The Laplacian on a Riemannian Manifold

Rosenberg, Steven

doi:10.1017/cbo9780511623783.002

Cited by 147 publications

(126 citation statements)

References 0 publications

Supporting

Mentioning

126

Contrasting

Order By: Relevance

“…For small σ , using the parametrix expansion [12], we can approximate heat kernel locally using the Gaussian kernel for small bandwidth:

K_{σ} (p, q) = \frac{1}{\sqrt{4 π σ}} exp [- \frac{d^{2} (p, q)}{4 σ}] [1 + O (σ^{2})],

where d ( p , q ) is the geodesic distance between p and q . For sufficiently small bandwidth, all the kernel weights are concentrated near the center, so the first neighbors of a given vertex in a mesh is used in the approximation.…”

Section: Resultsmentioning

confidence: 99%

“…

K_{σ} (p, q) = \sum_{j = 0}^{\infty} e^{- λ_{j} σ} ψ_{j} false(p false) ψ_{j} false(q false),

where σ is the bandwidth of the kernel [3,12]. Then heat kernel smoothing of Y is given analytically as…”

Section: Heat Kernel Smoothingmentioning

confidence: 99%

“…This is taken as the estimate for θ . Since the expansion (4) is a unique solution to isotropic heat diffusion [3,12], we can avoid the need to solve the diffusion equation using less stable numerical schemes such as the finite difference method [1,8,14]. …”

Section: Heat Kernel Smoothingmentioning

confidence: 99%

See 2 more Smart Citations

Heat Kernel Smoothing Using Laplace-Beltrami Eigenfunctions

Seo

Chung

Vorperian

2010

Medical Image Computing and Computer-Assisted Intervention – MICCAI 2010

View full text Add to dashboard Cite

Abstract. We present a novel surface smoothing framework using the Laplace-Beltrami eigenfunctions. The Green's function of an isotropic diffusion equation on a manifold is constructed as a linear combination of the Laplace-Beltraimi operator. The Green's function is then used in constructing heat kernel smoothing. Unlike many previous approaches, diffusion is analytically represented as a series expansion avoiding numerical instability and inaccuracy issues. This proposed framework is illustrated with mandible surfaces, and is compared to a widely used iterative kernel smoothing technique in computational anatomy. The MATLAB source code is freely available at http://brainimaging.waisman.wisc. edu/~chung/lb.

show abstract

“…For small σ , using the parametrix expansion [12], we can approximate heat kernel locally using the Gaussian kernel for small bandwidth:

K_{σ} (p, q) = \frac{1}{\sqrt{4 π σ}} exp [- \frac{d^{2} (p, q)}{4 σ}] [1 + O (σ^{2})],

Section: Resultsmentioning

confidence: 99%

“…

K_{σ} (p, q) = \sum_{j = 0}^{\infty} e^{- λ_{j} σ} ψ_{j} false(p false) ψ_{j} false(q false),

where σ is the bandwidth of the kernel [3,12]. Then heat kernel smoothing of Y is given analytically as…”

Section: Heat Kernel Smoothingmentioning

confidence: 99%

See 1 more Smart Citation

Heat Kernel Smoothing Using Laplace-Beltrami Eigenfunctions

Seo

Chung

Vorperian

2010

Medical Image Computing and Computer-Assisted Intervention – MICCAI 2010

View full text Add to dashboard Cite

show abstract

“…The resulting adjacency graph approximates the underlying manifold of the dependence structure of the sample. The eigenfunctions of the Laplace-Beltrami operator [18] on the manifold are generalized geometric harmonic functions, which contain useful intrinsic geometric structure information on the population. The eigenvectors of the associated graph Laplacian matrix (see Methods ) are first-order linear approximations of the Laplacian eigenfunctions, and they relate to the intrinsic dependence structure of the data.…”

Section: Introductionmentioning

confidence: 99%

Laplacian Eigenfunctions Learn Population Structure

2009

View full text Add to dashboard Cite

Principal components analysis has been used for decades to summarize genetic variation across geographic regions and to infer population migration history. More recently, with the advent of genome-wide association studies of complex traits, it has become a commonly-used tool for detection and correction of confounding due to population structure. However, principal components are generally sensitive to outliers. Recently there has also been concern about its interpretation. Motivated from geometric learning, we describe a method based on spectral graph theory. Regarding each study subject as a node with suitably defined weights for its edges to close neighbors, one can form a weighted graph. We suggest using the spectrum of the associated graph Laplacian operator, namely, Laplacian eigenfunctions, to infer population structure. In simulations and real data on a ring species of birds, Laplacian eigenfunctions reveal more meaningful and less noisy structure of the underlying population, compared with principal components. The proposed approach is simple and computationally fast. It is expected to become a promising and basic method for population genetics and disease association studies.

show abstract

“…The zeta function (i.e. the sum of the eigenvalues raised to a non-integer power), on the other hand also leads to invariants [12] that are more trackable. From the Mellin transform, it follows that the zeta-function is the moment generating function of the heat kernel trace.…”

Section: Introductionmentioning

confidence: 99%

Object recognition using graph spectral invariants

Xiao

Wilson

Hancock

2008

2008 19th International Conference on Pattern Recognition

View full text Add to dashboard Cite

Graph structures have been proved important in high level-vision since they can be used to represent structural and relational arrangements ofobjects in a scene. One ofthe problems that arises in the analysis ofstructural abstractions of object is graph clustering. In this paper, we explore how permutation invariants computed from the trace of the heat kernel can be used to characterize graphs for the purposes ofmeasuring similarity and clustering. We explore three different approaches to characterize the heat kernel trace as afunction oftime. These are the heat kernel trace moments, heat content invariants and symmetric polynomials with Laplacian eigenvalues as inputs. Experiments on the COIL 100 and Caltech 256 databases reveal that the proposed invariants are effective and outperform the tradition methods.

show abstract

The Laplacian on a Riemannian Manifold

Cited by 147 publications

References 0 publications

Heat Kernel Smoothing Using Laplace-Beltrami Eigenfunctions

Heat Kernel Smoothing Using Laplace-Beltrami Eigenfunctions

Laplacian Eigenfunctions Learn Population Structure

Object recognition using graph spectral invariants

Contact Info

Product

Resources

About