Genus Zero Surface Conformal Mapping and Its Application to ...

IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 23, NO. 7, JULY 2004 1

Genus Zero Surface Conformal Mapping and ItsApplication to Brain Surface MappingXianfeng Gu, Yalin Wang, Tony F. Chan, Paul M. Thompson, Shing-Tung Yau

Abstract— It is well known that any genus zero surface canbe mapped conformally onto the sphere and any local portionthereof onto a disk. However, it is not trivial to find a generalmethod which computes a conformal mapping between twogeneral genus zero surfaces. We propose a new variationalmethod which can find a unique mapping between any two genuszero manifolds by minimizing the harmonic energy of the map.We demonstrate the feasibility of our algorithm by applying it tothe cortical surface matching problem. We use a mesh structureto represent the brain surface. Further constraints are addedto ensure that the conformal map is unique. The conformalmapping is also used for spherical harmonic transformation forthe purpose of brain surface compression and rotation-invariantshape analysis. Empirical tests on MRI data show that themappings preserve angular relationships, are stable in MRIsacquired at different times, and are robust to differences in datatriangulation, and resolution. Compared with other brain surfaceconformal mapping algorithms, our algorithm is more stable andhas good extensibility.

Index Terms— Conformal Map, Brain Mapping, LandmarkMatching, Spherical Harmonic Transformation.

I. I NTRODUCTION

RECENT developments in brain imaging have acceleratedthe collection and databasing of brain maps. Nonetheless,

computational problems arise when integrating and comparingbrain data. One way to analyze and compare brain data is tomap them into a canonical space while retaining geometricinformation on the original structures as far as possible [1]–[5]. Fischl et al. [1] demonstrate that surface based brainmapping can offer advantages over volume based brain map-ping, especially when localizing cortical deficits and functionalactivations. Thompson et al. [4], [5] introduce a mathematicalframework based on covariant partial differential equations,and pull-backs of mappings under harmonic flows, to helpanalyze signals localized on brain surfaces.

A. Previous work

Conformal surface parameterizations have been studied in-tensively. Most works on conformal parameterizations dealwith surface patches homeomorphic to topological disks. For

X. Gu is with the department of Computer and Information Science andEngineering, University of Florida, FL 32611, [email protected].

Y. Wang is with the Mathematics Department, UCLA, CA 90095, [email protected].

T. F. Chan is with the Mathematics Department, UCLA, CA 90095,[email protected].

P. M. Thompson is with the Laboratory of Neuro Imaging, Department ofNeurology, UCLA School of Medicine, [email protected].

S.-T. Yau is with the Department of Mathematics, Harvard University,[email protected]

surfaces with arbitrary topologies, Gu and Yau [6] introduce ageneral method for global conformal parameterizations basedon the structure of the cohomology group of holomorphicone-forms. They generalize the method for surfaces withboundaries in [7].

For genus zero surfaces, there are five basic approaches toachieve conformal parameterizations.

1) Harmonic energy minimization.Eck et al. [8] introducethe discrete harmonic map, which approximates thecontinuous harmonic map [9] by minimizing ametricdispersion criterion. Desbrun et al. [10], [11] com-pute the discrete Dirichlet energy and apply confor-mal parameterization to interactive geometry remeshing.Pinkall and Polthier compute the discrete harmonic mapand Hodge star operator for the purpose of creatinga minimal surface [12]. Kanai et al. use a harmonicmap for geometric metamorphosis in [13]. Gu and Yauin [6] introduce a non-linear optimization method tocompute global conformal parameterizations for genuszero surfaces. The optimization is carried out in thetangent spaces of the sphere.

2) Cauchy-Riemann equation approximation.Levy et al.[14] compute a quasi-conformal parameterization oftopological disks by approximating the Cauchy-Riemannequation using the least squares method. They show rig-orously that the quasi-conformal parameterization existsuniquely, and is invariant to similarity transformations,independent of resolution, and orientation preserving.

3) Laplacian operator linearization.Haker et al. [3], [15]use a method to compute a global conformal mappingfrom a genus zero surface to a sphere by representingthe Laplace-Beltrami operator as a linear system.

4) Angle based method.Sheffer et al. [16] introduce anangle based flattening method to flatten a mesh to a 2Dplane so that it minimizes the relative distortion of theplanar angles with respect to their counterparts in thethree-dimensional space.

5) Circle packing. Circle packing is introduced in [2],[17]. Classical analytic functions can be approximatedusing circle packing. For general surfaces inR3, thecircle packing method considers only the connectivitybut not the geometry, so it is not suitable for ourparameterization purpose.

Bakircioglu et al. use spherical harmonics to compute a flowon the sphere in [18] in order to match curves on the brain.Thompson and Toga use a similar approach in [19]. This flowfield can be thought of as the variational minimizer of the


integral over the sphere ofLu, with L some power of theLaplacian, andu the deformation. This is very similar to thespherical harmonic map used in this paper.

B. Basic Idea

Define a mappingf : M1 → M2 between two surfacesM1

andM2, where the first fundamental forms of the two surfacesareds2

1 =∑

gijdxidxj andds22 =

∑gijdxidxj . We sayf is

a conformal mappingif f∗ds22 = λds2

1, whereλ is a positivescalar andf∗ is the pull back function.

f∗ds22 =

∑mn

(∑

ij

gij(f(x))∂f i

∂xm

∂f j

∂xn)dxmdxn

Intuitively, all the angles onM1 are preserved onM2. Figure 1shows a conformal mapping example. Figure 1(a) shows a realmale face. We conformally map it to a square as in 1(c) andget its conformal parameterization. We illustrate the conformalparameterization via the texture mapping of a checkerboard inFigure 1.

(a) (b) (c)

Fig. 1. Conformal surface parameterization examples. (a) is a real male face.(c) is a square into which the human face is conformally mapped. (b) is theconformal parameterization illustrated by the texture map. As shown, the rightangles on the checkboard are well preserved on the surface in (b).

It is well known that any genus zero surface can be mappedconformally onto the sphere and any local portion thereofonto a disk. This mapping, a conformal equivalence, is one-to-one, onto, and angle-preserving. Moreover, the elementsof the first fundamental form remain unchanged, except fora scaling factor (the so-calledConformal Factor). For thisreason, conformal mappings are often described as beingsimilarities in the small. Since the cortical surface of the brainis a genus zero surface, conformal mapping offers a convenientmethod to retain local geometric information, when mappingdata between surfaces. Indeed, several groups have createdflattened representations or visualizations of the cerebral cortexor cerebellum [2], [3] using conformal mapping techniques.However, these approaches are either not strictly angle preserv-ing [2], or there may be areas with large geometric distortions[3]. In this paper, we propose a new genus zero surfaceconformal mapping algorithm [6] and demonstrate its usein computing conformal mappings between brain surfaces.Our algorithm depends only on the surface geometry andis invariant to changes in image resolution and the specificsof the data triangulation. Our experimental results show thatour algorithm has advantageous properties for cortical surfacematching.

SupposeK is a simplicial complex, andf : |K| → R3,which embeds|K| in R3; then(K, f) is called a mesh. Giventwo genus zero meshesM1,M2, there are many conformalmappings between them. Our algorithm for computing confor-mal mappings is based on the fact that for genus zero surfacesS1, S2, f : S1 → S2 is conformal if and only iff is harmonic.All conformal mappings betweenS1, S2 form a group, the so-called Mobius group. Figure 2 show some examples of Mobiustransformations. We can conformally map the surface of thehead of Michelangelo’s David to a sphere. When we drawthe longitude and latitude lines on the sphere, we can inducecorresponding circles on the original surface (a) and (b). Weapply a Mobius transformation to the sphere and make thetwo eyes become north and south poles. When we draw thelongitude and latitude lines again (c), we get an interestingresult shown in (d). Note all the right angles between the linesare well preserved in (b) and (d). This example demonstratesthat all the conformal mapping results form a Mobius group.

(a) (b)

(c) (d)

Fig. 2. Mobius transformation example. We conformally map the surfaceof the head of Michelangelo’s David to a sphere. In (a), we select the nosetip as the north pole and draw longitude and latitude lines on the sphere.(b) shows the results on the original David head model. We apply a Mobiustransformation on the sphere in (a) and make the two eyes become the northand south poles. When drawing the longitude and latitude lines on the sphere(c), we get an interesting configuration for the lines on the original surface(d).

Our method is as follows: we first find a homeomorphismhbetweenM1 andM2, then deformh such thath minimizes theharmonic energy. To ensure the convergence of the algorithm,constraints are added; this also ensures that there is a uniqueconformal map.

This paper is organized as follows. In Section II, we givethe definitions of a piecewise linear function space, innerproduct and piecewise Laplacian. In Section III, we describethe steepest descent algorithm which is used to minimizethe string energy. In Section IV, we detail our conformalspherical mapping algorithms. In Section V, the conformalparameterization is optimized by integrating landmark infor-mation. Section VI applies conformal mapping for spherical


harmonic transformation, and rotation-invariant shape analysis.Experimental results on conformal mapping for brain surfacesare reported in Section VII. In Section VIII, we compare ouralgorithm with other conformal mapping approaches used inneuroimaging. We conclude the paper in Section IX.

II. PIECEWISEL INEAR FUNCTION SPACE, INNER

PRODUCT AND LAPLACIAN

For the diffeomorphisms between genus zero surfaces, ifthe map minimizes the harmonic energy, then it is conformal.Based on this fact, the algorithm is designed as a steepestdescent method.

This section formulates the mathematical concepts in arigorous way. The major concepts, the harmonic energy of amap and its derivative, are defined. Because all the calculationis carried out on surfaces, we use the absolute derivative.Furthermore, for the purpose of implementation, we introducethe definitions in discrete form.

We useK to represent the simplicial complex,u, v to denotethe vertices, andu, v to denote the edge spanned byu, v.We usef, g to represent the piecewise linear functions definedon K, use~f to represent vector value functions. We use∆PL

to represent the discrete Laplacian operator.Definition 1: All piecewise linear functions defined onK

form a linear space, denoted byCPL(K).In practice, we useCPL(K) to approximate all functions

defined onK. So the final result is an approximation to theconformal mapping. The higher the resolution of the mesh is,the more accurate the approximated conformal mapping is.

Definition 2: Suppose a set of string constantsku,v areassigned for each edgeu, v, the inner product onCPL isdefined as the quadratic form

< f, g >=12

∑

u,v∈K

ku,v(f(u)− f(v))(g(u)− g(v)) (1)

The energy is defined as the norm onCPL.Definition 3: Supposef ∈ CPL, the string energy is

defined as:

E(f) =< f, f >=∑

u,v∈K

ku,v||f(u)− f(v)||2 (2)

By changing the string constantsku,v in the energy formula,we can define different string energies.

Definition 4: If string constantsku,v ≡ 1, the string energyis known as the Tuette energy.

Definition 5: Suppose edgeu, v has two adjacent facesTα, Tβ , with Tα = v0, v1, v2, define the parameters

aαv1,v2

=12

(v1 − v3) · (v2 − v3)|(v1 − v3)× (v2 − v3)| (3)

aαv2,v3

=12

(v2 − v1) · (v3 − v1)|(v2 − v1)× (v3 − v1)| (4)

aαv3,v1

=12

(v3 − v2) · (v1 − v2)|(v3 − v2)× (v1 − v2)| (5)

(6)

Tβ is defined similarly. Ifku,v = aαu,v +aβ

u,v, the string energyobtained is called the harmonic energy.

The string energy is always a quadratic form. By carefullychoosing the string coefficients, we make sure the quadraticform is positive definite. This will guarantee the convergenceof the steepest descent method.

Definition 6: The piecewise Laplacian is the linear operator∆PL : CPL → CPL on the space of piecewise linearfunctions onK, defined by the formula

∆PL(f) =∑

u,v∈K

ku,v(f(v)− f(u)) (7)

If f minimizes the string energy, thenf satisfies the condition∆PL(f) = 0. SupposeM1,M2 are two meshes and the map~f : M1 → M2 is a map between them,~f can be treated as amap fromM1 to R3 also.

Definition 7: For a map~f : M1 → R3, ~f = (f0, f1, f2),fi ∈ CPL, i = 0, 1, 2, we define the energy as the norm of~f :

E(~f) =2∑

i=0

E(fi) (8)

The Laplacian is defined in a similar way,Definition 8: For a map ~f : M1 → R3 , the piecewise

Laplacian of~f is

∆PL~f = (∆PLf0,∆PLf1, ∆PLf2) (9)

A map ~f : M1 → M2 is harmonic, if and only if∆PL~f only

has a normal component, and its tangential component is zero.

∆PL(~f) = (∆PL~f)⊥ (10)

III. STEEPESTDESCENTALGORITHM

Suppose we would like to compute a mapping~f : M1 →M2 such that~f minimizes a string energyE(~f). This can besolved easily by the steepest descent algorithm:

d~f(t)dt

= −∆~f(t) (11)

~f(M1) is constrained to be onM2, so−∆~f is a section ofM2’s tangent bundle.

Specifically, suppose~f : M1 → M2, and denote theimage of each vertexv ∈ K1 as ~f(v). The normal onM2 at~f(v) is ~n(~f(v)). Define the normal component as

Definition 9: The normal component

(∆~f(v))⊥ =< ∆~f(v), ~n(~f(v)) > ~n(~f(v)), (12)

where<, > is the inner product inR3.Definition 10: The absolute derivative is defined as

D~f(v) = ∆~f(v)− (∆~f(v))⊥ (13)Then equation (14) isδ ~f = −D~f × δt.

IV. CONFORMAL SPHERICAL MAPPING

SupposeM2 is S2, then a conformal mapping~f : M1 → S2

can be constructed by using the steepest descent method. Themajor difficulty is that the solution is not unique but forms aMobius group.

Definition 11: Mapping f : C → C is a Mobius transfor-mation if and only if

f(z) =az + b

cz + d, a, b, c, d ∈ C, ad− bc 6= 0, (14)


whereC is the complex plane.All M obius transformations form the Mobius transformation

group. In order to determine a unique solution we can adddifferent constraints. In practice we use the following twoconstraints: the zero mass-center constraint and a landmarkconstraint.

Definition 12: Mapping ~f : M1 → M2 satisfies the zeromass-center condition if and only if∫

M2

~fdσM1 = 0, (15)

whereσM1 is the area element onM1.All conformal maps fromM1 to S2 satisfying the zero mass-center constraint are unique up to the Euclidean rotation group(which is 3 dimensional). We use the Gauss map as the initialcondition.

Definition 13: A Gauss mapN : M1 → S2 is defined as

N(v) = ~n(v), v ∈ M1, (16)

where~n(v) is the normal atv.Algorithm 1: Spherical Tuette Mapping

Input (meshM ,step lengthδt, energy difference thresholdδE), output(~t : M → S2) where ~t minimizes the Tuetteenergy.

1) Compute Gauss mapN : M → S2. Let~t = N , computeTuette energyE0.

2) For each vertexv ∈ M , compute Absolute derivativeD~t.

3) Update~t(v) by δ~t(v) = −D~t(v)δt.4) Compute Tuette energyE.5) If E − E0 < δE, return~t. Otherwise, assignE to E0

and repeat steps2 through to5.Because the Tuette energy has a unique minimum, the algo-rithm converges rapidly and is stable. We use it as the initialcondition for the conformal mapping.

Algorithm 2: Spherical Conformal Mapping

Input (meshM ,step lengthδt, energy difference thresholdδE), output(~h : M → S2). Here~h minimizes the harmonicenergy and satisfies the zero mass-center constraint.

1) Compute Tuette embedding~t. Let~h = ~t, compute TuetteenergyE0.

2) For each vertexv ∈ M , compute the absolute derivativeD~h.

3) Update~h(v) by δ~h(v) = −D~h(v)δt.4) Compute Mobius transformation~ϕ0 : S2 → S2, such

that

Γ(~ϕ) =∫

S2~ϕ ~hdσM1 , ~ϕ ∈ Mobius(CP 1)(17)

~ϕ0 = min~ϕ||Γ(~ϕ)||2 (18)

where σM1 is the area element onM1. Γ(~ϕ) is themass center,~ϕ minimizes the norm in the mass centercondition.

5) compute the harmonic energyE.6) If E − E0 < δE, return~t. Otherwise, assignE to E0

and repeat step2 through to step6.

Step4 is non-linear and expensive to compute. In practice weuse the following procedure to replace it:

1) Compute the mass center~c =∫

S2~hdσM1 ;

2) For all v ∈ M , ~h(v) = ~h(v)− ~c;

3) For all v ∈ M , ~h(v) =~h(v)

||~h(v)|| .This approximation method is good enough for our purpose.By choosing the step length carefully, the energy can bedecreased monotonically at each iteration.

V. OPTIMIZE THE CONFORMAL PARAMETERIZATION BY

USING LANDMARKS

In order to compare two brain surfaces, it is desirable toadjust the conformal parameterization and match the geometricfeatures on the brains as well as possible. We define an energyto measure the quality of the parameterization. Suppose twobrain surfacesS1, S2 are given, conformal parameterizationsare denoted asf1 : S2 → S1 andf2 : S2 → S2, thematchingenergyis defined as

E(f1, f2) =∫

S2||f1(u, v)− f2(u, v)||2dudv (19)

We can composite a Mobius transformationτ with f2, suchthat

E(f1, f2 τ) = minζ∈Ω

E(f1, f2 ζ), (20)

where Ω is the group of Mobius transformations. We uselandmarks to obtain the optimal Mobius transformation. Land-marks are commonly used in brain mapping. We manuallylabel the landmarks on the brain as a set of uniformlyparametrized sulcal curves [4], as shown in Figure 7. Firstwe conformally map two brains to the sphere, then we pursuean optimal Mobius transformation to minimize the Euclideandistance between the corresponding landmarks on the spheres.Suppose the landmarks are represented as discrete point sets,and denoted aspi ∈ S1 and qi ∈ S2, pi matchesqi,i = 1, 2, . . . , n. The landmark mismatch functional foru ∈ Ωis defined as

E(u) =n∑

i=1

||pi − u(qi)||2, u ∈ Ω, pi, qi ∈ S2 (21)

In general, the above variational problem is a nonlinear one. Inorder to simplify it, we convert it to a least squares problem.First we project the sphere to the complex plane, then theMobius transformation is represented as a complex linearrational formula, Equation 14. We add another constraint foru, so thatu maps infinity to infinity. That means the northpoles of the spheres are mapped to each other. Thenu can berepresented as a linear formaz + b. Then the functional ofucan be simplified as

E(u) =n∑

i=1

g(zi)|azi + b− τi|2 (22)

wherezi is the stereo-projection ofpi, τi is the projection ofqi, g is the conformal factor from the plane to the sphere, itcan be simplified as

g(z) =4

1 + zz. (23)

So the problem is a least squares problem.


VI. SPHERICAL HARMONIC ANALYSIS

Let L2(S2) denote the Hilbert space of square integrablefunctions on theS2. In spherical coordinates,θ is taken asthe polar (colatitudinal) coordinate withθ ∈ [0, π], andφ asthe azimuthal (longitudinal) coordinate withφ ∈ [0, 2π). Theusual inner product is given by

< f, h >=∫ π

0

[∫ 2π

0

f(θ, φ)h(θ, φ)dφ] sin θdθ.

A function f : S2 → R is called aSpherical Harmonic,if it is an eigenfunction of Laplace-Beltrami operator, namely∆f = λf , whereλ is a constant. There is a countable setof spherical harmonics which form an orthonormal basis forL2(S2).

For any nonnegative integerl and integerm with |m| ≤ l,the (l, m)−spherical harmonicY m

l is a harmonic homoge-neous polynomial of degreel. The harmonics of degreelspan a subspace ofL2(S2) of dimension2l + 1 which isinvariant under the rotations of the sphere. The expansion ofany functionf ∈ L2(S2) in terms of spherical harmonics canbe written

f =∑

l≥0

∑

|m|≤l

f(l, m)Y ml

and f(l, m) denotes the(l,m) Fourier coefficient, equal to< f, Y m

l >. Spherical harmonicY ml has an explicit formula

Y ml (θ, φ) = kl,mPm

l (cosθ)eimφ,

where Pml is the associated Legendre functionof degreel

and orderm and kl,m is a normalization factor. The detailsare explained in [20].

Once the brain surface is conformally mapped toS2,the surface can be represented as three spherical functions,x0(θ, φ), x1(θ, φ) and x2(θ, φ). The function xi(θ, φ) ∈L2(S2) is regularly sampled and transformed toxi(l,m) usingFast Spherical Harmonic Transformation as described in [21].

Many processing tasks that use the geometric surface ofthe brain can be accomplished in the frequency domain moreefficiently, such as geometric compression, matching, surfacedenoising, feature detection, and shape analysis [22], [23].

A. Brain Geometry Compression

Similar to image compression using Fourier analysis, geo-metric brain data can be compressed using spherical harmonicanalysis [22]. Global geometric information is concentratedin the low frequency components, whereas noise and locallydetailed information is concentrated in the high frequency part.By using low pass filtering, we can keep the major geometricfeatures and compress the brain surface without losing toomuch information.

B. Rotation Invariant Shape Descriptor

The geometric representation(x1(θ, φ), x2(θ, φ), x3(θ, φ))depends on the orientation of the brain. Brain registrationhas to be applied first in order to compare the geometricrepresentations of two different brains. A rotation-invariantshape descriptor can be formulated based on the frequency

coefficients. Because the harmonics of degreel span therotation invariant subspace ofL2(S2), the following shapedescriptor is also rotation invariant

s(l) =∑

i

∑

|m|≤l

||xi(l,m)||2. (24)

Given two brain surfaces, we can compute their shape de-scriptor from their spherical harmonic spectrum, and comparethem directly without any registration.

Figure 10 illustrate the shape descriptors for the same brainwith different orientations. It is clear that the shape descriptoris totally rotation invariant [24].

Spherical harmonic analysis can be applied to brain surfacedenoising and feature extraction also. In order to compare twobrain surfaces, it can be more efficient to compare their lowband frequency coefficients.

VII. E XPERIMENTAL RESULTS

The 3D brain meshes are reconstructed from 3D256x256x124 T1 weighted SPGR (spoiled gradient) MRIimages, by using an active surface algorithm that deforms atriangulated mesh onto the brain surface [5]. Figure 3(a) and(c) show the same brain scanned at different times [4]. Becauseof the inaccuracy introduced by scanner noise in the input data,as well as slight biological changes over time, the geometricinformation is not exactly the same. Figure 3(a) and (c) revealminor differences.

(a) (b)

(c) (d)

Fig. 3. Reconstructed brain meshes and their spherical harmonic mappings.(a) and (c) are the reconstructed surfaces for the same brain scanned atdifferent times. Due to scanner noise and inaccuracy in the reconstructionalgorithm, there are visible geometric differences. (b) and (d) are the sphericalconformal mappings of (a) and (c) respectively; the normal information ispreserved. By the shading information, the correspondence is illustrated.

The conformal mapping results are shown in Figure 3(b)and (d). From this example, we can see that although the brainmeshes are slightly different, the mapping results look quitesimilar. The major features are mapped to the same position


on the sphere. This suggests that the computed conformalmappings continuously depend on the geometry, and canmatch the major features consistently and reproducibly. Inother words, conformal mapping may be a good candidatefor a canonical parameterization in brain mapping.

(a) (b)

Fig. 4. Conformal texture mapping. (a) Texture mapping of the sphere;(b) Texture mapping of the brain. The conformality is visualized by texturemapping of a checkerboard image. The sphere is mapped to the plane bystereographic projection, then the planar coordinates are used as the texturecoordinates. This texture parameter is assigned to the brain surface throughthe conformal mapping between the sphere and the brain surface. All the rightangles in the texture are preserved on the brain surface.

Figure 4 shows the mapping is conformal by texture map-ping a checkerboard to both the brain surface mesh and aspherical mesh. Each black or white square in the texture ismapped to sphere by stereographic projection, and pulled backto the brain. Note that the right angles are preserved both onthe sphere and the brain.

(a) (b)

Fig. 5. Conformal mappings of surfaces with different resolutions. (a).Surfacewith 20, 000 faces; (b) Surface with50, 000 faces. The original brain surfacehas 50, 000 faces, and is conformally mapped to a sphere, as shown in(a). Then the brain surface is simplified to20, 000 faces, and its sphericalconformal mapping is shown in (b).

Conformal mappings are stable and depend continuouslyon the input geometry but not on the triangulations, and areinsensitive to the resolutions of the data. Figure 5 shows thesame surface with different resolutions, and their conformalmappings. The mesh simplification is performed using astandard method. The refined model has 50k faces, coarse onehas 20k faces. The conformal mappings map the major featuresto the same positions on the spheres.

In order to measure the conformality, we map the iso-polarangle curves and iso-azimuthal angle curves from the sphereto the brain by the inverse conformal mapping, and measurethe intersection angles on the brain. The distribution of theangles of a subject (A) are illustrated in Figure 6. The anglesare concentrated about the right angle.

0 20 40 60 80 100 120 140 160 180 2000

1000

2000

3000

4000

5000

6000

7000

8000Angle Distribution

Angles

Fre

quen

cy

(a) (b)

Fig. 6. Conformality measurement. (a) Intersection angles; (b) Angledistribution. The curves of iso-polar angle and iso-azimuthal angle are mappedto the brain, and the intersection angles are measured on the brain. Thehistogram is illustrated.

Subject Vertex # Face # Before AfterA 65,538 131,072 - -B 65,538 131,072 604.134 506.665C 65,538 131,072 414.803 365.325

TABLE I

MATCHING ENERGY FOR THREE SUBJECTS. SUBJECTA WAS USED AS THE

TARGET BRAIN. FOR SUBJECTSB AND C, WE FOUND M OBIUS

TRANSFORMATIONS THAT MINIMIZED THE LANDMARK MISMATCH

FUNCTIONS, RESPECTIVELY.

Figure 7 shows the landmarks, and the result of the opti-mization by a Mobius transformation. We also computed thematching energy, following Equation 19. We did our testing onthree example subjects. Their information is shown in Table I.We took subject A as the target brain. For each new subjectmodel, we found a Mobius transformation that minimizedthe landmark mismatch energy on the maximum intersectionsubsets of it and A. As shown in Table I, the matching energieswere reduced after the Mobius transformation.

Figure 8 illustrates the geometric compression results usingspherical harmonic compression. Figure 9 shows theL2 errorsof the compression result. The low pass filters are appliedto remove high frequency components, and the surfaces arereconstructed from the remaining low frequency components.The surface is normalized such that the total area is4π, thentheL2 error between the reconstructed surface and the originalsurface is computed. The curve shows the normalizedL2 errorvs the ratio of retained low frequency components. The figureillustrates that the major geometric information is encoded inthe low frequency part.

Figure 10 illustrates the relative error between the rotation-invariant shape descriptors for the original brain surface andfor the rotated brain surface. Because the first30 low fre-quency components generate more than99% of the totalenergy, only the first30 shape descriptor errors are shown inthe figure. From the figure, it is clear that the relative errorsare less than one percent, and therefore the shape descriptorsare rotation invariant.

The method described in this work is quite general. Wetested the algorithm on other genus zero surfaces, includingthe hand and foot surface. Some of the experimental resultsare illustrated in Figure 11.


(a) (b)

(c) (d)

Fig. 7. Mobius transformation to minimize the deviations between landmarks.The blue curves are the landmarks. The correspondence between curves hasbeen preassigned. The desired Mobius transformation is obtained to minimizethe matching error on the sphere.

(a) (b)

(c) (d)

Fig. 8. This figure illustrates the geometric compression results usingspherical harmonics. After we conformally map the brain to a sphere, wecan use spherical harmonics to compress the geometry. (a) is the originalbrain surface. (b), (c) and (d) are brain surfaces reconstructed from sphericalharmonics with1

8, 1

64and 1

256of the original low frequency coefficients,

separately.

VIII. C OMPARISON WITH OTHER WORK

Several other studies of conformal mappings between brainsurfaces are reported in [2], [3], [15], [17]. In [2], [17], Hurdalet al. used the circle packing theorem and the ring lemma toestablish a theorem: there is a unique circle packing in theplane (up to certain transformations) which is quasi-conformal(i.e. angular distortion is bounded) for a simply-connectedtriangulated surface. They demonstrated their experimentalresults for the surface of the cerebellum. This method onlyconsiders the topology without considering the brain’s geo-metric structure. Given two different mesh structures of thesame brain, one can predict that their methods may generatetwo different mapping results. Compared with their work, ourmethod really preserves angles and establishes a good mappingbetween brains and a canonical space.

Haker et al. [3], [15] built a finite element approximation ofthe conformal mapping method for brain surface parameteriza-tion. They selected a point as the north pole and conformallymapped the cortical surface to the complex plane. In theresulting mapping, the local shape is preserved and distancesand areas are only changed by a scaling factor. Based onHaker et al. [3], Joshi et al. [25] obtained a unique conformalmapping by fixing three point correspondences between twobrains. Since stereo projection is involved, there is significantdistortion around the north pole areas, which brings instabilityto this approach. Compared with their work, our method ismore accurate, with no regions of large area distortion. It isalso more stable and can be readily extended to compute mapsbetween two general manifolds.

Finally, we note that Memoli et al. [26] mentioned theywere developing implicit methods to compute harmonic mapsbetween general source and target manifolds. They used levelsets to represent the brain surfaces. Due to the extensivefolding of the human brain surface, these mappings have tobe designed very carefully.

IX. CONCLUSION AND FUTURE WORK

In this paper, we propose a general method which findsa unique conformal mapping between genus zero manifolds.Specifically, we demonstrate its feasibility for brain surfaceconformal mapping research. Our method only depends onthe surface geometry and not on the mesh structure (i.e.gridding) and resolution. Our algorithm is efficient and stable

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

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Fig. 9. This figure illustrates the normalizedL2 errors of the compression.


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Fig. 10. The brain surface is rotated90 degree with respect to x-axis. Theshape descriptors defined in 24 are computed for both the original surfaceand the rotated surface, denoted ass(l) and s′(l) respectively. The relativeerrors(s(l)−s′(l))/s(l) is illustrated as a function ofl. Because the first30s(l) generate almost all the energy, the curve is truncated atl = 30. Fromthe curve it can be verified that the relative error is less than one percent, andthus the shape descriptors are rotation invariant.

Fig. 11. Spherical conformal mapping of genus zero surfaces. Extrudingparts (such as fingers and toes) are mapped to denser regions on the sphere.

in reaching a solution. Landmark information are integratedto optimize the brain surface conformal parameterization.We apply conformal mapping to the brain surface sphericalharmonic transformation, which is used for geometry com-pression and rotation-invariant shape analysis. In the futureresearch, we will study several applications of these mappingalgorithms, such as providing a canonical space for automatedfeature identification, brain to brain registration, brain structuresegmentation, brain surface denoising, shape analysis andconvenient surface visualization, among others. We are tryingto generalize this approach to compute conformal mappingsbetween non-zero genus surfaces.

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