3D Anatomical Shape Atlas Construction Using

Mesh Quality Preserved Deformable Models

Xinyi Cui1, Shaoting Zhang1,�, Yiqiang Zhan2,Mingchen Gao1, Junzhou Huang3, and Dimitris N. Metaxas1

1 Dept. of Computer Science, Rutgers Univ., Piscataway, NJ, USA2 CAD R&D, Siemens Healthcare, Malvern, PA, USA

3 Dept. of Computer Science and Engineering, Univ. of Texas at Arlington, TX, USAshaoting@cs.rutgers.edu

Abstract. The construction of 3D anatomical shape atlas has beenextensively studied in medical image analysis research for a variety ofapplications. Among the multiple steps of shape atlas construction, es-tablishing anatomical correspondences across subjects is probably themost critical and challenging one. The adaptive focus deformable model(AFDM) [16] was proposed to tackle this problem by exploiting cross-scale geometry characteristics of 3D anatomy surfaces. Although the ef-fectiveness of AFDM has been proved in various studies, its performanceis highly dependent on the quality of 3D surface meshes. In this paper,we propose a new framework for 3D anatomical shape atlas construc-tion. Our method aims to robustly establish correspondences across dif-ferent subjects and simultaneously generate high-quality surface mesheswithout removing shape detail. Mathematically, a new energy term isembedded into the original energy function of AFDM to preserve surfacemesh qualities during the deformable surface matching. Shape detailsand smoothness constraints are encoded into the new energy term usingthe Laplacian representation An expectation-maximization style algo-rithm is designed to optimize multiple energy terms alternatively untilconvergence. We demonstrate the performance of our method via two di-verse applications: 3D high resolution CT cardiac images and rat brainMRIs with multiple structures.

Keywords: Shape atlas, one-to-one correspondence, mesh quality,Laplacian surface, deformable models.

1 Introduction

The 3D shape based reconstruction of anatomy has been of particular interest andits importance has been emphasized in a number of recent studies [3,6,10,11,8]. Itprovides a reference shape and variances for a population of shapes. Such shapeinformation can be useful in numerous applications such as, but not limited to,statistical analysis of populations [5], the segmentation of the structures of in-terest [22,23], and the detection of disease regions [21]. Besides shape modeling,

� Corresponding author.

J.A. Levine, R.R. Paulsen, Y. Zhang (Eds.): MeshMed 2012, LNCS 7599, pp. 12–21, 2012.c© Springer-Verlag Berlin Heidelberg 2012

Mesh Quality Preserved Deformable Models 13

image atlases have also been extensively investigated [9]. Since an image atlas isnot easily adapted to a surface atlas, we will only focus on the methods for theconstruction of shape models in this paper.

Target mesh After deformation Source mesh

Fig. 1. The source mesh is registered to the targetone. both of them are already smoothed. The re-sulting mesh contains many artifacts because of themesh degeneration during the deformation.

A shape has several differ-ent representations. Cootes etal. proposed a diffeomorphicstatistical shape model whichanalyzes the parameters ofthe deformation field [4].Styner et al. used a character-istic 3D shape model dubbedM-Rep to construct the at-las [7,20]. In [3], distancetransform was used to createa shape complex atlas. Themost widely used 3D shaperepresentation is probably feature point sets or landmarks from a polygon mesh.Using this representation, the mean shape and its variances can be easily com-puted using generalized Procrustes analysis and Principal Component Analysis(PCA). Furthermore, such representations have been used in many segmentationalgorithms, e.g., Active Shape Model (ASM) [5]. In this context, the main chal-lenge is to robustly discover geometric correspondences for all vertices among allsample shapes. One widely used approach is to register a reference shape to tar-get ones. The deformed reference shapes are expected to have identical geometriccharacteristics as the targets shapes and same topology and connectivity of thereference one. Hence, the geometric correspondences between reference shapeand target shapes are generated by simply matching the closest vertices betweendeformed reference shapes and target ones. These shape registration techniqueshas been widely investigated in biomedical applications [1,13,15,17,18,19]. Theadaptive focus deformable model (AFDM) [16] is a very effective approach todo shape registration, since its attribute vectors reflect the geometric structureof the model from a local to global level. Although it has been widely used inmany clinical applications, this method is sensitive to mesh qualities becausedegenerate mesh or skinny triangles can adversely affect the registration per-formance. Even if preprocessing is applied to smooth the mesh, it may still bedegenerate during the deformation (Fig. 1). Furthermore, the resulting mesh isnot guaranteed to be smooth, while a high-quality mesh is usually preferred sinceit benefits the performance of many applications, such as statistical analysis andsegmentation.

In this paper, we propose a unified framework to compute geometry correspon-dence for all vertices among sample shapes, and generate high quality mesheswithout significantly sacrificing shape details. A new type of energy term isincorporated into the AFDM framework to preserve mesh quality during defor-mation. Combining this quality energy with the model energy in the originalAFDM, our method is able to robustly discover one-to-one correspondence even

14 X. Cui et al.

for very complex or degenerate shapes. The whole energy function is optimizedusing an Expectation-maximization type of algorithm. Three energy terms areminimized alternately until converge.

Our algorithm is evaluated in two diverse applications: 1) high resolutioncardiac shape model in CT images, and 2) multiple structure shape model ofrat brains in dense brain MRI. For cardiac application, our method can findthe anatomical point correspondence both among multiple instances of the samephase of a cardiac cycle and sequential phases of one cycle. The ability to fitthe atlas to all temporal phases of a dynamic study can benefit the automaticfunctional analysis. For the rat brain application, our approach can discover thespatial relationship among multiple structures and construct a shape atlas forall structures.

The major contributions are: 1) propose a unified framework to improve thetraditional AFDM by incorporating an energy term to preserve mesh qualityduring runtime. The resulting meshes are generally smooth without significantlysacrificing shape details. It can also robustly discover the one-to-one correspon-dence for very complex data (e.g., the shapes from high resolution cardiac CTimages); 2) this approach enables us to solve two diverse and challenging tasks.Specifically, we create a high resolution cardiac shape atlas with many complexshape features such as papillary muscles and the trabeculae. We also effectivelyconstruct the atlas of multiple rat brain structures using a small set of samples.

2 Methodology

2.1 Algorithm Framework

As we discussed, the performance of the AFDM relies on themesh quality. Thoughpre-processing techniques using mesh smoothing methods such as [2,24,25] can al-leviate the problem to some degree, it is still highly possible that themesh is degen-erate during shape deformation. Thus it is desirable to design a unified frameworkto consider shape deformation and mesh quality simultaneously.

We define this shape modeling problem as an energy minimization procedure.Three energy functions are introduced to control the model deformation andpreserve the mesh quality: model energy, external energy and quality energy.Here we use Mo, Mt and Md to denote the original mesh, the target mesh andthe deformed mesh. Our goal is to deform an original mesh Mo to a target mesh.Md is the deformed mesh we want to compute. It needs to be close to the targetmesh Mt on the boundaries and keep the geometric characteristics and meshquality similar to the original mesh Mo. Md is initialized as Mo. The energyfunction is defined as:

E = Emodel (Md,Mo) + Eext (Md,Mt) + Equality(Md,Mo) (1)

The model energy Emodel (Md,Mo) reflects the geometric differences betweenMo

and Md. The external energy Eext(Md,Mt) drives the model deforming towardsthe boundaries of the target modelMt. By jointly minimizing these two terms, the

Mesh Quality Preserved Deformable Models 15

model will deform to the boundaries of the target and still preserve its geometriccharacteristics. In our study, an additional termQuality energy Mquality(Md,Mo)is designed to ensure that vertices are evenly distributed and shape details areroughly preserved. Using only Emodel and Eext produces similar results as theAFDM, which makes it sensitive to mesh quality. The quality energyMquality weintroduce here ensures that the mesh quality is improved during deformation pro-cedure, making the whole model more robust to handle diverse input.

The minimization of external energy Eext is fairly standard. First, a distancetransform is applied to Mt to obtain a binary distance 3D image It, whichis the implicit embedding space of the target mesh. Then the deformed meshMd is placed in the embedding space It. The standard gradient on Md verticesis computed from It. The gradient force drives Md to be close to Mt on theboundaries. The details of Emodel and Equality are introduced in the next twosubsections.

2.2 Model Energy

The model energy measures the geometric differences between the original modeland its deformed version. It is defined by the differences of geometric attribute vec-tors. An attribute vector is attached to each vertex of the model, which reflects thegeometric structure of the model from a local to global level. For a particular ver-tex Vi in 3D, each attribute is defined as the volume of a tetrahedron on that ver-tex. The other three vertices form the tetrahedron are randomly chosen from thelth level neighborhood of Vi. Smaller tetrahedrons reflect the local structure neara vertex while larger tetrahedrons reflect a more global information around a ver-tex. The attribute vector, if sufficient enough, uniquely characterizes the differentparts of a boundary surface. The volume of a tetrahedron is defined as fl(Vi). Theattribute vector on a vertex is defined as F (Vi) = [f1(Vi), f2(Vi), ..., fR(Vi)(Vi)],where R(Vi) is the neighborhood layers we want to use around Vi.

The model energy term is defined by the difference of the attribute vectorsbetween the original model and the deformed model:

Emodel(Md,Mo) =

N∑

i=1

R(Vi)∑

l=1

δl(fd,l(Vi)− fo,l(Vi))2, (2)

where fd,l(Vi) and fo,l(Vi) are components of attribute vectors of the deformedmodel and the model at vertex Vi, respectively. δl here denotes the importance ofthe lth neighborhood layers. R(Vi) is the number of neighborhood layers aroundvertex Vi.

Emodel is optimized by exploiting the affine invariant property of the geometricattribute vector fl(Vi). According to the proof provided by [16], δl(fd,l(Vi) −fo,l(Vi)) ≡ 0 under affine transformation. Therefore, during registration process,we divide a surface into a set of segments and deform them using a local affinetransformation. In this way, the geometric properties of the original surface ispreserved while it still deforms to the target one.

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2.3 Quality Energy

The Quality Energy is introduced to ensure that vertices are evenly distributedand shape details are roughly preserved during the deformation procedure. Inmany cases, there is a tradeoff between smoothing and keeping shape details. Theenergy term we use here is able to smooth the shape without losing importantdetails. We extend the Laplacian coordinate [12] to achieve this. The mesh M

of the shape is denoted by a pair (V,E), where V = {v1, ..., vn} denotes thegeometric positions of the vertices in R

3 and E describes the connectivity. Theneighborhood ring of a vertex i is the set of adjacent vertices Ni = {j|(i, j) ∈ E}.The degree di of this vertex is the number of elements in Ni. Instead of usingabsolute coordinates V, the mesh geometry is described as a set of differentialsΔ = {δi}. Specifically, coordinate i is represented by the difference between viand the weighted average of its neighbors using δi = vi−

∑j∈Ni

wijvj . This δi canapproximate the normal vector of vi, if we use the cotangent weight as wij [14].Assume V is the matrix representation of V. Using a small subset A ⊂ V of manchor points, a mesh can be reconstructed from connectivity information alone.The x, y and z positions of the reconstructed object (V ′

p = [v′1p, ..., v′np]

T , p ∈{x, y, z}) can be solved separately by minimizing the quadratic energy:

Equality(Md,Mo) = ‖Md − L(Mo)‖ = ‖LuV′p −Δ‖2 +Σa∈A‖v′ap − vap‖2, (3)

where L denotes the Laplacian representation of the original shape. Lu is theLaplacian matrix computed by using uniform weights (i.e., wij = 1

di), and the

vap are anchor (landmark) points. ‖LV ′p − Δ‖2 tries to smooth the mesh when

keeping it similar to the original shape, and∑

a∈A‖v′ap − vap‖2 keeps the anchor

points unchanged. Since the cotangent weights approximate the normal direction,and the uniform weights point to the centroid, minimizing the difference of thesetwo (i.e., LuV

′ and Δ) means to move the vertices along the tangential direction.Since this scheme prevents the movement along the vertices’ normal directions,the shape is smoothed without significantly losing the detail. With m anchors,(3) can be rewritten as a (n + m) × n overdetermined linear system AV ′

p = b as

[L Iap]T · V ′

p = [Δ Vap]T . m can be chosen as 4 to n ( n

10 in our applications).Since AV ′

p = b is overdetermined, this can be always solved in the least squares

sense using the method of normal equations V ′p = (ATA)−1AT b [12]. The conju-

gate gradient method is used in our system to solve it efficiently. The first n rowsofAV ′

p = b are the Laplacian constraints, corresponding to ‖LV ′p −Δ‖2, while the

lastm rowsare the positional constraints, corresponding to∑

a∈A ‖v′ap−vap‖2. Iapis the index matrix of Vap, which maps each V ′

ap to Vap. The reconstructed shape isgenerally smooth, while shape details are still preserved.

2.4 Optimization Framework

To optimize this energy function, we use an expectation-maximization (EM)type of algorithm. During the “E” step, the model energy and external energyare minimized using similar approach as the AFDM. Thus the reference shape is

Mesh Quality Preserved Deformable Models 17

deformed to fit the target one, although this deformation may not be accuratedue to the mesh quality. In the “M” step, the mesh quality is improved by mini-mizing the quality energy. This step is formulated as a least square problem andsolved efficiently. Two procedures are alternately employed to robustly registerthe reference model to the target model. Note that theocratically this EM algo-rithm may lead to local minima since this is not a convex problem. However, inour extensive experiments in Sec. 3 , we did not observe this situation yet.

3 Experiments

We validate our method using two applications: 1) high resolution shapemodel in CT images, and 2) multiple structure shape model of rat brains inMR Microscopy.

3.1 High Resolution Cardiac Model

Recent developments on the CT technologies are able to produce 4D high res-olution cardiac images in a single heart beat of humans. The reconstruction ofthe endocardial surface of the ventricles with incorporation of finer details cangreatly assist doctors in diagnosis and functional assessment. In this experiment,we applied our framework to create the shape atlas from 4D cardiac reconstruc-tions, which captures a whole cycle of cardiac contraction from high resolutionCT images. The CT data were acquired on a 320-MDCT scanner using a con-ventional ECG-gated contrast-enhanced CT angiography protocol. The imagingprotocol parameters include: prospectively triggered, single-beat, volumetric ac-quisition; detector width 0.5 mm, voltage 120 KV, current 200 − 550 mA. Re-constructions were done at 10 equally distributed time frames in a cardiac cycle.The resolution of each time frame is 512 by 512 by 320.

Fig. 2 shows the reconstruction results of high resolution cardiac CT imagesat different time frames. The three-dimensional structures, their relationshipand their movement during the cardiac cycle are much more readily appreciatedfrom the shape model than from the original volumetric image data. Since theseshapes are independently reconstructed, there is no one-to-one correspondencefor vertices, and they may have different topologies for some small details, whichprevent the statistical analysis. Shape registration can be applied to establishthe one-to-one correspondence relations. However, this task is very challengingbecause of the complex shape details and the presence of degenerate skinnytriangles. Our proposed model successfully and robustly registers these meshestogether, by starting from the the middle frame and propagating in two direc-tions. Once one-to-one correspondence is obtained, PCA can be applied straight-forwardly to obtain shape statistics. Fig. 3 visualizes the shape variation alongthe first and second principal directions. The first mode represents the changingof the volume magnitude, and the second mode captures the changing of shapedetails such as the papillary muscles and the trabeculae. Such information canbe used in clinical applications to categorize the cardiac properties.

18 X. Cui et al.

Fig. 2. The cardiac shapes extracted from high resolution CT images. The complexshape details are captured, such as the papillary muscles and the trabeculae.

Fig. 3. Modes with largest variances, from −3σ to 3σ. The first mode represents thechanging of the volume size. The second mode is the changing of shape details suchas papillary muscles. For better visualization, please refer to the video sequence in thesupplementary materials.

Table 1 shows the quantitative comparisons of two methods, by evaluatingthe mesh quality, registration accuracy, and running time. To evaluate the meshquality, radius ratio is computed: ti = 2 r

R , ti ∈ [0, 1], where R and r are theradii of the circumscribed and inscribed circles respectively. ti = 1 indicates awell shaped triangle, while small values mean degenerate meshes. The min andmean values of radius ratio are reported, which are two important measurementsfor robust modeling and simulation. To evaluate the accuracy, we compute themean and standard deviation of voxel distances between two shapes. In general,the proposed method achieves better mesh quality and accuracy than the tradi-tional AFDM, showing that our method improves the mesh quality with detailpreserved and without sacrificing the accuracy. Furthermore, its computationalcost is also comparable to the AFDM, even with an extra mesh quality energyterm. It is a C++ implementation on a Quad CPU 2.4GHZ PC. The reason isthat mesh quality constraint aims to produce evenly distributed vertices, whichalso speeds up the convergence of the AFDM.

Mesh Quality Preserved Deformable Models 19

Table 1. Quantitative comparisons of the proposed method and the AFDM, includingthe mesh quality measured by the min and mean values of radius ratio (Qmean, Qmin),the accuracy of registration measured by the mean and standard deviation of voxeldistances between the target and deformed shapes (Distance), and the running time(T ime). #V denotes the number of vertices of one shape.

High Resolution Cardiac Rat Brain Structures#V Qmin Qmean Distance T ime #V Qmin Qmean Distance T ime

AFMD 20K 0.00 0.73 2.03 ± 0.72 8′37′′ 2.5K 0.05 0.81 0.58 ± 0.15 17′′

Ours 20K 0.02 0.97 0.47 ± 0.13 7′56′′ 2.5K 0.10 0.93 0.21 ± 0.08 19′′

3.2 Multiple Structures of Rat Brains

Reconstruction of brain images is important to understand the relationship be-tween anatomy and mental diseases in brains. Volumetric analysis of variousbrain structures such as the cerebellum plays a critical role in studying thestructural changes in brain regions. Rat brains images are often used as modelsfor human disease since they exhibit key features of abnormal neurological condi-tions and are served as a convenient starting point for novel studies. In this study,we use the proposed method to create a 3D shape atlas of multiple brain struc-tures based on MR images of the rat brain. In our experiments, 11 adult maleSprague-Dawley rats were transcardially perfused with 4% paraformaldehyde.Heads were stored in paraformaldehyde and scanned for MRI. Brains remainin skulls during scanning in order to avoid tissue and shape distortions duringbrain extraction. The heads were scanned on a 21.1T Bruker Biospin Avancescanner (Bruker Biospin Corporation, Massachusetts, USA). The protocol con-sisted of a 3D T2-weighted scan with echo-time (TE) 7.5ms, repetition time(TR) 150ms, 27.7 kHz bandwidth, field of view (FOV) of 3.4×3.2×3.0mm, andvoxel size 0.08mm, isotropic. Because of this high resolution, real 3D annotationis performed manually by multiple clinical experts. We focus on three complexstructures of the rat brain: a) the cerebellum, b) the left and right striatum, andc) the left and right hippocampus.

Fig. 4 shows the shape variation along the first and second principal direc-tions. Our method is able to capture the spatial relationships among different

Fig. 4. Modes with largest variances, from −3σ to 3σ. The first mode represents thechanging of the size. For better visualization, please refer to the video sequence in thesupplementary materials.

20 X. Cui et al.

neighboring structures. The first mode is the changing of the volume size. Thesecond mode is the changing of the local details, such as two protruding parts ofthe cerebellum. Table 1 shows the quantitative comparisons of our method andthe AFDM, which follows the same pattern as the previous experiment.

4 Conclusions

In this paper we presented an algorithm to robustly discover geometric corre-spondence and effectively model 3D shapes. This method improves the tradi-tional adaptive focus deformable model by incorporating a quality energy, whichensures the mesh quality during the deformation. Thus the shape registrationis robust and the defomed mesh is generally smooth without sacrificing shapedetails. After registration, one-to-one correspondence can be obtained for allvertices among sample shapes. Such correspondences can be used to generatePDM for many segmentation methods, and can also be used to obtain high tem-poral resolution using interpolations. We extensively validated this method intwo applications. In the future, we plan to use these generated shape models tofacilitate the segmentation algorithms by using it as the shape prior information.

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