Diffeomorphic Surface Flows: A Novel Method of Surface Evolution (original) (raw)
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Lecture Notes in Computer Science, 2019
Deformable shape modeling approaches that describe objects in terms of their medial axis geometry (e.g., m-reps [10]) yield rich geometrical features that can be useful for analyzing the shape of sheet-like biological structures, such as the myocardium. We present a novel shape analysis approach that combines the benefits of medial shape modeling and diffeomorphometry. Our algorithm is formulated as a problem of matching shapes using diffeomorphic flows under constraints that approximately preserve medial axis geometry during deformation. As the result, correspondence between the medial axes of similar shapes is maintained. The approach is evaluated in the context of modeling the shape of the left ventricular wall from 3D echocardiography images. 1
The surface area preserving mean curvature flow
2003
Let M 0 be a compact, strictly convex hypersurface of dimension n ≥ 2, without boundary, smoothly embedded in R n+1 and represented locally by some diffeomorphism F 0 : R n ⊃ U → F 0 (U ) ⊂ M 0 ⊂ R n+1 . Under the surface area preserving mean curvature flow, formulated by Pihan in [P], the family of maps F t = F (·, t) evolves according to
Finite topology self-translating surfaces for the mean curvature flow in R^3
2015
Finite topology self translating surfaces to mean curvature flow of surfaces constitute a key element for the analysis of Type II singularities from a compact surface, since they arise in a limit after suitable blow-up scalings around the singularity. We find in R^3 a surface M orientable, embedded and complete with finite topology (and large genus) with three ends asymptotically paraboloidal, such that the moving surface Σ(t) = M + te_z evolves by mean curvature flow. This amounts to the equation H_M = ν· e_z where H_M denotes mean curvature, ν is a choice of unit normal to M, and e_z is a unit vector along the z-axis. The surface M is in correspondence with the classical 3-end Costa-Hoffmann-Meeks minimal surface with large genus, which has two asymptotically catenoidal ends and one planar end, and a long array of small tunnels in the intersection region resembling a periodic Scherk surface. This example is the first non-trivial one of its kind, and it suggests a strong connection...
Brain Surface Conformal Parameterization With the Ricci Flow
IEEE Transactions on Medical Imaging, 2000
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 processing. By solving the Yamabe equation with the Ricci flow method, we can conformally parameterize a brain surface via a mapping to a multi-hole disk. The resulting parameterizations do not have any singularities and are intrinsic and stable. To illustrate the technique, we computed parameterizations of cortical surfaces in MRI scans of the brain. We also show the parameterization results are consistent with constraints imposed on the mappings of selected landmark curves, and the resulting surfaces can be matched to each other using constrained harmonic maps. Unlike previous planar conformal parameterization methods, our algorithm does not introduce any singularity points.
Coarse-to-Fine Hamiltonian Dynamics of Hierarchical Flows in Computational Anatomy
2020 IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops (CVPRW), 2020
We present here the Hamiltonian control equations for hierarchical diffeomorphic flows of particles. We define the controls to be a series of multi-scale vector fields, each with their own reproducing kernel Hilbert space norm. The hierarchical control is connected across scale through successive refinements that refine as they ascend the hierarchy with commensurately higher bandwidth Green's kernels. Interestingly the geodesic equations do not separate, with fine scale motions determined by all of the particle information simultaneously, from coarse to fine. Additionally, the hierarchical conservation law is derived, defining the geodesics and demonstrating the constancy of the Hamiltonian. We show results on one simulated example and one example from histological images of an Alzheimer's disease brain. We introduce the varifold action to transport the weights of micro-scale particles for mapping to sub millimeter scale cortical folds.
Lectures on Mean Curvature Flow
2020
ABSTRACT. In this series of lectures I will introduce the mean curvature flow of a compact hypersurface in the Euclidean space with particular attention to the cases of curves and surfaces. The basic properties and the main analytic and geometric techniques used in the analysis of this flow will be discussed, for instance maximum and comparison principles. Moreover, I will present the fundamental Huisken's monotonicity formula and the Harnack inequality by Hamilton, which are the key tools in the study of the singularity formation. Up to now, the classification of possible asymptotic shape of a singularity is almost complete for some classes of evolving hypersurfaces. For others it seems difficult and quite far off. Time allowing, in the last lectures I will try to outline an up-to-date scenario of the "state of the art".
The normalized mean curvature flow for a small bubble in a Riemannian manifold
Journal of Differential Geometry, 2003
The evolution of an embedded surface under the normalized mean curvature flow is the result of a complicated interaction between the geometry of the evolving surface and the geometry of the ambient space, and is not well understood in the context of a general Riemannian manifold. In the present paper we identify a class of initial conditions, that we call "bubbles", whose dynamics is primarily determined by the ambient space. A bubble is an embedded surface that is close to a small geodesic ball; we find that its shape is robust along the evolution. Moreover, under a relatively tight condition relating shape to size, we show that the velocity of the center of the bubble is given, to principal order, by the gradient of the scalar curvature. Finally under natural conditions of compactness and nondegeneracy we show that such solutions converge, as t tends to infinity, to surfaces of constant mean curvature.
On the Laplace-Beltrami Operator and Brain Surface Flattening
IEEE Transactions on Medical Imaging, 1999
In this paper, using certain conformal mappings from uniformizationtheory, we give an explicit method for flattening the brain surfacein a way which preserves angles. From a triangulated surface representationof the cortex, we indicate how the procedure may be implementedusing finite elements. Further, we show how the geometry ofthe brain surface may be studied using this approach.Keywords: Brain flattening, functional MRI,
NeuroImage, 2011
This paper introduces a novel large deformation diffeomorphic metric mapping algorithm for whole brain registration where sulcal and gyral curves, cortical surfaces, and intensity images are simultaneously carried from one subject to another through a flow of diffeomorphisms. To the best of our knowledge, this is the first time that the diffeomorphic metric from one brain to another is derived in a shape space of intensity images and point sets (such as curves and surfaces) in a unified manner. We describe the Euler-Lagrange equation associated with this algorithm with respect to momentum, a linear transformation of the velocity vector field of the diffeomorphic flow. The numerical implementation for solving this variational problem, which involves largescale kernel convolution in an irregular grid, is made feasible by introducing a class of computationally friendly kernels. We apply this algorithm to align magnetic resonance brain data. Our whole brain mapping results show that our algorithm outperforms the image-based LDDMM algorithm in terms of the mapping accuracy of gyral/sulcal curves, sulcal regions, and cortical and subcortical segmentation. Moreover, our algorithm provides better whole brain alignment than combined volumetric and surface registration (Postelnicu et al., 2009) and hierarchical attribute matching mechanism for elastic registration (HAMMER) (Shen and Davatzikos, 2002) in terms of cortical and subcortical volume segmentation.