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Brain Surface Conformal Parameterization with
Algebraic Functions
Yalin Wang^1 ,^2 , Xianfeng Gu^3 , Tony F. Chan^1 , Paul M. Thompson^2 , and Shing-Tung Yau^4 (^1) Mathematics Department, UCLA, Los Angeles, CA 90095, USA, (^2) Lab. of Neuro Imaging, UCLA School of Medicine, Los Angeles, CA 90095, USA, (^3) Comp. Sci. Department, SUNY at Stony Brook, Stony Brook, NY 11794, USA, (^4) Department of Mathematics, Harvard University, Cambridge, MA 02138, USA, {ylwang}@math.ucla.edu
Abstract. In medical imaging, parameterized 3D surface models are of great interest for anatomical modeling and visualization, statistical comparisons of anatomy, and surface-based registration and signal pro- cessing. Here we introduce a parameterization method based on algebraic functions. By solving the Yamabe equation with the Ricci flow method, we can conformally map a brain surface to a multi-hole disk. The re- sulting parameterizations do not have any singularities and are intrinsic and stable. To illustrate the technique, we computed parameterizations of several types of anatomical surfaces in MRI scans of the brain, in- cluding the hippocampi and the cerebral cortices with various landmark curves labeled. For the cerebral cortical surfaces, we show the parameter- ization results are consistent with selected landmark curves and can be matched to each other using constrained harmonic maps. Unlike previous planar conformal parameterization methods, our algorithm does not in- troduce any singularity points. It also offers a method to explicitly match landmark curves between anatomical surfaces such as the cortex, and to compute conformal invariants for statistical comparisons of anatomy.
1 Introduction
Surface-based modeling is valuable in brain imaging to help analyze anatomical shape, to statistically combine or compare 3D anatomical models across subjects, and to map and compare functional imaging parameters localized on anatomical surfaces. Parameterization of these surface models involves computing a smooth (differentiable) one-to-one mapping of regular 2D coordinate grids onto the 3D surfaces, so that numerical quantities can be computed easily from the result- ing models [1]. The mesh-based work contrasts with implicit methods, which typically define a surface as the level set of a higher-dimensional function [2]. Relative to level set methods, surface meshes can allow regular 2D grids to be imposed on complex structures, transforming a difficult 3D problem into a 2D planar problem, with simpler data structures, discretization schemes, and rapid data access and navigation. Here we present a new method to parameterize brain
2 Wang, Gu, Chan, Thompson and Yau
surfaces based on algebraic functions. We find a planar conformal parameteri- zation without any singularities by solving the Yamabe equation with the Ricci flow method. This method can compute conformal invariants of brain surfaces which can be used to compare and classify brain surface structures. Compared with previous brain conformal parametrization work [3, 4], the parameterization provided by our algorithm does not have any zero points so there is less area distortion. By solving a harmonic map in the parameter domain, our algorithm provides smooth correspondence fields for matching of different brain surfaces while explicitly matching labeled sets of landmark curves.
1.1 Previous Work
Brain surface parameterization has been studied intensively. Schwartz et al. [5], and Timsari and Leahy [6] computed quasi-isometric flat maps of the cerebral cortex. Drury et al. [7] presented a multiresolution method for flattening the cerebral cortex. Hurdal and Stephenson [8] report a discrete mapping approach that uses circle packings to produce “flattened” images of cortical surfaces on the sphere, the Euclidean plane, and the hyperbolic plane. The obtained maps are quasi-conformal approximations of classical conformal maps. Haker et al. [9] implement a finite element approximation for parameterizing brain surfaces via conformal mappings. They select a point on the cortex to map to the north pole of the Riemann sphere and conformally mapped the rest of the cortical surface to the complex plane by stereographic projection of the Riemann sphere to the com- plex plane. Gu et al. [10] propose a method to find a unique conformal mapping between any two genus zero manifolds by minimizing the harmonic energy of the map. They demonstrate this method by conformally mapping a cortical surface to a sphere. Ju et al. [11] present a least squares conformal mapping method for cortical surface flattening. Joshi et al. [12] propose a scheme to parameterize the surface of the cerebral cortex by minimizing an energy functional in the pth norm. Wang et al. [3, 4] have used holomorphic 1-forms to parameterize anatom- ical surfaces with complex (possibly branching) topology. Recently, Ju et al. [13] reported the results of a quantitative comparison of FreeSurfer [14], CirclePack, and least squares conformal mapping (LSCM) with respect to geometric distor- tion and computational speed. They found that FreeSurfer performs best with respect to a global measurement of metric distortion, whereas LSCM performs best with respect to angular distortion and best in all but one case with a lo- cal measurement of metric distortion. Among the three approaches, FreeSurfer provides a more homogeneous distribution of metric distortion across the whole cortex than CirclePack and LSCM. LSCM is the most computationally efficient algorithm for generating spherical maps, while CirclePack is extremely fast for generating planar maps from patches.
1.2 Theoretical Background and Definitions
Given a multi-hole genus zero punctured surface, it is conformally equivalent to a special type of algebraic function,
ω^2 = (z − z 0 )(z − z 1 )(z − z 2 )...(z − z 2 g+1), (1)
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Algorithm 1 Conformal Mapping to a Multi-hole Punctured Disk Input: mesh M , step length ≤, energy difference threshold δK; Output: h : M → D. Here D ∈ R^2 , and D is a multi-hole disk.
- Computing initial radii γi for each vertex, and angle φij for each edge eij , such that lij = γ i^2 + γ^2 j − 2 γiγj cosφij.
- Compute boundary loops, denoted as Γ 0 , Γ 1 , ..., Γn. The Γ 0 is the exterior boundary.
- Set target Gaussian curvature of each interior vertex to be zero, K˜i = 0.
- For any vertex on vk ∈ Γ 0 , set its target Gaussian curvature to K˜k = (^) |^2 Γπi| , where |Γ 0 | denotes the number of vertices in Γ 0.
- For any vertex on vk ∈ Γi, i 6 = 0, set its target Gaussian curvature to K^ ˜k = − 2 π |Γi| , where^ |Γi|^ denotes the number of vertices in^ Γi.
- Update the vertex radii with the Ricci flow,
γi(t + 1) = γi(t) + ≤ × ( K˜i(t) − Ki(t)) × γi(t).
- Update the target Gaussian curvature for boundary vertices, suppose vk ∈ Γi, suppose ek− 1 ,k, ek,k+1 ∈ Γi, then let Si = Σepq ∈Γi lpq , then
K^ ˜k = lk−^1 ,k^ +^ lk,k+ 2 Si × Ci,
where Ci is Ci =
2 π, i = 0 − 2 π, i 6 = 0
- Repeat step 6 and 7 until the maximal Gaussian curvature error, maxi|Ki − K^ ˜i|, is less than δK.
2.2 Surface Matching with Punctured Disk Parameterization
After the computation of conformal parameterizations for open boundary genus zero surfaces with a multiple-hole punctured disk, we can compute the direct correspondence of two surfaces by solving a constrained harmonic mapping prob- lem [4]. Given two surfaces S 1 and S 2 , their punctured disk parameterizations are τ 1 : S 1 → R^2 and τ 2 : S 2 → R^2 , we want to compute a map, φ : S 1 → S 2. Instead of directly computing of φ, we can easily find a harmonic map between the parameter domains. We look for a harmonic map, τ : R^2 → R^2 , such that τ ◦ τ 1 (S 1 ) = τ 2 (S 2 ), τ ◦ τ 1 (∂S 1 ) = τ 2 (∂S 2 ), ∆τ = 0. Then the map φ can be obtained by φ = τ 1 ◦ τ ◦ τ 2 − 1. Since τ is a harmonic map while τ 1 and τ 2 are conformal map, the resulting φ is a harmonic map.
3 Experimental Results
We applied our algorithms to parameterize various anatomic surfaces extracted from 3D MRI scans of the brain. We tested our algorithm on a left hippocam- pal surface, a structure in the medial temporal lobe of the brain. The original surface is shown in Figure 1(a). We leave two holes on the front and back of
Brain Surface Conformal Parameterization with Algebraic Functions 5
Fig. 1. Illustrates conformal maps from hippocampal and cortical surfaces to multi-hole disks. (a) shows the front view of a hippocampal surface and (d) shows its conformal map to a 1-hole disk. (b) and (c) are two cerebral cortical surfaces. Two central sulci are labeled as yellow curves on each of them. After cutting along the landmark curves, each of these two surfaces can be conformally mapped to a 1-hole disk ((e) and (f)). The radii of the inner circles are conformal invariants of two surfaces and can be used as shape index to compare and classify brain surfaces. (g)-(j) show a conformal map from a left hemisphere cortex with 5 labeled landmarks to a 4-hole disk. (g) and (h) show the front and back side of the surface. (j) shows its conformal map on a 4-hole disk.
the hippocampal surface, representing its anterior junction with the amygdala, and its posterior limit as it turns into the white matter of the fornix. It can be logically represented as an open boundary genus one surface, i.e., a cylinder (note that spherical harmonic representations would also be possible, if ends were closed [19]). Its conformal map to a 1-hole disk is illustrated in Figure 1(d). For the two boundaries of the hippocampal surface, one boundary is mapped to the exterior circle and the other is mapped to the internal circle.
We also applied our algorithm to parameterize the surface of the cerebral cor- tex. The cerebral cortex and landmark data are the same ones used in [20]. We tested our algorithm with different landmark sets. Figure 1(b) and (c) show two cortical surfaces with two central sulci on two hemispheres. After we cut the cor- tical surface open along the two central sulci, the cortical surface is topologically equivalent to an open boundary genus one surface. Figure 1(e) and (f) show their conformal parameterizations in a 1-hole disk. One of the two landmark curves
Brain Surface Conformal Parameterization with Algebraic Functions 7
Fig. 2. Illustrates direct surface matching between two different cerebral cortical surfaces while explicitly matching landmark curves. (a)-(b) show a left cerebral cortex with four labeled landmarks and (c) shows its conformal map to a 3-hole disk. (d)-(e) show another left hemisphere model of the cerebral cortex with the same landmarks labeled and (f) shows its conformal map to a 3-hole disk. (g) is the parameterization of surface (d)-(e) after a constrained harmonic map from (f) to (c) is built. (h)-(m) show a morphing sequence from surface (a)-(b) to surface (d)-(e). (j)-(l) are the intermediate shapes when we linearly interpolate surface correspondence vector field between two surfaces (h) and (m). Although the cortical surface shape changes considerably, the relative positions of the selected landmark curves do not change.
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abe equation to obtain a conformal deformation that conformally maps open boundary surfaces to multi-hole disk. We tested our algorithm on the hippocam- pus and surface models of the cerebral cortex. Used as a canonical space, the multi-hole disk conformal parameterization provides a brain surface matching approach that can exactly match landmark curves lying on the surfaces. Com- pared with other work which conformally maps brain surfaces to parallelograms, our algorithm offers some advantages because it does not introduce any singular points. Our future work will include algebraic function computation based on the multi-hole disk parameterization and empirical application of these Yamabe flow concepts to medical applications in computational anatomy.
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