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IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 26, NO. 6, JUNE 2007 853
Brain Surface Conformal Parameterization Using
Riemann Surface Structure
Yalin Wang* , Member, IEEE , Lok Ming Lui, Xianfeng Gu, Kiralee M. Hayashi, Tony F. Chan, Arthur W. Toga,
Paul M. Thompson, and Shing-Tung Yau
Abstract— In medical imaging, parameterized 3-D surface models are useful for anatomical modeling and visualization, statistical comparisons of anatomy, and surface-based registration and signal processing. Here we introduce a parameterization method based on Riemann surface structure, which uses a special curvilinear net structure (conformal net) to partition the surface into a set of patches that can each be conformally mapped to a parallelogram. The resulting surface subdivision and the pa- rameterizations of the components are intrinsic and stable (their solutions tend to be smooth functions and the boundary conditions of the Dirichlet problem can be enforced). Conformal parameteri- zation also helps transform partial differential equations (PDEs) that may be defined on 3-D brain surface manifolds to modified PDEs on a two-dimensional parameter domain. Since the Jacobian matrix of a conformal parameterization is diagonal, the modified PDE on the parameter domain is readily solved. To illustrate our techniques, we computed parameterizations for several types of anatomical surfaces in 3-D magnetic resonance imaging scans of the brain, including the cerebral cortex, hippocampi, and lateral ventricles. For surfaces that are topologically homeomorphic to each other and have similar geometrical structures, we show that the parameterization results are consistent and the subdivided surfaces can be matched to each other. Finally, we present an automatic sulcal landmark location algorithm by solving PDEs on cortical surfaces. The landmark detection results are used as constraints for building conformal maps between surfaces that also match explicitly defined landmarks.
Manuscript received June 12, 2006; revised January 30, 2007. This work was supported in part by the National Institutes of Health through the NIH Roadmap for Medical Research under Grant U54 RR021813, in part by NIH/NCRR under Resource Grant P41 RR013642, in part by the National Science Foundation (NSF) under Contract DMS-0610079 and ONR Contract N0014-06-1-0345. The work of X. Gu was supported in part by the NSF under CAREER Award CCF-0448339, Grant DMS-0528363, and Grant NSF DMS-0626223. The work of P. M. Thompson was supported by the National Institute for Biomedical Imaging and Bioengineering, National Center for Research Resources, National Institute for Neurological Disorders and Stroke, National Institute on Aging, and National Institute for Child Health, and Development under Grant EB01651, Grant RR019771, AG016570, Grant NS049194, and Grant HD050735. Asterisk indicates corresponding author. *Y. Wang is with the Laboratory of Neuro Imaging, Department of Neu- rology, University of California—Los Angeles School of Medicine, Los An- geles, CA 90095 USA and with the Department of Mathematics, University of California, Los Angeles, CA 90095 USA (e-mail: [email protected]). L. M. Lui is with the Department of Mathematics, University of California, Los Angeles, CA 90095 USA (e-mail: [email protected]). X. Gu is with the Department of Computer Science, Stony Brook University, Stony Brook, NY 11794 USA (e-mail: [email protected]). K. M. Hayashi, A. W. Toga, and P. M. Thompson are with the Laboratory of Neuro Imaging, Department of Neurology, University of California—Los Angeles School of Medicine, Los Angeles, CA 90095 USA (e-mail: [email protected]; [email protected]; [email protected]). T. F. Chan is with the National Science Foundation, Arlington, VA 22230 USA, on leave from the Department of Mathematics, University of California, Los Angeles, CA 90095 USA (e-mail: [email protected]). S.-T. Yau is with the Department of Mathematics, Harvard University, Cam- bridge, MA 02138 USA (e-mail: [email protected]).
Digital Object Identifier 10.1109/TMI.2007.
Index Terms— Brain mapping, conformal parameterization, partial differential equation, Riemann surface structure.
I. INTRODUCTION
S
URFACE-BASED modeling is valuable in brain imaging to help analyze anatomical shape, to statistically combine or compare 3-D anatomical models across subjects, and to map and compare functional imaging parameters localized on anatom- ical surfaces. Parameterization of these surface models involves computing a smooth (differentiable) one-to-one mapping of reg- ular 2-D coordinate grids onto the 3-D surfaces, so that nu- merical quantities can be computed easily from the resulting models [1]–[3]. Mesh-based work on surface parameterization contrasts with implicit methods, which often define a surface as the level set of a higher dimensional function [4], [5]. Relative to level set methods, surface meshes can allow regular 2-D grids to be imposed on complex structures, transforming a difficult 3-D problem into a 2-D planar problem with simpler data struc- tures, discretization schemes, and rapid data access and naviga- tion. Even so, it is often difficult to smoothly deform a complex 3-D surface to a sphere or 2-D plane without substantial angular or areal distortion. Here we present a new method to parame- terize brain surfaces based on their Riemann surface structure. By contrast with variational approaches based on surface infla- tion, our method can parameterize surfaces with arbitrary com- plexity, while formally guaranteeing minimal distortion, defined using an appropriate energy functional. This includes branching surfaces not topologically homeomorphic to a sphere (higher genus objects), which can be expressed using a graph structure of connected surface components. Based on our conformal pa- rameterization, a set of simple covariant differential operators are developed and applied to assist in the problem of automati- cally detecting sulcal surface landmarks lying on the cortex.
A. Previous Work Brain surface parameterization has been studied inten- sively. Schwartz et al. [6] and Timsari and Leahy [7] compute quasi-isometric flat maps of the cerebral cortex. Drury et al. [8] present a multiresolution flattening method for mapping the cerebral cortex to a 2-D plane [9]. Hurdal and Stephenson [10] report a discrete mapping approach that uses circle packing [11] 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. [12] implement a finite-element
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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 map the rest of the cortical surface to the complex plane by invoking the standard stereographic projection of the Riemann sphere to the complex plane. Gu et al. [13] 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. [14] present a least squares conformal mapping method for cortical surface flattening. Joshi et al. [15] propose a scheme to parameterize the surface of the cerebral cortex by minimizing an energy functional in the th norm. Recently, Ju et al. [16] report the results of a quantitative comparison of FreeSurfer [17], CirclePack [10], and least squares conformal mapping (LSCM) [14] with respect to geometric distortion and computational speed. Their research results indicate that FreeSurfer performs best with respect to a global measurement of metric distortion, whereas LSCM performs best with respect to minimizing angular distortion and best in all but one case with a local measurement of metric distortion. Among the three approaches, FreeSurfer provides more homogeneous distribution of metric distortion across the whole cortex than CirclePack and LSCM. LSCM is the most computationally effi- cient algorithm for generating spherical maps, while CirclePack is extremely fast for generating planar maps from patches. Solving partial differential equations (PDEs) on surfaces has also been widely studied. Turk [18] proposed a method to generate textures on arbitrary surfaces using reaction diffusion, which requires the PDEs to be solved on a surface. Dorsey et al. [19] also solve PDEs on a surface in a “virtual weathering” application. Both of them solve the PDE directly on the trian- gulated surface, involving the discretization of equations on a general polygonal grid. Stam et al. [20] proposed a method to simulate fluid flow on a surface via solving the Navier–Stokes equation. They achieve this by combining the two-dimensional stable fluid solver with an atlas of parametrizations of a Cat- mull–Clark surface. Clarenza et al. [21] propose an algorithm for solving finite-element-based PDEs on surfaces defined by a set of sampled points. They construct a number of local fi- nite-element matrices to represent surface properties over small point neighborhoods. These matrices are next assembled into a single matrix that allows the PDE to be discretized and solved on the entire surface. Bertalmio et al. [22] and Mémoli et al. [5] implement a framework for solving PDEs on the surface via the level set method. They represent the surface implicitly by the zero-level set of an embedding function and extend the data on the surface to the 3-D volume. This allows them to perform all the computation on the fixed Cartesian grid. In related work using PDEs to match parameterized models of the cortex, Thompson et al. [2] discretize the Navier elasticity equation on a surface using a covariant PDE approach. The method uses Christoffel symbols to adjust the differential operators for changes in the base vectors of the parameterization. A related covariant approach has also been used for Laplace–Beltrami smoothing of data defined on a surface [23] and registering and analyzing functional neuroimaging data constrained to the cortical surface of the brain [24].
Fig. 1. Structure of a manifold. An atlas is a family of charts that jointly form an open covering of the manifold [25].
B. Theoretical Background and Definitions A manifold of dimension is a connected Hausdorff space for which every point has a neighborhood that is home- omorphic to an open subset of. Such a homeomorphism is called a coordinate chart. An atlas is a family of charts for which constitutes an open covering of . As shown in Fig. 1, suppose and are two charts on a manifold ; then the chart transition function, is defined as
An atlas on a manifold is called differentiable if all chart transition functions are differentiable of class. A chart is called compatible with a differentiable atlas if adding this chart to the atlas still yields a differentiable atlas. The set of all charts compatible with a given differentiable atlas yields a differentiable structure. A differentiable manifold of dimension is a manifold of dimension together with a differentiable structure. For a manifold with an atlas , if all chart transition functions, defined by (1), are holomorphic, then is a conformal atlas for. A chart^ is^ compatible^ with an atlas if the union is still a conformal atlas. Two conformal atlases are compatible if their union is still a conformal atlas. Each conformal compatible equivalence class is a conformal structure. A two-manifold with a conformal structure is called a Riemann surface. It has been proven that all metric orientable surfaces are Riemann surfaces and admit conformal structures [26]. Holomorphic and meromorphic functions and differential forms can be generalized to Riemann surfaces by using the notion of conformal structure. For example, a holomorphic 1-form is a complex differential form, such that in each local frame , , where , the parametric representation is , where is a holomorphic function. On a different chart , with another local frame , where
856 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 26, NO. 6, JUNE 2007
of solving the PDE on the manifold. To do this, we have to use the manifold version of differential operators (tensor calculus). Using the conformal parametrization, these differential opera- tors on the manifold can be described more easily on the pa- rameter domain by simple formulas. This will be explained briefly as below. Let be a manifold and be the global con- formal parametrization of. With the conformal parametriza- tion, we can do calculus on surfaces similar to what we do on
. Suppose is a smooth map. We will first de- fine partial derivative of. On , we usually define the partial derivative by taking the limit. For example, . With the conformal parametrization, we can define the partial deriva- tive on scalar functions in the same manner. Because of the stretching effect, we define
where is the conformal factor. is defined similarly. Now, the gradient of a function , is characterized by. Simple checking gives us where is the inverse of the Riemannian metric. With the conformal parametrization, we can de- fine the gradient of similar to the definition on. Namely
where
Suppose is a smooth function. With this definition of gradient, we still have the following useful fact as in :
Length of
where is the delta function and is the Heaviside function. Next, we give a definition of a differential operator on a vector field. It can be done by the covariant differentiation
TABLE I LIST OF F ORMULAS FOR S OME S TANDARD D IFFERENTIAL O PERATORS ON G ENERAL M ANIFOLD
, where is called the direction of the differentiation. This operation is based on the tensor calculus [31]. Given a Riemannian manifold , where is the Riemannian metric, suppose is the coordinate basis of the vector field. A simple verification will tell us the covariant derivative can be calculated by the following formula:
Suppose now that the parametrization is conformal. The Rie- mannian metric is simply , where are the conformal factor and the identity matrix respectively. We then have the following formula:
With this formula, we can calculate easily. More inter- esting formulas for the manifold differential operators [32] on the parameter domain can be found in Table I.
II. H OLOMORPHIC F LOW S EGMENTATION To compute the holomorphic flow segmentation of a surface, first we compute the conformal structure of the surface; then we select a holomorphic differential form and locate the zero points on it (details are given below). By tracing horizontal trajectories through the zero points, the critical graph can be constructed and
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the surface is divided into several patches. Each patch can then be conformally mapped to a planar parallelogram by integrating the holomorphic differential form. In this paper, surfaces are represented as triangular meshes, namely, piecewise polygonal surfaces. The computations with differential forms are based on solving elliptic partial differ- ential equations on surfaces using finite-element method. Sup- pose is a simplicial complex, and a mapping embeds in ; then is called a triangular mesh. , where are the sets of -simplices. We use to denote an -simplex, where.
A. Computing Conformal Structures
A method for computing the conformal structure of a surface was introduced in [33]. Suppose is a closed genus surface with a conformal atlas. The conformal structure induces holomorphic 1-forms; all holomorphic 1-forms form a linear space of dimension 2 over real numbers that is isomorphic to the first cohomology group of the surface
. The set of holomorphic 1-forms determines the conformal structure. Therefore, computing conformal structure of is equivalent to finding a basis for. The holomorphic 1-form basis is com- puted as follows: compute the homology basis, find the dual co- homology basis, diffuse the cohomology basis to a harmonic 1-form basis, and then convert the harmonic 1-form basis to holomorphic 1-form basis by using the Hodge star operator. The details of the computation are given in [33]. 3 In terms of data structure, a holomorphic 1-form is represented as a vector- valued function defined on the edges of the mesh .
B. Canonical Conformal Parameterization Computation
Given a Riemann surface , there are infinitely many holo- morphic 1-forms, but each of them can be expressed as a linear combination of the basis elements. We define a canonical con- formal parameterization as any linear combination of the set of holomorphic basis functions , where. They satisfy , where are ho-
mology bases and is the Kronecker symbol. Then we com- pute a canonical conformal parameterization. We select a specific parameterization that maximizes the uni- formity of the induced grid over the entire domain using the algorithms introduced in [34], for the purpose of locating zero points in the next step.
C. Locating Zero Points
For a close surface with genus , any holomorphic 1-form has 2 2 zero points. For an open boundary surface with genus , any holomorphic 1-form has 1 zero points. The horizontal trajectories through the zero points will partition the surface into several discrete patches. Each patch is either a topological disk or a cylinder and can be conformally parameterized by a parallelogram.
(^3) For surfaces with boundaries, we apply the conventional double covering technique, which glues two copies of the same surface along their corresponding boundaries to form a symmetric closed surface. Then we apply the above pro- cedure to find the holomorphic 1-form basis.
1) Estimating the Conformal Factor: Suppose we already have a global conformal parameterization, induced by a holo- morphic 1-form. Then we can estimate the conformal factor at each vertex, using the following formulas:
where and are the set of 0-simplices and 1-simplices, respectively; is the valence of vertex. 2) Locating Zero Points: We find the cluster of vertices with relatively small conformal factors (the lowest 5–6%). These are candidates for zero points. We cluster all the candidates using the metric on the surface. For each cluster, we pick the vertex that is on the surface and closest to the center of gravity of the cluster, using the surface metric to define geodesic distances. Because the triangulation is finite and the computation is an approximation, the number of zero points may not equal to the Euler number. In this case, we refine the triangulation of the neighborhood of the zero point candidate and refine the holo- morphic 1-form. 4 According to the Riemann surface theory, zero points are iso- lated in general. In very rare cases, zero points can be merged. However, the probability of those cases is zero. The config- uration of zero points is completely determined by the coho- mology class of the holomorphic 1-forms. By choosing coho- mology class, we can control the distances among zero points. In practice, with topological optimization method [34], we can adaptively change the surface cohomology class of the holo- morphic 1-form by changing the surface topological structure, e.g., cutting open surfaces along some feature lines. As shown in Fig. 2(b), by choosing the canonical conformal parameteri- zation result, the zero points are well separated by the cutting curves.
_D. Holomorphic Flow Segmentation
- Tracing Horizontal Trajectories:_ Once the zero points are located, the horizontal trajectories through them can be traced. First we choose a neighborhood of a vertex , where represents a zero point and is a set of neighboring faces of. Then we map it to the parameter plane by integrating. Suppose a vertex and a path composed by a sequence of edges on the mesh is ; then the parameter location of is. The conformal parameterization is a piecewise linear map. Then the horizontal trajectory is mapped to the horizontal line in the parameter domain. We slice using the line by edge splitting operations. Sup- pose the boundary of intersects constant at a point ; then we choose a neighborhood of and repeat the process. Each time we extend the horizontal trajectory and encounter edges intersecting the trajectory. We insert new vertices at the intersection points, until the trajectory reaches another zero point (for a closed surface) or the boundary of the mesh (for an open boundary surface). Each zero point connects to four (^4) It is worth noting that the same research group has proposed to use a new mathematical tool, the Ricci flow, to eliminate the numerical errors in the zero point location computation [35]. The detailed development will be reported in our future work.
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where is the level set function and and is the mean curvature function on the cortical surface. With the conformal parametrization, we have
For simplicity, we let and. Fixing , we must have
Fixing , the Euler–Lagrange equation becomes
Locating sulcal landmark features using geometric properties of the anatomical features has been widely studied [39], [40], [43]–[48]. In general, sulcal landmarks on the cortex lie in re- gions with relatively high mean curvature [49], [50]. Using the CV segmentation, we can consider the signal of interest, defined on the cortex, as the mean curvature. The sulci position on the cortical surface can then be located by extracting out the high curvature region. Inside each region, a good initialization of the landmark curve is considered as the shortest path that connects two anchor points when the region is considered as a graph. We select two umbilic points as the anchor points. By definition, an umbilic point on a manifold is a location at which the two principal curvatures are the same. If there are multiple umbilic points are found in one region, we select the two that are furthest apart.
IV. O PTIMIZATION OF B RAIN C ONFORMAL M APPING U SING
S ULCAL L ANDMARKS
In brain-mapping research, cortical surface data are often mapped conformally to a parameter domain such as a sphere, providing a common coordinate system for data integration [13], [17], [51]. As an application of our automatic landmark tracking algorithm, we use the automatically labelled landmark curves to generate an optimized conformal mapping on the sur- face. This matching of cortical patterns improves the alignment of data across subjects. It is done by minimizing the compound energy functional
Fig. 3. The holomorphic flow segmentation results of (a)–(d) a hippocampal surface and (e)–(l) two lateral ventricular surfaces. (b) is the conformal net of the hippocampal surface in (a). (c) is an isoparameter curve used to unfold the surface. (d) is the rectangle to which the surface is conformally mapped. (e)–(h) show lateral ventricles parameterized using holomorphic 1-forms for a 65-year-old subject with HIV/AIDS; (i)–(l) show the same maps computed for a healthy 21-year-old control subject. (e) and (i) show that five cuts are auto- matically introduced and convert the lateral ventricular surface into a genus 4 surface. Other pictures show the (f) and (j) computed conformal net, (g) and (k) holomorphic flow segmentation, and (h) and (l) their associated parameter do- mains (the texture mapped into the parameter domain here simply corresponds to the intensity of the surface rendering, which is based on the surface normals).
where is the harmonic energy of the parameteri- zation [13] and
is the landmark mismatch energy [52], where the norm rep- resents distance between automatic traced landmark curves and , represented on the parameter domain. Here, automatically traced landmark (continuous) curves are used and the correspondence is obtained on the curves using the unit length and unit speed reparametrization. We can minimize the energy by steepest descent method. The resulting brain conformal mapping improves cortical surface registration sig- nificantly while preserving conformality to the highest possible degree [52]. Note that we may also describe the landmark matching problem on the sphere within the large deformation diffeomorphism framework [53].
V. EXPERIMENTAL R ESULTS
A. Hippocampal Surface Parameterization Fig. 3(a)–(d) shows experimental results for a hippocampal surface, a structure in the medial temporal lobe of the brain. The original surface is shown in (a). We leave two holes at the front and back of the hippocampal surface, representing its an- terior junction with the amygdala and its posterior limit as it
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turns into the white matter of the fornix. The hippocampus can be logically represented as an open boundary genus one sur- face, i.e., a cylinder (note that spherical harmonic representa- tions would also be possible if the ends were closed [54]–[57]). The computed conformal net is shown in (b), where the red and blue curves are the vertical trajectories and the horizontal tra- jectories of the net, respectively. A horizontal trajectory curve is shown in (c). Cutting the surface along this curve, we can then conformally map the hippocampus to a rectangle. The de- tailed surface information is well preserved in (d). Compared with other spherical parameterization methods, which may have high-valence nodes and dense tiles at the poles of the spherical coordinate system, our parameterization can represent the sur- face with minimal distortion. 5
B. Lateral Ventricular Surface Parameterization
Shape analysis of the lateral ventricles—a fluid-filled struc- ture deep in the brain—is of great interest in the study of psychiatric illnesses, including schizophrenia [58], [59], and in degenerative diseases such as Alzheimer’s disease [60], [61]. These structures are often enlarged in disease and can provide sensitive measures of disease progression. To model the lateral ventricular surface in each brain hemisphere, we introduce five cuts. Although this seems arbitrary, the cuts are motivated by examining the topology of the lateral ventricles, in which several horns are joined together at the ventricular “atrium” or “trigone.” We call this topological change topological opti- mization. The topological optimization helps to get a uniform parameterization on some areas that otherwise are very difficult for usual parameterization methods to capture. A systematic topological optimization method was introduced in [34]. With this method, we automatically located initial points, which are simply the extremal points of the anterior/posterior coordinate, for these cuts at a specific set of extreme points. Specifically, they are at the ends of the frontal, occipital, and temporal horns of the ventricles. Fig. 3(e) and (i) shows five cuts introduced on two subjects’ ventricular surfaces. After the cutting, the surfaces become open-boundary genus 4 surfaces. The rest of Fig. 3 shows parameterizations of the lateral ventricles of the brain. Fig. 3(e)–(h) shows the results of parameterizing a ventricular surface for a 65-year-old patient with HIV/AIDS (note the disease-related enlargement) and Fig.3 (i)–(l) shows the results for the ventricular model of a 21-year-old control subject (surface data are from [60]). The computed conformal structures are shown in Fig. 3(f) and (j) by texture mapping a checkerboard image to the surface. There are a total of three zero points on each of the ventricular surfaces. Two of them are located at the middle part of the two “arms” (where the temporal and occipital horns join at the ventricular atrium), as shown by the large black dots in (f) and (j). The third zero point is located in the middle of the model, where the frontal horns are closest to each other. Based on the computed conformal structure, we can partition the surface into six patches. Each patch can be
(^5) When matching two hippocampal surfaces with our conformal parameteri- zation, we need to find a consistent trajectory to cut them open. A possible way to do it is to use principal axis registration first, i.e., get the inertia matrix of one surface and apply its inverse to the other one. After that, we can find a consistent trajectory that guarantees the cutting results have consistent boundaries.
Fig. 4. Illustrates the parameterization of cortical surfaces using the holomor- phic 1-form approach. The thick lines are landmark curves, including several major sulci lying in the cortical surface. These sulcal curves are always mapped to a boundary in the parameter space images.
conformally mapped to a rectangle. Although the two brain ventricle shapes are very different, the segmentation results are consistent in that the surfaces are partitioned into patches with the same relative arrangement and connectivity. Thus our method provides a basis to perform direct surface matching be- tween any two ventricles. Note that this is somewhat a different approach to surface matching than those typically adopted. Rather than use a single parameterization for the entire surface, and match landmarks that lie within it, instead a topological de- composition of the surface is used and the patches are piecewise matched and guaranteed to be conformal. The advantage of this approach is that it can handle the matching of objects with nonspherical (but consistent) topology, which would otherwise be extremely difficult to parameterize and match.
C. Cerebral Cortical Surface Parameterization For the surface of the cerebral cortex, our algorithm also provides a way to perform surface matching while explicitly matching sulcal curves or other landmarks lying in the surface. Note that typically two surfaces can be matched by using a landmark-driven flow in their parameter spaces [2], [13], [17], [62], [63]. An alternative approach is to supplement the critical graph with curved landmarks that can then be forced to lie on the boundaries of rectangles in the parameter space. This has the advantage that conformal grids are still available on both surfaces, as is a correspondence field between the two con- formal grids. Fig. 4 shows the results for the cortical surfaces of two left hemispheres. As shown in the first row, we selected four major landmark curves, for the purpose of illustrating
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Fig. 6. An average cortical surface of the brain derived by using optimized conformal parametrization with the automatically traced landmarks. The major sulcal landmarks are well preserved (see the areas inside green circles in (B) and (C). In (D), the sulci at the back of the brain, in the occipital lobe surface, are averaged out because no landmark constraint was added in that region, for purposes of illustration.
our algorithm can help by improving the alignment of major anatomical features in the cortex. Further validation work, of course, would be necessary to assess whether this results in greater detection sensitivity in computational anatomy studies of the cortex, but the greater reinforcement of features suggests that landmark alignment error is substantially reduced, and this is one major factor influencing signal detection in multisubject cortical studies.
VI. C ONCLUSION AND F UTURE W ORK
In this paper, we presented a brain surface parameterization method that invokes the Riemann surface structure to generate conformal grids on surfaces of arbitrary complexity (including branching topologies). For high genus surfaces, a global con- formal parameterization induces a canonical segmentation, i.e., there is a discrete partition of the surface into conformally pa- rameterized patches. We tested our algorithm on hippocampal and lateral ventricular surfaces and on surface models of the cerebral cortex. The grid generation algorithm is intrinsic (i.e., it does not depend on any initial choice of surface coordinates) and is stable, as shown by grids induced on ventricles of var- ious shapes and sizes. With the conformal parameterization, we employed a set of formulas for the covariant differential opera- tors on the manifold based on tensor calculus. Because of this conformal parameterization, the extension of well-known 2-D PDE solvers to general 3-D surfaces becomes relatively simple. We illustrate the use of the conformal grid for hosting a sur- face-based PDE by presenting a brain landmark detection al- gorithm. The landmark detection results are examined by com- puting an average cortical surface after conformal parameter- izations are optimized using sulcal features as landmarks. Our future work will focus on signal processing on brain surfaces, as
well as brain surface registration with surface metric proposed in [30], and shape and asymmetry analysis for subcortical struc- tures.
A PPENDIX A Claim 1: Suppose is a smooth function. Then
Proof: Recall that the Co-area formula reads
where is the Hausdorff measure. Let be the conformal parametrization of the surface and . Then
where is the parametrization of. Q.E.D.
A PPENDIX B Case 2: Let. The geodesic curvature of . Proof: Recall that the geodesic curvature of a curve is
Let the parametrization of the zero level set of be
. Then. This implies
and
WANG et al. : BRAIN SURFACE CONFORMAL PARAMETERIZATION 863
Now
Thus, for conformal parametrization, we have
and
Combining these, we have
and
So
Q.E.D.
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