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BRAIN SURFACE CONFORMAL PARAMETERIZATION WITH THE SLIT MAPPING
Yalin Wang^1 ,^2 , Xianfeng Gu^3 , Tony F. Chan^1 , Paul M. Thompson^2 , Shing-Tung Yau^4
1 Mathematics Department, UCLA
2 Lab. of Neuro Imaging and Brain Research Institute, UCLA School of Medicine
3 Computer Science Department, Stony Brook University
4 Mathematics Department, Harvard University
ABSTRACT
Brain surface conformal mapping has been studied inten- sively. In this paper, we propose a method that computes a conformal mapping from a multiply connected mesh to the so-called slit domain, which consists of a canonical rectangle or disk in which 3D curved landmarks on the original surfaces are mapped to concentric or parallel lines in the slit domain. After cutting along some landmark curve features on surface models of the cerebral cortex, we obtain multiple connected domains. By computing exact harmonic one-forms, closed harmonic one-forms, and holomorphic one-forms, we are able to build a circular slit mapping that conformally maps the surface to an annulus with some concentric arcs and a rectangle with some slits. The whole algorithm is based on solving linear systems so it is very stable. In the slit domain parameterization results, the feature curves are either mapped to straight lines or a concentric arcs. This representation is convenient for anatomical visualization, and may assist sta- tistical comparisons of anatomy, surface-based registration and signal processing. Preliminary experimental results pa- rameterizing various brain anatomical surfaces are presented. Index Terms— Biomedical Imaging, Brain Mapping, Surface Parameterization, Slit Mapping
1. INTRODUCTION
Parameterization of anatomical surfaces in brain imaging is valuable to help analyze anatomical shape and statisti- cally combine or compare 3D anatomical models across subjects. Conformal parameterization can provide a one-to- one, onto, and angle-preserving map from a general manifold to a canonical space, such as a sphere, disk or other subre- gion of the plane. In this mapping, the elements of the first fundamental form remain unchanged, except for a scaling This work was funded by the National Institutes of Health through the NIH Roadmap for Medical Research, Grant U54 RR021813 entitled Center for Computational Biology (CCB). The work was performed while the third author was on leave at the National Science Foundation as Assistant Director of the Directorate for Mathematics & Physical Sciences.
factor. The mapping preserves similarities in small regions; it can also be used to simplify covariant derivative formulas on a general manifold, which is useful for solving variational problems on surfaces, such as signal denoising or compu- tation of correspondence vector fields that match different surfaces. In this paper, we introduce a new method to conformally map a multiply connected domain to an annulus with multi- ple concentric arcs (called the circular slit map) or to a rect- angle with multiple straight lines (the parallel slit map). It is a global conformal parameterization method without seg- mentation. First, it computes exact harmonic one-forms and closed harmonic one-forms. Secondly, it computes all bases of holomorphic one-forms. Given appropriate boundary con- ditions, it can compute a unique circular slit map up to a ro- tation around the center. The slit mapping computes the in- trinsic structure of the given surface, which can be reflected in the shape of the target domain. This work continues our group’s previous research on brain surface conformal parameterization. The new work has the same motivation, i.e., to match brain landmarks along boundaries by global conformal parameterization, but it over- comes the singularity problem in the holomorphic flow seg- mentation algorithm [1]. The method is also more stable than the highly non-linear solution of the Ricci flow method [2].
1.1. Related Work
Brain surface parameterization has been studied intensively. Schwartz et al. [3], and Timsari and Leahy [4] computed quasi-isometric flat maps of the cerebral cortex. Drury et al. [5] presented a multiresolution method for flattening the cerebral cortex. Hurdal and Stephenson [6] reported a dis- crete mapping approach that uses circle packings to produce “flattened” images of cortical surfaces on the sphere, the Eu- clidean plane, and the hyperbolic plane. The maps obtained are quasi-conformal approximations of classical conformal maps. Haker et al. [7] implemented a finite element ap- proximation for parameterizing brain surfaces via conformal mappings. They select a point on the cortex to map to the
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north pole of the Riemann sphere and conformally map the rest of the cortical surface to the complex plane by stere- ographic projection of the Riemann sphere to the complex plane. Gu et al. [8] proposed a method to find a unique con- formal mapping between any two genus zero manifolds by minimizing the harmonic energy of the map. They demon- strate this method by conformally mapping a cortical surface to a sphere. Ju et al. [9] presented a least squares confor- mal mapping method for cortical surface flattening. Joshi et al. [10] proposed a scheme to parameterize the surface of the cerebral cortex by minimizing an energy functional in the pth^ norm. Wang et al. [1] have used holomorphic 1-forms to parameterize anatomical surfaces with complex (possibly branching) topology. Wang et al. [2] introduced a brain surface conformal mapping algorithm based on alge- braic functions. By solving the Yamabe equation with the Ricci flow method, it can conformally map a brain surface to a multi-hole disk. Recently, Ju et al. [11] reported the results of a quantitative comparison of FreeSurfer [12], CirclePack, and least squares conformal mapping (LSCM) with respect to geometric distortion and computational speed.
2. THEORETICAL BACKGROUND
Suppose S is a surface embedded in R^3 , with induced Eu- clidean metric g. S is covered by an atlas {(Uα, φα)}. Sup- pose (xα, yα) is the local parameter on the chart (Uα, φα). We say (xα, yα) is isothermal, if the metric has the represen- tation g = e^2 λ(xα,yα)(dx^2 α + dy^2 α). The Laplace-Beltrami operator is defined as Δg = 1 e^2 λ(xα,yα)^ (^
∂^2 ∂x^2 α^ +^
∂^2 ∂y^2 α^ ). A function^ f^ :^ S^ →^ R^ is^ harmonic, if Δgf ≡ 0. Suppose ω is a differential one-form with the represen- tation fαdxα + gαdyα in the local parameters (xα, yα), and fβ dxβ + gβ dyβ in the local parameters (xβ , yβ ). Then ( (^) ∂x α ∂xβ
∂yα ∂xβ ∂xα ∂yβ
∂yα ∂yβ
fα gα
fβ gβ
ω is a closed one-form, if on each chart (xα, yα), ∂f ∂yα −^
∂g ∂xα = 0.^ ω^ is an^ exact one-form, if it equals the gradient of some function. An exact one-form is also a closed one-form. If a closed one-form ω satisfies (^) ∂x∂fα + (^) ∂y∂gα = 0, then ω is a harmonic one-form. The gradient of a harmonic function is an exact harmonic one-form. The so-called Hodge star operator turns a one-form ω to its conjugate ∗ω, ∗ω = −gαdxα + fαdyα. A holomorphic one-form is a complex differential form ω +
− 1 ∗ω, where ω is a harmonic one-form. Suppose S is an open surface with n boundaries γ 1 , · · · , γn. We can uniquely find a holomorphic one-form ω, such that
∫
γk
ω =
2 π k = 1 − 2 π k = 2 0 otherwise
Circular Slit Mapping Fix a point p 0 on the surface, for any point p ∈ S, let γ be an arbitrary path connection p 0 and p, then the circular slit mapping is defined as φ(p) = e
R γ ω^.
Theorem 2.1 The function φ effects a one-to-one confor- mal mapping of M onto the annulus 1 < |z| < eλ^0 minus n − 2 concentric arcs situated on the circles |z| = eλi^ , i = 1 , 2 , · · · , n − 2.
The proof of the above theorem on slit mapping can be found in [13]. For a given choice of the inner and outer circle, the circular slit mapping is uniquely determined up to a rotation around the center. The parallel slit mapping can be defined in a similar way.
Parallel Slit Mapping Let S¯ be the universal covering space of the surface S, π : S¯ → S be the projection and ω¯ = π∗ω be the pull back of ω. Fix a point ¯p 0 on S¯, for any point p ∈ S¯, let γ¯ be an arbitrary path connection p¯ 0 and p¯, then the parallel slit mapping is defined as φ¯(¯p) =
¯γ ¯ω.
- ALGORITHM PIPELINE
Suppose the input mesh has n + 1 boundaries, ∂M = γ 0 − γ 1 − · · · − γn. Without loss of generality, we map γ 0 to the outer circle of the circular slit domain, γ 1 to the inner circle, and all the others to the concentric slits. The algorithm pipeline is as follows : 1 Compute the basis for all exact harmonic one-forms;
- Compute the basis for all closed harmonic one-forms;
- Compute the basis for all holomorphic one-forms;
- Construct the slit mapping.
3.1. Basis for Exact Harmonic One-forms
The first step of the algorithm is to compute the basis for exact harmonic one-forms. Let γk be an inner bound- ary, we compute a harmonic function fk : S → R by solving the following Dirichlet problem on the mesh{ M : Δfk ≡ 0 fk|γj = δkj^ , where^ δkj^ is the Kronecker function,^ Δ^ is the discrete Laplace-Beltrami operator using the co-tangent formula proposed in [14]. The exact harmonic one-form ηk can be computed as the gradient of the harmonic function fk, ηk = dfk, and {η 1 , η 2 , · · · , ηn} form the basis for the exact harmonic one- forms.
3.2. Basis for Harmonic One-forms After getting the exact harmonic one-forms, we will compute the closed one-form basis. Let γk (k > 0 ) be an inner bound- ary. Compute a path from γk to γ 0 , denote it as ζk. ζk cut the mesh open to Mk, while ζk itself is split into two boundary segments ζ k+ and ζ k− in Mk. Define a function gk : Mk → R
In Figure 1, Subfigure 3 demonstrates various parallel slit map results given different boundary conditions. As shown in Subfigure 3, four landmarks were cut open. After the cut, the surface turns into an open boundary genus three surface. For the Equation 1, we selected two different pairs of landmarks as the exterior and inner boundaries by putting the integration of different γk as 2 π and − 2 π. The second column shows the parameterization results when we use landmark a and d as the exterior and inner circular boundaries, respectively. The third column shows the parameterization results when we select the other pair of landmark curves as the boundary conditions.
5. CONCLUSIONS AND FUTURE WORK
In this paper, we presented a brain surface conformal param- eterization method based on the slit map, which transfers cor- tical geometry and any embedded landmarks into a canon- ical domain, with conformal coordinates. With fixed bound- ary conditions, our algorithm can compute unique circular slit maps and parallel slit maps, where the positions and lengths of the slits are determined by the conformal equivalence class of the surface. We tested our algorithm on hippocampal, lat- eral ventricular and cerebral cortical surfaces. Compared with our previous work [1, 2], our new work does not have any sin- gularities and more stable because only linear systems are in- volved. Another advantage is that mappings between surfaces could be readily set up via 2D matching in these domains, us- ing the conformal parameterization to conveniently discretize differential operators. Our future work will include empiri- cal application of the slit map algorithm to biomedical appli- cations in computational anatomy, including the detection of population differences and the tracking of brain change over time.
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Fig. 1. In Subfigure 1: (a) and (b) show the cortical surface with 6 landmarks cut open, including an open boundary at the corpus callosum (in green); (c) is the parallel slit map result; (f) is the circular slit map result; (d) and (e) show the confor- mal texture parameterized by the circular slit map (f). Sub- figure 2 illustrates the parallel slit mapping results for a hip- pocampal surface and for a surface of the left lateral ventricle. 5respectively. In Subfigure 3, conformal parameterization re- sults are shown with different boundary conditions. The first column shows a cerebral cortical surface with 4 landmarks in- troduced as cuts. The second column shows the circular slit map and parallel slit map results when a pair of landmarks are selected as boundaries The third column shows results the other pair of landmarks are selected as boundaries.
