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Brain Surface Parameterization Using Riemann
Surface Structure
Yalin Wang^1 , Xianfeng Gu^2 , Kiralee M. Hayashi^3 , Tony F. Chan^1 , Paul M. Thompson^3 , and Shing-Tung Yau^4 (^1) Mathematics Department, UCLA, Los Angeles, CA 90095, USA (^2) Comp. Sci. Department, SUNY at Stony Brook, Stony Brook, NY 11794, USA (^3) Lab. of Neuro Imaging, UCLA School of Medicine, Los Angeles, CA 90095, USA (^4) Department of Mathematics, Harvard University, Cambridge, MA 02138, USA {ylwang, chan}@math.ucla.edu, [email protected], {khayashi, thompson}@loni.ucla.edu, [email protected]
Abstract. We develop a general approach that uses holomorphic 1- forms to parameterize anatomical surfaces with complex (possibly branch- ing) topology. Rather than evolve the surface geometry to a plane or sphere, we instead use the fact that all orientable surfaces are Riemann surfaces and admit conformal structures, which induce special curvilin- ear coordinate systems on the surfaces. Based on Riemann surface struc- ture, we can then canonically partition the surface into patches. Each of these patches can be conformally mapped to a parallelogram. The re- sulting surface subdivision and the parameterizations of the components are intrinsic and stable. To illustrate the technique, we computed con- formal structures for several types of anatomical surfaces in MRI scans of the brain, including the cortex, hippocampus, and lateral ventricles. We found that the resulting parameterizations were consistent across subjects, even for branching structures such as the ventricles, which are otherwise difficult to parameterize. Compared with other variational ap- proaches based on surface inflation, our technique works on surfaces with arbitrary complexity while guaranteeing minimal distortion in the pa- rameterization. It also offers a way to explicitly match landmark curves in anatomical surfaces such as the cortex, providing a surface-based frame- work to compare anatomy statistically and to generate grids on surfaces for PDE-based signal processing.
1 Introduction
In brain imaging research, parameterization of various types of anatomical sur- face models in magnetic resonance imaging (MRI) scans of the brain involves computing a smooth (differentiable) one-to-one mapping of regular 2D coordi- nate grids onto the 3D surfaces, so that numerical quantities can be computed easily from the resulting models [1,2]. Even so, it is often difficult to smoothly deform a complex 3D surface to a sphere or 2D plane without substantial an- gular or area distortion. Here we present a new method to parameterize brain surfaces based on their Riemann surface structure. By contrast with variational
J. Duncan and G. Gerig (Eds.): MICCAI 2005, LNCS 3750, pp. 657–665, 2005. ©c Springer-Verlag Berlin Heidelberg 2005
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approaches based on surface inflation, our method can parameterize surfaces with arbitrary complexity including branching surfaces not topologically home- omorphic to a sphere (higher-genus objects) while formally guaranteeing minimal distortion.
1.1 Previous Work
Brain surface parameterization has been studied intensively. Schwartz et al. [3], and Timsari and Leahy [4] compute quasi-isometric flat maps of the cerebral cortex. Hurdal and Stephenson [5] 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. Angenent et al. [6] represent the Laplace-Beltrami operator as a linear system and implement a finite element approximation for parameterizing brain surfaces via conformal mapping. Gu et al. [7] propose a method to find a unique conformal mapping between any two genus zero manifolds by minimizing the harmonic energy of the map.
1.2 Theoretical Background and Definitions
We begin with some formal definitions that will help to formulate the param- eterization problem(for further reading, please refer to [8]). For a manifold M with an atlas A = {Uα, φα}, if all chart transition functions φαβ = φβ ◦ φ− α 1 : φα(Uα ∩ Uβ ) → φβ (Uα ∩ Uβ ) are holomorphic, A is a conformal atlas for M. A chart {U (^) α′, φ′ α} is compatible with an atlas A, if the union A ∪ {U (^) α′, φ′ α} is still a conformal atlas. Each conformal compatible equivalence class is a conformal structure. A 2-manifold with a conformal structure is called a Riemann surface. It has been proven that all metric orientable surfaces are Riemann surfaces. Holomorphic and meromorphic functions and differential forms can be gen- eralized to Riemann surfaces by using the notion of conformal structure. For example, a holomorphic one-form ω is a complex differential form, such that in each local frame zα = (uα, vα), the parametric representation is ω = f (zα)dzα, where f (zα) is a holomorphic function. On a different chart {Uβ , φβ }, ω = f (zα(zβ )) dz dzαβ dzβ. For a genus g closed surface, all holomorphic one-forms form a real 2g dimensional linear space. At a zero point p ∈ M of a holomorphic one-form ω, any local parametric representation ω = f (zα)dzα, f |p = 0. According to the Riemann-Roch theorem, in general there are 2g − 2 zero points for a holomorphic one-form defined on a surface of genus g. A holomorphic one-form induces a special system of curves on a surface, the so-called conformal net. A curve γ ⊂ M is called a horizontal trajectory of ω, if ω^2 (dγ) ≥ 0; similarly, γ is a vertical trajectory if ω^2 (dγ) < 0. The horizontal and vertical trajectories form a web on the surface. The trajectories that connect zero points, or a zero point with the boundary are called critical trajectories. The critical horizontal trajectories form a graph, which is called the critical graph. In general, the behavior of a trajectory may be very complicated, it may have infi- nite length and may be dense on the surface. If the critical graph is finite, then
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formal structure A induces holomorphic 1-forms; all holomorphic 1-forms form a linear space Ω(M ) of dimension 2g which is isomorphic to the first cohomology group of the surface H^1 (M, R). The set of holomorphic one-forms determines the conformal structure.
2.2 Canonical Conformal Parameterization Computation
Given a Riemann surface M , there are infinitely many holomorphic 1-forms, but each of them can be expressed as a linear combination of the basis elements. We define a canonical conformal parameterization as any linear combination of the set of holomorphic basis functions ωi, i = 1, ..., g. They satisfy
ζi ωj^ =^ δ
j i , where ζi, i = 1, ...n are homology bases and δji is the Kronecker symbol. Then we compute a canonical conformal parameterization ω =
∑n i=1 ωi.
2.3 Zero Points Location
For surface with genus g > 1, any holomorphic 1-form ω has 2g − 2 zero points. The horizontal trajectories through the zero points will partition the surface into several patches. Each patch is either a topological disk or a cylinder, and can be conformally parameterized by ω using φ(p) =
γ ω.
Estimating the Conformal Factor. Suppose we already have a global confor- mal parameterization, induced by a holomorphic 1-form ω. Then we can esti- mate the conformal factor at each vertex, using the following formulae: λ(v) = 1 n
[u,v]∈K 1
|ω([u,v])|^2 |r(u)−r(v)|^2 , u, v^ ∈^ K^0 , where^ n^ is the valence of vertex^ v.
Locating Zero Points. We find the cluster of vertices with relatively small con- formal 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 closest to the center of gravity of the cluster, using the surface metric to define geodesic distances.
2.4 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 Uv of a vertex v representing a zero point, Uv is a set of neighboring faces of v, then we map it to the parameter plane by integrating ω. Suppose a vertex w ∈ Uv, and a path composed by a sequence of edges on the mesh is γ, then the parameter location of w is φ(w) =
γ ω. The map φ(w) is a piecewise linear map. Then the horizontal trajectory is mapped to the horizontal line y = 0 in the plane. We slice φ(Uv ) using the line y = 0 by edge splitting operations. Suppose the boundary of φ(Uv ) intersects y = 0 at a point v′, then we choose a neighborhood of v′^ 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 or the boundary of the mesh. We repeat the tracing process until each zero point connects 4 horizontal trajectories.
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Critical Graph. Given a surface M and a holomorphic 1-form ω on M , we define the graph G(M, ω) = {V, E, F }, as the critical graph of ω. Here V is the set of zero points of ω, E is the set of horizontal trajectories connecting zero points or the boundary segments of M , and F is the set of surface patches segmented by E. Given two surfaces with similar topologies and geometries, by choosing ap- propriate holomorphic 1-forms, we can obtain isomorphic critical graphs, which will be used for patch-matching described in the next section.
3 Experimental Results
We tested our algorithm on various anatomic surfaces extracted from 3D MRI scans of the brain to illustrate the approach. Figure 1 (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). (b) shows the conformal mapping of (a) to a sphere with a variational method introduced in [7]. Since the shape of hippocampal surface is not quite similar to a sphere, lots of distortion has been introduced. In our method, we leave two holes on the front and back of 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, a cylinder (note that spherical harmonic representations would also be possible, if the ends were closed). The computed conformal structure is shown in (c). Then we can conformally map the hippocampus to a rectangle (d). Since the surface of rectangle is similar to the one of hippocampus, the detailed surface information is well preserved in (d). Compared with other spherical pa- rameterization methods (e.g. (b)), which may have high-valence nodes and dense tiles at the poles of the spherical coordinate system, our parameterization can represent the surface with minimal distortion. Shape analysis of the lateral ventricles is of great interest in the study of psychiatric illnesses, including schizophrenia, and in degenerative diseases such as Alzheimer’s disease. These structures are often enlarged in disease and can provide sensitive measures of disease progression. We can optimize the conformal parameterization by topology modification. For the lateral ventricle surface in each brain hemisphere, we introduce five cuts. Since these cutting positions are at the end of the frontal, occipital, and temporal horns of the ventricles, they can potentially be located automatically. The second row in Figure 1 shows 5 cuts introduced on three subjects ventricular surfaces. After the cutting, the surfaces become open boundary genus 4 surfaces. Figure 1 (e)-(g) show parameterizations of the lateral ventricles of the brain. (e) shows the results of parameterizing a ventricular surface for a 65-year-old patient with HIV/AIDS (note the disease-related enlargement), (f) the results for the ventricular model of a 21-year-old control subject, and (g) the results for a 28-year-old control subject. The surfaces are initially generated by using an unsupervised tissue classifier to isolate a binary map of the cerebrospinal fluid in
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the MR image, and tiling the surface of the largest connected component inside the brain. Based on the computed conformal structure, we can partition the surface into 6 patches. Each patch can be conformally mapped to a rectangle. Although the three 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 way for direct surface matching between any two ventricles. 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. An alternative ap- proach is to supplement the critical graph with curved landmarks that can then
Fig. 2. Illustrates the parameterization of cortical surfaces using the holomorphic 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.
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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 conformal grids. Figure 2 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 the approach (thick lines show the precental and postcentral sulci, and the superior temporal sulcus, and the perimeter of the corpus callosum at the midsagittal plane). By cutting the surface along the landmark curves, we obtain a genus 3 open boundary surface. There are therefore two zero points (observable as a large white region and black region in the conformal grid; an illustration of the conformal structure is shown in the first panel the first row). We show corti- cal surfaces from two different subjects in Figure 2 (these are extracted using a deformable surface approach, but are subsequently reparameterized using holo- morphic 1-forms). The second and fourth rows show the segmented patches for each cortical surface. The rectangles that these patches conformally map to are shown on the third and fifth row, respectively. Since the landmark curves lie on the boundaries of the surface patches, they can be forced to lie on an iso- parameter curve and can be constrained to map to rectangle boundaries in the parameter domain. Although the two cortex surfaces are different, the selected sulcal curves are mapped to the rectangle boundaries in the parameter domain. This method therefore provides a way to warp between two anatomical surfaces while exactly matching an arbitrary number of landmark curves lying in the surfaces. This is applicable to tracking brain growth or degeneration in serial scans, and composite maps of the cortex can be made by invoking the consistent parameterizations. Lamecker et al’s work [10] has the similar motivation as ours for the cortex case, which is to partition a surface into canonical patches and pa- rameterize the patches with minimal distortion. However, our partition method is based on intrinsic Riemann surface structure and theirs is based on shortest paths along lines of high curvature. Thus our method is global and more stable.
4 Conclusion and Future Work
In this paper, we presented a brain surface parameterization method that in- vokes the Riemann surface structure to generate conformal grids on surfaces of arbitrary complexity (including branching topologies). We tested our algo- rithm on the hippocampus, lateral ventricle surfaces and on surface models of the cerebral cortex. The grid generation algorithm is intrinsic (i.e. does not depend on any initial choice of surface coordinates) and is stable, as shown by grids induced on ventricles of various shapes and sizes. Compared with other work conformally mapping brain surfaces to sphere, our work may intro- duce less distortion and may be especially convenient for other post-processing work such as surface registration and landmark matching. Our future work in- clude automatic location of cutting positions and more experiments on disease assessment.
