisdescription.

Nain et al. [6] aim to perform localized statisticalshape analysis. That is, rather than looking at globalshape variations, they are interested in localized shapechanges on the surface. To compute these local-ized shape statistics, they employ spherical wavelets.While this tool allows for localized statistical shapeanalysis, it cannot directly be used to overcome theproblems caused by posture changes.

Wang et al. [7] propose an approach called mul-tivariate tensor-based morphometry. This approachmaps each surface to a canonical representation calleda holomorphic form and computes a parameterizationin this canonical space. The method then performsstatistics based on the deformation in the canonicalspace. This approach therefore emphasizes deforma-tions of the models and marks them as unusual. In thispaper, we aim to perform statistics on the shape onlyfor a set of models given in varying postures.

Vasilescu and Terzopoulos [8] propose a nonlinear,multifactor model that generalizes Independent Com-ponent Analysis (ICA). The multilinear ICA learns

the statistically independent components of multiplefactors related to two-dimensional images. They ap-ply the multilinear analysis to sets of facial imagesthat combine different facial geometries, expressions,head poses, and lighting conditions. Allen et al. [4],Anguelov et al. [9], and Hasler et al. [10] propose sta-tistical models to encode human body shape. Allen etal. only encode the shape variation in a standard pos-ture. Anguelov et al. treat human body pose and hu-man body shape independently, and perform PCA oneach space. Hasler et al. propose a representation thatjointly encodes both human body shape and pose andthat is locally translation and rotation invariant. By us-ing this model to perform statistical analysis, humanbody shape and pose are treated as correlated entities.While these methods allow to analyze shape and pos-ture, they cannot be used to compute shape variationsfrom a population of shapes in varying postures.

For human bodies, much research is focused on es-timating the pose, i.e. the articulated skeletal struc-ture, of the human body from partial information, suchas a single image [11]. While for human populations,an estimate of the pose could be used to differentiatebetween shape and posture, our approach operates onthe surface directly without the need for an estimatedpose of the subject.

We propose to use a method based on the Laplaceoperator to perform posture-invariant statistical shapeanalysis. The Laplace operator has previously beenused for mesh smoothing and regularization [12], forsurface editing [13], and for 3D mesh fitting [14]. Fur-thermore, the spectrum of the Laplace operator hasbeen used for mesh processing in a similar way tothe Fourier transforms of images [15], and for shapematching and dissimilarity computation of volumet-ric data [16] and of triangle meshes [17]. Baran etal. [18] used differential coordinates over local con-nected patches of a mesh to define a shape represen-tation that is used to transfer mesh deformations fromone character to another one while preserving the se-mantic meaning.

Our method uses local frames to derive a posture-invariant representation of a surface. Yu et al. [19]used a similar representation for mesh editing. Theirframework allows both local and global mesh editing.The method uses a mesh solver based on the Poissonequation. Local frames are used to represent bound-ary conditions since this allows to encode the differ-ence between an undeformed and a deformed frameuniquely using a rotation and a scale factor.

3. Approach

We propose an approach to perform posture-invariant shape analysis. The input to our algorithmis a database S 1, S 2, . . . , S m of possibly incomplete

2

3D models. We start by parameterizing the shapes us-ing a template-based approach to obtain a set of 3Dshapes X1, X2, . . . , Xm represented by triangular mani-fold meshes with the same connectivity.

The approach then proceeds in three steps. First,the meshes are represented in a posture-invariant way.Second, shape analysis is performed on the posture-invariant representations. Third, posture-invariantrepresentatives are used to reconstruct mesh shapes tovisualize the shape variation.

3.1. Computing CorrespondencesThis section outlines the approach for parameteriz-

ing the shapes S 1, S 2, . . . , S m over a common domain.Computing correspondences between pairs or a pop-ulation of shapes is a challenging problem, especiallywhen the shapes deform in a highly non-rigid way.Furthermore, the accuracy of the correspondence in-formation has a large influence on the quality of theresulting shape variations.

In this work, we focus on statistical shape analysisof human shapes. Several methods compute the cor-respondence information of a set of human models bydeforming a template model to each 3D model of a hu-man subject. Allen et al. [4] compute the correspon-dence information by deforming a template shape us-ing an energy optimization method. This approach re-quires a sparse set of labeled marker positions and forthe subjects of the database to be in a standard posture.Xi et al. [20] improve this approach by adding a steptransforming the template model using a radial basisfunction based on the known marker positions. Hasleret al. [10] extend Allen et al.’s approach to work formodels in varying postures. As before, a set of labeledmarker positions is required for this method.

In our experiments, we use two databases: theCAESAR database [1] and the MPI database [10].The MPI database contains 520 models of 111 dif-ferent individuals in up to 35 different postures.The models are parameterized using the approach byHasler et al. [10] and each parameterized model con-tains 6449 vertices. The markers used for parameteri-zation were placed manually on the scans.

The CAESAR database contains scans of about5000 different models and all of the subjects wereasked to maintain a standard posture. The only pos-ture variation in this database is a slight variation ofthe arm position. The models are parameterized usingthe approach by Xi et al. [20] and each parameterizedmodel contains 10002 vertices. The markers used forthe parameterization were placed by anthropometricexperts prior to scanning the subjects. In our experi-ments, we use 500 models of the CAESAR database.

As these approaches deform a template model toeach scan S i, the parameterized models share the sametopology and the same spatial resolution in corre-sponding areas of the models. Hence, in the follow-

ing, we do not need to consider problems caused bytopological noise or non-isotropic meshes.

3.2. Posture-Invariant RepresentationWe use a method based on the Laplace operator to

find posture-invariant representations of the parame-terized shapes. The approach proceeds by computingthe combinatorial Laplace matrix L of Xk. The com-binatorial Laplace matrix L contains the elements

Li j =

−1, if i = j,

1deg

(v(k)

i

) , if v(k)j ∈ N1

(v(k)

i

),

0, otherwise,

where v(k)i denotes the position vector of a vertex of

Xk, N1

(v(k)

i

)denotes the one-ring neighborhood of

v(k)i , and deg

(v(k)

i

)denotes the number of vertices in

N1

(v(k)

i

).

Since all of the meshes Xk are parameterized, thecombinatorial Laplace matrices for all of the meshesare the same. This will play a crucial role duringthe reconstruction step. Note that other isometry-invariant operators, such as the Laplace-Beltrami op-erator, do not have this property.

Given the Laplace matrix, we can compute theLaplace offsets ∆

(k)i for Xk as ∆

(k)1. . .

∆(k)n

= L

v(k)1. . .

v(k)n

.These offsets are not posture-invariant. Hence, we

express each offset with respect to the local coordinatesystem shown in Figure 1 as follows. At each vertexv(k)

i , we pick an arbitrary but fixed neighbor v(k)j as the

first neighbor (we choose the same first neighbor forall of the parameterized meshes). We then computea local orthonormal coordinate system at v(k)

i usingthe normal vector at v(k)

i , the normalized projection ofthe difference vector v(k)

j − v(k)i to the tangent plane of

v(k)i , and the cross product of the previous two vectors.

The normal vector is computed as the weighted aver-age of the normal vectors of v(k)

i ’s incident triangles,where the weights are proportional to the triangle ar-eas. We denote the three vectors defining the localorthonormal coordinate system by f1

(v(k)

i

), f2

(v(k)

i

),

and f3(v(k)

i

). Since the local coordinate system is or-

thonormal, we can express ∆(k)i in this coordinate sys-

tem as∆

(k)i = ω1(k)

i f1(v(k)

i

)+ ω2(k)

i f2(v(k)

i

)+ ω3(k)

i f3(v(k)

i

).

The local coordinates ωj(k)i are designed to be

invariant with respect to rigid transformations ofN1

(v(k)

i

). In many cases, such as the motion of hu-

mans, when the posture of a shape changes, most ar-eas of the shape deform in a locally near-rigid way.

3

v(k)i

v(k)j

f1(v(k)i )

f2(v(k)i )

f3(v(k)i )

Figure 1: Illustration of the local frame of a vertex v(k)i .

Hence, most of the local coordinates ω j(k)i are not af-

fected by such motions. We therefore propose to per-form shape analysis on the local coordinates ω j(k)

i toobtain a posture-invariant shape analysis.

To account for global scaling of the shape, we alsostore a coefficient s(k) related to the scale of the shape.More specifically, s(k) is computed as the averagegeodesic distance between any two vertices on X(k).Note that s(k) depends on the resolution of X(k). In ourapproach, this does not cause problems because all ofthe parameterized models X(k) have the same resolu-tion. We compute the geodesics using the fast march-ing technique [21].

3.3. Shape Analysis

We perform statistical shape analysis by comput-ing the PCA on the vectors ω j(k)

i , s(k). Compared toperforming PCA on the coordinates v(k)

i directly, thisoffers the advantage of being mostly posture-invariantsince the local coordinates ω j(k)

i , s(k) only depend onthe one-ring neighborhood of the vertex v(k)

i .Using PCA, we map each vector of local coordi-

nates[ω1(k)

1 , ω2(k)1 , ω3(k)

1 , . . . , ω1(k)n , ω2(k)

n , ω3(k)n , s(k)

]Tto

a vector W of PCA weights. Note that we normalizethe PCA space, such that each coordinate axis has unitlength. This non-uniform scaling allows us to con-duct fair comparisons between different PCA spacesin Section 4. We can visualize the shape variation ofthe shapes Xk by sampling the learned PCA space. Itremains to discuss how to compute a mesh based on agiven set of ordered local coordinates ω j

i , s.

3.4. ReconstructionTo visualize the shape variation, we aim to find a

mesh X for a given set of ordered local coordinatesω

ji , s. Recall that for any mesh

L

v1. . .vn

=

ω11 f1(v1) + ω2

1 f2(v1) + ω31 f3(v1)

. . .ω1

n f1(vn) + ω2n f2(vn) + ω3

n f3(vn)

. (1)

Here, L is known since it is identical for all of theparameterized shapes and ω j

i and s are given. Hence,we aim to find vertex positions vi that satisfy the aboveequation.

Note that the solution to this problem is not uniquesince different postures of the same shape may sat-isfy this equation. We aim to solve the problem us-ing an optimization technique. A good initializationis needed to solve this problem.

We use the following heuristic to solve for the ver-tex positions vi. We choose a mesh Xk as templateshape and we initialize vi to the vertex positions ofthe template shape. This allows for the computationof the frames f j(vi). Equation 1 implies that vi =∑

v j∈N1(vi)1

deg(vi)v j− (ω1

i f1(vi)+ω2i f2(vi)+ω3

i f3(vi)). Toimprove the initialization, we use this equation to it-eratively move vi towards the desired position d(vi) =∑

v j∈N1(vi)1

deg(vi)v j − (ω1

i f1(vi) + ω2i f2(vi) + ω3

i f3(vi)).More specifically, in our implementation, we updatethe vertex positions to vt+1

i = 0.5vti + 0.5d(vt

i), where tis the number of iterations and where v0

i is the vertexcoordinate of the template shape. The frames f j(vi)are recomputed in each iteration. We set the numberof iterations to 50.

Once a good initialization is found, we solve Equa-tion 1 by minimizing the energy

E =∑

i

(vi −∑

v j∈N1(vi)

1deg(vi)

v j−(ω1

i f1(vi) + ω2i f2(vi) + ω3

i f3(vi)))2

using a quasi-Newton approach [22].For this approach, we compute the gradient of E

with respect to the vertex positions vi. We use numer-ical gradients of the local frames for this computation.This is efficient since every local frame only dependson the vertex coordinates of vi and its neighboring ver-tices. Hence, on average, the numerical derivative ofeach local frame depends on a small constant numberof vertex positions.

The formulation of E and its gradient are simplebecause we choose the matrix L to be identical forall of the parameterized shapes X(k). If L is chosendifferently, for instance as the Laplace-Beltrami ma-trix, then each mesh may have a different matrix Lthat depends on the vertex positions vi. Hence, in eachgradient computation, we would need to compute thederivative of L with respect to vi, which would lead toa more complex and less efficient algorithm.

After minimizing E, the result has the shape of Xin the posture of Xk. Finally, we scale the result to thedesired scale factor s.

4. Experiments

We analyze the accuracy of the reconstruction stepof the algorithm. Furthermore, we perform statisti-cal shape analysis on a population of shapes and com-pare the results to the results obtained using a standardPCA approach.

4

4.1. Reconstruction AccuracyThe first experiment demonstrates the accuracy of

the reconstruction step. We choose one mesh of theMPI database, denoted by T , as template mesh andanother mesh of the MPI database, denoted by X, tocompute the local frames ω j

i and the scale factor s.We then compute the reconstruction using the coordi-nates of T as initial coordinates and ω

ji and s as the

desired local coordinates and scale. Figure 2 showsthe results. The reconstruction is the body shape of Xin the posture of T . Note that small geometric details,such as skin folding, are captured in the reconstruc-tion. This shows that the reconstruction algorithm canbe used to reposition a body to a standard posture.

Most models of the MPI database were scanned ina standard posture and in multiple other postures. Wechoose a model in standard posture as template T . Weevaluate the accuracy of the models by comparing theresult of our method to the shape of subject X scannedin standard posture. We denote this model by M. Thecolor-coded signed distances (in meters) between thereconstruction and M are shown in Figure 2. Note thatmost errors on the torso are under 2cm. Larger errorson the arms and legs are due to the slight posture dif-ference between T and M.

Figure 3 shows the back of the model used in thereconstruction shown in the middle of Figure 2. Theleft side of the figure shows the model X used as in-put for the reconstruction and the right side of the fig-ure shows the resulting reconstruction. Note that Xcontains areas with noise, skinny triangles, and self-intersections. While the reconstruction has similar lo-calized problems in the corresponding areas, the over-all reconstruction of the shape is correct and areas thatwere not noisy in X are not affected in the reconstruc-tion. This shows that the method is robust with respectto noise. Furthermore, this shows that the method pre-serves local features present on the mesh.

Figure 3: Left: mesh X used as input for reconstruction containsareas with noise, skinny triangles, and self-intersections. Right:reconstruction only contains problems in areas that were noisy inX.

4.2. Influence of JointsThe second experiment shows the influence of the

movement near joints on the local coordinates ω ji . Re-

call that ω ji do not change when the neighborhood of

vertex vi is transformed rigidly. The one-ring neigh-borhood of a vertex near a joint of a human modelmay not be transformed rigidly when the posture ofthe model changes. Instead, some of the triangles ad-jacent to that vertex may stretch. However, if the reso-lution of the model is high compared to the size of thedeformation, we expect that the stretch is distributedover several vertices, and that the change of a specificω

ji is small.In the following, we show experimentally that a

posture change such as those in the MPI databasedoes not significantly alter the local coordinates in thearea around a joint position. Consider the area Ablue

around the right elbow shown in blue in the left of Fig-ure 4. For each model of the MPI database, we com-pute the local coordinates of the vertices in Ablue, andwe reduce the dimensionality of this data using multi-dimensional scaling (MDS). This allows us to visual-ize the data in a way that shows the distances betweenthe data points, as shown in the right of Figure 4. Eachpoint corresponds to the local coordinates in Ablue onone model of the MPI database. The red points show20 different postures of the same subject. For someof the red points, a screen shot of the correspondingposture is shown and we can observe that the postureof the right elbow varies significantly. Note that thered points form a cluster in the MDS space. This im-plies that the distances between the local coordinatesin Ablue are small for models of the same subject indifferent postures.

Figure 4: Left: template mesh with highlighted area Ablue aroundthe right elbow. Right: visualization of local coordinates in Ablueover all models of the MPI database. Red points show differentpostures of the same subject and for some of the red points, a screenshot of the corresponding posture is shown.

4.3. Shape AnalysisThe third experiment conducts shape analysis using

our representation. The most common way to con-duct shape analysis on databases containing models

5

Figure 2: Each sub-figure shows one reconstruction. From left to right: template mesh T , mesh X used to compute the local frames ω ji ,

reconstruction, visualization of error (in meters).

in standard posture is standard PCA. We compare theresults obtained using our approach to the results ob-tained using standard PCA. Recall that our approachcomputes a PCA space by performing PCA on the rep-resentationω j(k)

i , s(k). For our approach, the meshes donot need to be spatially aligned. Standard PCA com-putes a PCA space by performing PCA on the ver-tex coordinates of the meshes after rigidly aligning themeshes using generalized Procrustes analysis [3].

First, we show the result for the CAESAR database.Figure 5 shows the computed PCA spaces. The leftside of the figure shows the PCA space obtained usingour approach and the right side of the figure shows thePCA space obtained using standard PCA. Note thatstandard PCA captures arm and body posture move-ments in the principal components (rotation of shoul-der along the first principal component and bending ofthe elbow and posture change along the second princi-pal component), while our approach only finds shapevariations. This shows that our approach contains no-tably less posture variation than standard PCA evenfor databases containing subjects in standard posture.

−6λi Mean 6λi −6λi Mean 6λi

Figure 5: Statistical shape analysis of CAESAR database. The leftside shows the results using our approach and the right side showsthe results using standard PCA. Each row shows the impact of oneprincipal component (the shapes at distance 6λi from the mean andthe mean, where λi is the i-th eigenvalue).

For this data set, we evaluate the statistical mod-els generated using our approach and using standardPCA with the help of the three error measures sug-gested by Styner et al. [23]: compactness, general-ization, and specificity. A compact model has littlevariance and it requires few principal components todescribe the model, a general model has the ability todescribe instances that are not in the training set, anda specific model only generates shapes that are similarto the ones in the training set.

More specifically, compactness is defined as C(k) =∑ki=1 λi/

∑mi=1 λi, where k is the number of principal

components that are kept, m is the number of mod-els in the training set, and where λi is the i-th eigen-value. The compactness error is computed as σC(k) =√

2/mC(k). To measure generalization G(k), we com-pute a statistical model using the first k eigenvectorswith all but one of the corresponded scans of thetraining set and fit the model to the excluded scan.Generalization measures the accuracy of the descrip-tion of the excluded scan and is computed as the dis-tance from the original model to its projection ontoPCA space, averaged over the complete set of trials.The standard error is computed using the standard de-viation σ of errors of the complete set of trials asσG(k) = σ/

√m − 1. Specificity S (k) is defined as the

average distance of S rand to its nearest neighbor in thetraining set, where S rand is a shape computed usingthe statistical model from a uniformly distributed ran-dom sample in PCA shape space. The standard erroris computed using the standard deviation σ of the dis-tances as σS (k) = σ/

√m.

Figure 6 gives the error measures. Note that ourmethod leads to lower errors for all three evaluationmeasures. This is to be expected because our methodrepresents the statistical model in a posture-invariantway while standard PCA encodes posture variationsin the statistical model.

Second, we show the result for the MPI database.Figure 7 shows the computed PCA spaces. The leftside of the figure shows the PCA space obtained us-ing our approach and the right side of the figure showsthe PCA space obtained using standard PCA. Whenanalyzing the shape variations found using our ap-proach, we can see that the principal components cap-ture shape differences rather than posture differences

6

Figure 6: Error evaluation. The result using standard PCA is shown in blue and the result using our method is shown in red. The x-axis showsthe number of principal components and the y-axis shows values of the error measures.

and that visually realistic body shapes are obtained.This shows that our approach is suitable for analyzingthe shape changes of databases in varying posture.

However, it is well-known that standard PCA is notsuitable for analyzing the shape changes of databasesin varying postures. This can be seen when analyz-ing the shape variations found using standard PCA.We can see that the standard PCA approach mostlycaptures posture changes. Furthermore, the standardPCA approach leads to unrealistic body shapes, as canbe seen when looking at the arms in the first row of theright side of the figure.

−λi Mean λi −λi Mean λi

Figure 7: Statistical shape analysis of MPI database. The left sideshows the results using our approach and the right side shows theresults using standard PCA. Each row shows the impact of one prin-cipal component (the shapes at distance λi from the mean and themean, where λi is the i-th eigenvalue).

4.4. Application to Estimating Body Shape of DressedSubject

Statistical shape models are often useful for syn-thesizing new shapes from partial information. As anapplication of our learned shape space, we estimatethe body shape from a scan of a dressed subject. Toobtain a parameterization that is consistent with ourtraining data, we first fit a template model to the scanshown on the left of Figure 8 using an approach sim-ilar to the one used by Hasler et al. [10] to register adatabase of scans. The result is shown in the middleof Figure 8, and we can see that the surface includesthe shape of the clothing. To estimate the body shapeof the subject, we compute the local coordinates of thedeformed template, project the result onto to the PCAspace learned using the MPI database (using the first15 principal components), and use the point in PCAspace to reconstruct the body shape. The template fit-ted to the scan is used as initialization when recon-structing the body shape. The right of Figure 8 showsthe result. Table 1 gives an evaluation of the accuracyof the model using a set of the measurements. Allof the measurements except the waist girth are within3cm of the true measurements, which is fairly accu-rate. The waist girth of our estimate is higher thanthe true waist girth because the subject was scannedwhile wearing a loose sweater. The loose clothingleads to a surface that is far from the true body surfacein the area of the waist. Hence, the scan observationcould be explained by human body shapes with dif-ferent waist girths. Our estimate is a solution that isvisually consistent with the scan. This demonstratesthat our learned shape space can be applied to synthe-size new shapes based on partial observations.

Height Arm Length Leg Length Waist GirthGround truth 182 57 79 74Estimate 185 60 77 85

Table 1: Biometric measurements (in cm). The first row shows theground truth measurements and the second row shows the corre-sponding measurements on the estimated body shape.

7

Figure 8: Left: scan data from [24]. Middle: template model fittedto scan. Right: estimated body shape.

5. Conclusion

We proposed an approach to perform statisticalshape analysis in a posture-invariant way. This ap-proach was shown to allow for the analysis and visu-alization of shape variations in a database of humanmodels in varying postures. This is useful for designapplications, where one is interested in visualizing theshape variations of a population in varying postureswhile ignoring posture variations.

The reconstruction of a triangle mesh based on a setof ordered local coordinates is currently solved using aheuristic energy optimization scheme. An interestingquestion for future work is to find more robust waysto solve this problem.

Acknowledgments

We thank Nils Hasler for providing us with the MPIdatabase and the scanned clothed subject. We thankJonathan Boisvert and Alan Brunton for helpful dis-cussions on the topic, and the anonymous reviewersfor insightful comments. This work has partially beenfunded by the Cluster of Excellence Multimodal Com-puting and Interaction within the Excellence Initiativeof the German Federal Government.

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