Stochastic Heat Kernel Estimation on Sampled Manifolds

The heat kernel is a fundamental geometric object associated to every Riemannian manifold, used across applications in computer vision, graphics, and machine learning. In this article, we propose a novel computational approach to estimating the heat kernel of a statistically sampled manifold (e.g. m...

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Vydáno v:Computer graphics forum Ročník 36; číslo 5; s. 131 - 138
Hlavní autoři: Aumentado‐Armstrong, T., Siddiqi, K.
Médium: Journal Article
Jazyk:angličtina
Vydáno: Oxford Blackwell Publishing Ltd 01.08.2017
Témata:
Brownian motion
Categories and Subject Descriptors (according to ACM CCS)
Computer simulation
Computer vision
Density
Economic models
G.3 [Mathematics of Computing]: Probability and Statistics—Probabilistic Algorithms
I.3.5 [Computer Graphics]: Computational Geometry and Object Modelling—Geometric Algorithms, Languages, and Systems
Machine learning
Manifolds (mathematics)
Numerical integration
Parallel processing
Riemann manifold
Statistical analysis
Statistical methods
Three dimensional models
Trajectory analysis
ISSN:0167-7055, 1467-8659
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Shrnutí:The heat kernel is a fundamental geometric object associated to every Riemannian manifold, used across applications in computer vision, graphics, and machine learning. In this article, we propose a novel computational approach to estimating the heat kernel of a statistically sampled manifold (e.g. meshes or point clouds), using its representation as the transition density function of Brownian motion on the manifold. Our approach first constructs a set of local approximations to the manifold via moving least squares. We then simulate Brownian motion on the manifold by stochastic numerical integration of the associated Ito diffusion system. By accumulating a number of these trajectories, a kernel density estimation method can then be used to approximate the transition density function of the diffusion process, which is equivalent to the heat kernel. We analyse our algorithm on the 2‐sphere, as well as on shapes in 3D. Our approach is readily parallelizable and can handle manifold samples of large size as well as surfaces of high co‐dimension, since all the computations are local. We relate our method to the standard approaches in diffusion geometry and discuss directions for future work.
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ISSN:0167-7055
1467-8659
DOI:10.1111/cgf.13251