BRAIN SURFACE CONFORMAL PARAMETERIZATIONYalin Wang

Mathematics Department, UCLAemail: ylwang@math.ucla.edu

Xianfeng GuComputer Science Department

SUNY at Stony Brookemai: gu@cs.sunysb.edu

Kiralee M. HayashiLaboratory of Neuro Imaging

UCLA School of Medicineemail: khayashi@loni.ucla.edu

Tony F. ChanMathematics Department, UCLA

email: chan@math.ucla.edu

Paul M. ThompsonLaboratory of Neuro ImagingUCLA School of Medicine

email: thompson@loni.ucla.edu

Shing-Tung YauDepartment of Mathematics

Harvard Universityemail: yau@math.harvard.edu

ABSTRACTWe develop a general approach that uses holomorphic 1-forms to parameterize anatomical surfaces with complex(possibly branching) topology. Rather than evolve the sur-face geometry to a plane or sphere, we instead use the factthat all orientable surfaces are Riemann surfaces and ad-mit conformal structures, which induce special curvilinearcoordinate systems on the surfaces. We can then automat-ically partition the surface using a critical graph that con-nects zero points in the conformal structure on the surface.The trajectories of iso-parametric curves canonically parti-tion a surface into patches. Each of these patches is eithera topological disk or a cylinder and can be conformallymapped to a parallelogram by integrating a holomorphic1-form defined on the surface. The resulting surface subdi-vision and the parameterizations of the components are in-trinsic and stable. To illustrate the technique, we computedconformal structures for several types of anatomical sur-faces in MRI scans of the brain, including the cortex, hip-pocampus, and lateral ventricles. We found that the result-ing parameterizations were consistent across subjects, evenfor branching structures such as the ventricles, which areotherwise difficult to parameterize. Compared with othervariational approaches based on surface inflation, our tech-nique works on surfaces with arbitrary complexity whileguaranteeing minimal distortion in the parameterization. Italso generates grids on surfaces for PDE-based signal pro-cessing.

KEY WORDSBrain Mapping, Riemann Surface Structure, ConforamlNet, Critical Graph

1 Introduction

Surface-based modeling is valuable in brain imaging tohelp analyze anatomical shape, to statistically combine orcompare 3D anatomical models across subjects, and to mapfunctional imaging parameters onto anatomical surfaces.Multiple surfaces can be registered nonlinearly to constructa mean shape for a group of subjects, and deformation map-pings can encode shape variations around the mean. This

type of deformable surface registration has been used to de-tect developmental and disease effects on brain structuressuch as the corpus callosum and basal ganglia [1], the hip-pocampus [2], and the cortex [3]. Nonlinear matching ofbrain surfaces can also be used to track the progressionof neurodegenerative disorders such as Alzheimer’s dis-ease [2], to measure brain growth in development [1], andto reveal directional biases in gyral pattern variability [4].

Parameterization of anatomical surface models in-volves computing a smooth (differentiable) one-to-onemapping of regular 2D coordinate grids onto the 3D sur-faces, so that numerical quantities can be computed eas-ily from the resulting models. Even so, it is often diffi-cult to smoothly deform a complex 3D surface to a sphereor 2D plane without substantial angular or area distortion.Here we present a new method to parameterize brain sur-faces based on their Riemann surface structure. By contrastwith variational approaches based on surface inflation, ourmethod can parameterize surfaces with arbitrary complex-ity including branching surfaces not topologically homeo-morphic to a sphere (higher-genus objects) while formallyguaranteeing minimal distortion.

1.1 Previous Work

Brain surface parameterization has been studied inten-sively. Schwartz et al. [5], and Timsari and Leahy [6]computed quasi-isometric flat maps of the cerebral cor-tex. Hurdal and Stephenson [7] report a discrete map-ping approach that uses circle packings to produce “flat-tened” images of cortical surfaces on the sphere, the Eu-clidean plane, and the hyperbolic plane. The obtained mapsare quasi-conformal approximations of classical conformalmaps. Haker et al. [8] implemented a finite element ap-proximation for parameterizing brain surfaces via confor-mal mappings. They selected a point on the cortex to mapto the north pole and conformally mapped the rest of thecortical surface to the complex plane by stereographic pro-jection of the Riemann sphere to the complex plane. Gu etal. [9] proposed a method to find a unique conformal map-ping between any two genus zero manifolds by minimizingthe harmonic energy of the map. They demonstrated this

478-043 76

method by conformally mapping the cortical surface to asphere.

In computational anatomy, 3D shape analyses are of-ten performed to analyze the geometry of specific brainstructures, and compute statistical information on anatom-ical variability and group differences. Joshi et al. [10] an-alyzed the shape of the hippocampus via a spatially nor-malizing elastic transformation. Group differences in sur-face shape were identified by comparing the transforma-tions required to map individual surfaces to a group aver-age surface [2]. Kelemen et al. [11] studied 3D hippocam-pal shapes based on a boundary description using spheri-cal harmonic basis functions (SPHARM). The SPHARMshape description creates an implicit boundary correspon-dence between shapes. Later, Gerig et al. [12] usedSPHARM to study ventricular size and shape in 3D MRIscans of monozygotic and dizygotic twin pairs. Pizer etal. [13] proposed a method to apply sampled medial mod-els (M-reps) to shape analysis. By holding the topology ofthe model fixed, an implicit correspondence between sur-face boundary points and an underlying medial curve is es-tablished and can be applied to model anatomy in radiationoncology and for shape analysis.

1.2 Theoretical Background and Definitions

A manifold of dimension � is a connected Hausdorff space�for which every point has a neighborhood � that is

homeomorphic to an open subset � of ��� . Such a homeo-morphism ����� � is called a coordinate chart. An atlasis a family of charts ������������� for which ��� constitute anopen covering of

�.

PSfrag replacements

��� ���

���

�

���������! � � ��� � � �"���� � �

Figure 1. The Structure of a Manifold. An atlas is a familyof charts that jointly form an open covering of the manifold.

Suppose ������������� and �����#���!�#� are two charts on amanifold

�, � �%$ � �'&(*) , then the chart transition is de-

fined as � �"� �+� � �,� �-$ � � .�/� � �,� �0$ � � . An atlas�"� � �1� � � on a manifold is called differentiable if all charttransitions are differentiable of class 2�3 . A chart is calledcompatible with a differentiable atlas if adding this chartto the atlas still yields a differentiable atlas. The set of allcharts compatible with a given differentiable atlas yields

a differentiable structure. A differentiable manifold of di-mension � is a manifold of dimension � together with adifferentiable structure.

For a manifold�

with an atlas 4 ( �"�5���1�6��� , if allchart transition functions

�6��� ( �!�.78��9#:� �;���<�,��� $ �=�> �� �!���,��� $ ���! are holomorphic, then 4 is a conformal atlas for

�. A

chart �"���#�����#� is compatible with an atlas 4 , if the union4@?A�"����������� is still a conformal atlas.Two conformal atlases are compatible if their union is

still a conformal atlas. Each conformal compatible equiv-alence class is a conformal structure. A 2-manifold witha conformal structure is called a Riemann surface. It hasbeen proven that all metric orientable surfaces are Riemannsurfaces.

Holomorphic and meromorphic functions and differ-ential forms can be generalized to Riemann surfaces byusing the notion of conformal structure. For example, aholomorphic one-form B is a complex differential form,such that in each local frame CD� ( �FE����HGD�6 , the para-metric representation is B (JI �KC��! MLC"� , where I �FC��! isa holomorphic function. On a different chart �"�����1�!�#� ,B (NI �FC"���FCO�! H DP1Q�RP1QMS LCO� . For a genus T closed surface, allholomorphic one-forms form a real UVT dimensional linearspace.

At a zero point WYX � of a holomorphic one-form B ,any local parametric representation B (ZI �FCD�! HL;C"��� I+[ \-(]_^

According to the Riemann-Roch theorem, in generalthere are UVTa`@U zero points for a holomorphic one-formdefined on a surface of genus T .

A holomorphic one-form induces a special system ofcurves on a surface, the so-called conformal net. A curveb'c �

is called a horizontal trajectory of B , if Bd�D�KL b fe]; similarly, b is a vertical trajectory if B5�;�FL b hg ]

. Thehorizontal and vertical trajectories form a web on the sur-face. The trajectories that connect zero points, or a zeropoint with the boundary are called critical trajectories. Thecritical horizontal trajectories form a graph, which is calledthe critical graph. In general, the behavior of a trajectorymay be very complicated, it may have infinite length andmay be dense on the surface. If the critical graph is finite,then all the horizontal trajectories are finite. The criticalgraph partitions the surface into a set of non-overlappingpatches that jointly cover the surface, and each patch is ei-ther a topological disk or a topological cylinder. Each patchi c � can be mapped to the complex plane using the fol-lowing formulae. Suppose we pick a base point W�jkX i ,and any path b that connects W6j to W . Then if we define���lW� (NmOn B , the map � is conformal, and ��� i is a par-allelogram. We say � is the conformal parameterizationof�

induced by B . � maps the vertical and the horizon-tal trajectories to iso-u and iso-v curves respectively on theparameter plane. The structure of the critical graph and theparameterizations of the patches are determined by the con-formal structure of the surface. If two surfaces share similar

77

topologies and geometries, they can support consistent crit-ical graphs and segmentations (i.e. surface partitions), andthe parameterizations are consistent as well. Therefore, bymatching their parameter domains, the entire surfaces canbe directly matched in 3D. This generalizes prior work inmedical imaging that has matched surfaces by computinga smooth bijection to a single canonical surface, such as asphere or disk.

A Riemannian metric is a differential quadratic formon a differential manifold. On each chart �"�d��������� , it canbe represented as

Lpo � (rq �FE<�sGp HLDE �5t U�u0�vE��HGw ML;E�LDG t'x �vE��HGw ML;G � ^A special conformal structure can be chosen, such thatthe local parametric representation of Riemannian metricis Lpo � (zy �vE��HGw {�FL;E � t LDG � ^ In this case, the local coordi-nates of each chart are also called isothermal coordinates,and y �vE��HGw is called the conformal factor.

This paper takes the advantage of conformal struc-tures of surfaces, consistently segments them and param-eterizes the patches using a holomorphic 1-form.

We call the process of finding critical graph and seg-mentation as the holomorphic flow segmentation, which iscompletely determined by the geometry of the surface andthe choice of the holomorphic 1-form. (Note that this dif-fers from the typical meaning of segmentation in medicalimaging, and is concerned with the segmentation, or parti-tioning, of a general surface). Computing holomorphic 1-forms is equivalent to solving elliptic differential equationson the surfaces, and in general, elliptic differential opera-tors are stable. Therefore the resulting surface segmenta-tions and parameterizations are intrinsic and stable, and areapplicable for matching noisy surfaces derived from medi-cal images.

2 Holomorphic Flow Segmentation

To compute the holomorphic flow segmentation of a sur-face, first we compute the conformal structure of the sur-face; then we select one holomorphic differential form, andlocate the zero points on it. By tracing horizontal trajec-tories through the zero points, the critical graph can beconstructed and the surface is divided into several patches.Each patch can then be conformally mapped to a planarparallelogram by integrating the holomorphic differentialform.

In our work, surfaces are represented as triangularmeshes, namely piecewise polygonal surfaces.The compu-tations with differential forms are based on solving ellipticpartial differential equations on surfaces using the finite el-ement method.

2.1 Computing Conformal Structures

A method for computing the conformal structure of a sur-face was introduced in [14]. Suppose

�is a closed genus

T}| ]surface with a conformal atlas 4 . The conformal

structure 4 induces holomorphic 1-forms; all holomorphic1-forms form a linear space

i � � of dimension U"T whichis isomorphic to the first cohomology group of the surface~ : � � �M�a . The set of holomorphic one-forms determinesthe conformal structure. Therefore, computing conformalstructure of

�is equivalent to finding a basis for

i � � .The holomorphic 1-form basis ��B+�M�H� (*� �1Uw�O��������UVT6�

is computed as follows: compute the homology basis, findthe dual cohomology basis, diffuse the cohomology basis toa harmonic 1-form basis, and then convert the harmonic 1-form basis to holomorphic 1-form basis by using the Hodgestar operator. The details of the computation are given in[14].

For surfaces with boundaries, we apply the conven-tional double covering technique, which glues two copiesof the same surface along their corresponding boundariesto form a symmetric closed surface. Then we apply theabove procedure to find the holomorphic 1-form basis.

In terms of data structure, a holomorphic 1-formis represented as a vector-valued function defined on theedges of the mesh B � �D� : � � � �H� (�� �1U8���O�{��U"T ,.

2.2 Selecting the Optimal Holomorphic 1-form

Given a Riemann surface�

, there are infinitely manyholomorphic 1-forms, but each of them can be expressedas a linear combination of the basis elements. We de-fine a canonical conformal parameterization as any linearcombination of the set of holomorphic basis functions B � ,� (�� � ^�^�^ �HT . They satisfy�w��� B�� (}� �� �where �{�H�s� (�� � ^�^�^ � are homology bases and � �� is the Kro-necker symbol. Then we compute a canonical conformalparameterization

B ( �� ��� : B��^

We select a specific parameterization one that max-imizes the uniformity of the induced grid over the entiredomain using the algorithms introduced in [15], for thepurpose of locating zero points in the next step.

2.3 Locating Zero Points

For surface with genus T*| � , any holomorphic 1-formB has UVT�`�U zero points. The horizontal trajectoriesthrough the zero points will partition the surface into sev-eral patches. Each patch is either a topological disk or acylinder, and can be conformally parameterized by B using���lW� (}mOn B .

78

Estimating the Conformal Factor Suppose we alreadyhave a global conformal parameterization, induced by aholomorphic 1-form B . Then we can estimate the confor-mal factor at each vertex, using the following formulae:

y �vGw ( �� �� �;� �����D�d� [ B��s� E��HG;�� [ �[   �FE� =`   �vGw [ � �HE��HG-X¡�0jD� (1)

where � is the valence of vertex G .

Locating Zero Points We find the cluster of vertices withrelatively small conformal factors (the lowest ¢Y`¤£p¥ ).These are candidates for zero points. We cluster all thecandidates using the metric on the surface. For each clus-ter, we pick the vertex that is closest to the center of gravityof the cluster, using the surface metric to define geodesicdistances.

Because the triangulation is finite and the computa-tion is an approximation, the number of zero points maynot equal the Euler number. In this case, we refine the tri-angulation of the neighborhood of the zero point candidateand refine the holomorphic 1-form B .

2.4 Holomorphic Flow Segmentation

Tracing Horizontal Trajectories Once the zero pointsare located, the horizontal trajectories through them can betraced.

First we choose a neighborhood � � of a vertex G rep-resenting a zero point, � � is a set of neighboring faces ofG , then we map it to the parameter plane by integrating B .Suppose a vertex ¦§Xz� � , and a path composed by a se-quence of edges on the mesh is b , then the parameter loca-tion of ¦ is ���v¦f ( m n B .

The map ���F¦¨ is a piecewise linear map. Then thehorizontal trajectory is mapped to the horizontal line © ( ]in the plane. We slice ����� � using the line © ( ]

by edgesplitting operations. Suppose the boundary of �<��� � inter-sects © ( ]

at a point Gpª , then we choose a neighborhoodof G ª and repeat the process.

Each time we extend the horizontal trajectory and en-counter edges intersecting the trajectory, we insert new ver-tices at the intersection points, until the trajectory reachesanother zero point or the boundary of the mesh. We repeatthe tracing process until each zero point connects « hori-zontal trajectories.

Critical Graph Given a surface�

and a holomorphic 1-form B on

�, we define the graph x � � �MB5 ( �"��� q �suh� ,

as the critical graph of B . Here � is the set of zero pointsof B , q is the set of horizontal trajectories connecting zeropoints or the boundary segments of

�, and u is the set of

surface patches segmented by q .

3 Experimental Results

We tested our algorithm on various surfaces, includinganatomic surfaces extracted from 3D MRI scans of thebrain, and synthetic geometric examples to illustrate theapproach. Figure 2 (a)-(d) shows a closed genus 2 sur-face. We visualized the conformal structure by projectinga checkerboard image back onto the surface (Figure 2 (a)).There is a zero point shown in Figure 2 (a). Another zeropoint is on the back of the ”figure-eight” shaped surfaceand is symmetric to this zero point. The traced horizontaland vertical trajectories are shown in Figure 2 (c). Fromthe computed conformal structure, the ”figure-eight” sur-face can be segmented into two patches (Figure 2 (c)). Eachpatch can then be conformally mapped to a rectangle (Fig-ure 2 (d)).

Figure 2 (e)-(g) shows experimental results for a hip-pocampal surface, a structure in the medial temporal lobeof the brain. The original surface is shown in (e). We leavetwo holes on the front and back of the hippocampal surface,representing its anterior junction with the amygdala, and itsposterior limit as it turns into the white matter of the fornix.It can be logically represented as an open boundary genusone surface, a cylinder (note that spherical harmonic repre-sentations would also be possible, if the ends were closed).The computed conformal structure is shown in (f). A hor-izontal trajectory curve is shown in (g). Cutting the sur-face along this curve, we can then conformally map thehippocampus to a rectangle. Since the surface of rectangleis similar to the one of hippocampus, the detailed surfaceinformation is well preserved in (h). Compared with otherspherical parameterization methods, which may have high-valence nodes and dense tiles at the poles of the sphericalcoordinate system, our parameterization can represent thesurface with minimal distortion.

Shape analysis of the lateral ventricles is of greatinterest in the study of psychiatric illnesses, includ-ing schizophrenia, and in degenerative diseases such asAlzheimer’s disease. These structures are often enlarged indisease and can provide sensitive measures of disease pro-gression. We can optimize the conformal parameterizationby topology modification. For the lateral ventricle surfacein each brain hemisphere, we introduce five cuts. Sincethese cutting positions are at the end of the frontal, occip-ital, and temporal horns of the ventricles, they can poten-tially be located automatically. The upper row in Figure 3shows 5 cuts introduced on two subjects ventricular sur-faces. After the cutting, the surfaces become open bound-ary genus 4 surfaces.

The middle two rows of Figure 3 show parameteriza-tions of the lateral ventricles of the brain. The second rowshows the results of parameterizing a ventricular surfacefor a 65-year-old patient with HIV/AIDS (note the disease-related enlargement)and the third row shows the results forthe ventricular model of a 21-year-old control subject. Thesurfaces are initially generated by using an unsupervisedtissue classifier to isolate a binary map of the cerebrospinal

79

Figure 2. The holomorphic flow segmentation results ona synthetic surface and a hippocampus surface. (a) con-formal parameterization of a two hole torus surface. (b)the result of iso-parameter tracing from the detected zeropoints. (d) the two rectangles to which two segments in (c)are conformally mapped. (f) a conformal parameterizationof the hippocampus surface in (e). (g) an iso-parametercurve used to unfold the surface. (h) the rectangle the sur-face is conformally mapped to.

fluid in the MR image, and tiling the surface of the largestconnected component inside the brain. There are a totalof 3 zero points on each of the ventricular surfaces. Twoof them are located at the middle part of the two ”arms”(where the temporal and occipital horns join at the ventric-ular atrium), as shown by the large black dots in the secondrow. The third zero point is located in the middle of themodel, where the frontal horns are closest to each other.Based on the computed conformal structure, we can parti-tion the surface into 6 patches. Each patch can be confor-mally mapped to a rectangle. Although the two brain ven-tricle shapes are very different, the segmentation results areconsistent in that the surfaces are partitioned into patcheswith the same relative arrangement and connectivity.

Finally, the conformal grids induced here are orthog-onal and therefore especially suitable for numerical dis-cretization of partial differential equation (PDEs) on brainsurfaces. Surface-based PDEs are useful for elastic regis-tration of surfaces, for EEG/MEG reconstruction, for sig-nal denoising and regularization, and for generation ofgeodesic paths on surfaces. By using global conformal pa-rameterization, these problems can be converted into solv-ing PDEs on planar domains. Using the conformal factorand the Riemannian metric tensor, we can implement co-variant differentiation easily. Figure 4 shows illustrativeexamples of fluid flow simulation by solving the Navier-Stokes equation on the hippocampus surface. Althoughthese are artificial examples, they illustrate the feasibility ofdiscretizing PDEs on the conformal grids developed here.Compared with other work [16], our approach is relatively

efficient and avoids some complexity in handling compli-cated boundary constraints. We are currently extending thiswork to implement more general differential operators onbrain surfaces, for using in nonlinear surface registrationand surface-based signal processing.

Figure 3. Illustrates surface parameterization results for thelateral ventricles. The upper row shows ¢ cuts are intro-duced and they convert the lateral ventricle surface into agenus 4 surface. The second row shows models parameter-ized using holomorphic 1-forms, for a 65-year- old subjectwith HIV/AIDS and the third row shows the same mapscomputed for a healthy 21-year-old control subject. Thecomputed conformal structure, holomorphic flow segmen-tation and their associated parameter domains are shown.

4 Conclusion and Future Work

In this paper, we presented a brain surface parameterizationmethod that invokes the Riemann surface structure to gen-erate conformal grids on surfaces of arbitrary complexity(including branching topologies). For high genus surfaces,a global conformal parameterization induces a canonicalsegmentation, i.e. there is a discrete partition of the sur-face into conformally parameterized patches. Each parti-tion is either a topological disk or a cylinder and can be con-formally mapped to a rectangle in the parameter domain.We demonstrated the parameterization for both closed andopen boundary surfaces. We tested our algorithm on thehippocampus and lateral ventricle surfaces. The grid gen-eration algorithm is intrinsic (i.e. does not depend on anyinitial choice of surface coordinates) and is stable, as shown

80

Figure 4. Illustrates solving the Navier-Stokes equation forfluid propagations on brain surfaces. Although these areartificial examples, they show the feasibility of discretizingPDEs on the conformal grid structure, which can ensurenumerical accuracy for various applications of PDE-basedsignal processing on anatomical surfaces.

by grids induced on ventricles of various shapes and sizes.Compared with other work conformally mapping brain sur-faces to sphere, our work may introduce less distortionand may be especially convenient for other post-processingwork such as surface registration and landmark matching.Our future work will focus on signal processing on brainsurfaces, as well as brain surface registration and shape andasymmetry analysis for subcortical structures.

References

[1] P.M. Thompson, J.N. Giedd, R.P. Woods, D. Mac-Donald, A.C. Evans, and A.W. Toga. Growth patternsin the developing brain detected by using continuum-mechanical tensor maps. Nature, 404(6774):190–193, March 2000.

[2] J.G. Csernansky, S. Joshi, L.E. Wang, J. Haller,M. Gado, J.P. Miller, U. Grenander, and M.I. Miller.Hippocampal morphometry in schizophrenia via highdimensional brain mapping. Proc. Natl. Acad. Sci.,95:11406–11411, September 1998.

[3] P.M. Thompson, R.P. Woods, M.S. Mega, and A.W.Toga. Mathematical/computational challenges in cre-ating population-based brain atlases. In Human BrainMapping, volume 9, pages 81–92, Feb. 2000.

[4] G.L. Kindlmann, D.M. Weinstein, A.D. Lee, A.W.Toga, and P.M. Thompson. Visualization of anatomiccovariance tensor fields. In Proc. IEEE Engineeringin Medicine and Biology Society (EMBS), San Fran-sisco, CA, Sept. 1-5 2004.

[5] E.L. Schwartz, A. Shaw, and E. Wolfson. A nu-merical solution to the generalized mapmaker’s prob-lem: Flattening nonconvex polyhedral surfaces. IEEETransactions on Pattern Analysis and Machine Intel-ligence, 11(9):1005–1008, Sep. 1989.

[6] B. Timsari and R. Leahy. An optimization method forcreating semi-isometric flat maps of the cerebral cor-

tex. In Proceedings of SPIE, Medical Imaging, SanDiego, CA, Feb. 2000.

[7] M. K. Hurdal and K. Stephenson. Cortical cartogra-phy using the discrete conformal approach of circlepackings. NeuroImage, 23:S119–S128, 2004.

[8] S. Angenent, S. Haker, A. Tannenbaum, and R. Kiki-nis. Conformal geometry and brain flattening. MIC-CAI, pages 271–278, 1999.

[9] X. Gu, Y. Wang, T.F. Chan, P.M. Thompson, and S.-T. Yau. Genus zero surface conformal mapping andits application to brain surface mapping. IEEE Trans.on Med. Imaging, 23(8):949–958, Aug. 2004.

[10] S. Joshi, M. Miller, and U. Grenander. On the geom-etry and shape of brain sub-manifolds. IEEE PAMI,11:1317–1343, 1997.

[11] A. Kelemen, G. Szekely, and G. Gerig. Elastic model-based segmentation of 3d neuroradiological data sets.IEEE Trans. Med. Imaging, 18(10):828–839, Oct.1999.

[12] G. Gerig, M. Styner, D. Jones, D. Weinberger, andJ. Lieberman. Shape analysis of brain ventricles usingspharm. In Proc. MMBIA 2001, pages 171–178. IEEEComputer Society, Dec. 2001.

[13] S. Pizer, D. Fritsch, P. Yushkevich, V. Johnson, andE. Chaney. Segmentation, registration, and measure-ment of shape variation via image object shape. IEEETrans. Med. Imaging, 18:851–865, Oct. 1999.

[14] X. Gu and S.T. Yau. Computing conformal structuresof surfaces. Communication of Information and Sys-tems, December 2002.

[15] M. Jin, Y. Wang, S.T. Yau, and X. Gu. Optimal globalconformal surface parameterization for visualization.In IEEE Visualization, pages 267–274, Austin, TX,Oct. 2004.

[16] J. Stam. Flows on surfaces of arbitrary topology. InACM Transactions on Graphics, volume 22, pages724–731, 2003.

81