The Laplacian on a Riemannian Manifold

Abstract. This paper presents a unified image processing and analysis framework for cortical thickness in characterizing a clinical population. The emphasis is placed on the development of data smoothing and analysis framework. The human brain cortex is a highly convoluted surface. Due to the convoluted non-Euclidean surface geometry, data smoothing and analysis on the cortex are inherently difficult. When measurements lie on a curved surface, it is natural to assign kernel smoothing weights based on the geodesic distance along the surface rather than the Euclidean distance. We present a new data smoothing framework that address this problem implicitly without actually computing the geodesic distance and present its statistical properties. Afterwards, the statistical inference is based on the random field theory based multiple comparison correction. As an illustration, we have applied the method in detecting the regions of abnormal cortical thickness in 16 high functioning autistic children.

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“…This is a well known result [17]. This theorem implies that the heat kernel smoothing isotropically assigns weights on ∂Ω.…”

Section: Theorem 1 K σ * Y Is the Unique Solution Of The Following Imentioning

confidence: 69%

“…where σ is the bandwidth of the kernel [2] [17] . When g ij = δ ij , the heat kernel becomes the Gaussian kernel, which is the probability density of N (0, σ 2 ).…”

Section: Heat Kernel Smoothingmentioning

confidence: 99%

Unified Statistical Approach to Cortical Thickness Analysis

Chung

Robbins

Evans

2005

Lecture Notes in Computer Science

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“…As in the flat space, a reactiondiffusion system on a surface S⊂R 3 is governed by mass conservation: (1) where C i is the concentration of the i-th membrane species, Ji = −D i grad S C i is the diffusive flux density of this species, D i is the diffusion coefficient, and the source term R i describes the effect of all reactions on the i-th species. The differential operators div S and grad S are defined on S and the second-order differential operator div S (grad S ) ≡ Δ S that appears in Eqs (1) is the Laplace-Beltrami operator (LBO) [4], which is a generalization of the Laplacian on manifolds 1 .…”

Section: Introductionmentioning

confidence: 99%

Diffusion on a curved surface coupled to diffusion in the volume: Application to cell biology

Novak

Gao

Choi

et al. 2007

Journal of Computational Physics

106

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An algorithm is presented for solving a diffusion equation on a curved surface coupled to diffusion in the volume, a problem often arising in cell biology. It applies to pixilated surfaces obtained from experimental images and performs at low computational cost. In the method, the Laplace-Beltrami operator is approximated locally by the Laplacian on the tangential plane and then a finite volume discretization scheme based on a Voronoi decomposition is applied. Convergence studies show that mass conservation built in the discretization scheme and cancellation of sampling error ensure convergence of the solution in space with an order between 1 and 2. The method is applied to a cellbiological problem where a signaling molecule, G-protein Rac, cycles between the cytoplasm and cell membrane thus coupling its diffusion in the membrane to that in the cell interior. Simulations on realistic cell geometry are performed to validate, and determine the accuracy of, a recently proposed simplified quantitative analysis of fluorescence loss in photobleaching. The method is implemented within the Virtual Cell computational framework freely accessible at www.vcell.org.

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“…From now on, we suppose that the dimension d is even. The first crucial step is McKean-Singer formula [34] (A simple proof of it can be found in [39], p. 113). We have χ(M) = M Str p t (x, x)dx, t > 0, where P t = e −t and p t is the corresponding Schwartz kernel (density).…”

Section: The Chern-gauss-bonnet Theoremmentioning

confidence: 99%