Stochastic Heat Kernel Estimation on Sampled Manifolds

dc.contributor.authorAumentado-Armstrong, Tristanen_US
dc.contributor.authorSiddiqi, Kaleemen_US
dc.contributor.editorBærentzen, Jakob Andreas and Hildebrandt, Klausen_US
dc.date.accessioned2017-07-02T17:37:53Z
dc.date.available2017-07-02T17:37:53Z
dc.date.issued2017
dc.description.abstractThe 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.en_US
dc.description.number5
dc.description.sectionheadersSpectra and Kernels
dc.description.seriesinformationComputer Graphics Forum
dc.description.volume36
dc.identifier.doi10.1111/cgf.13251
dc.identifier.issn1467-8659
dc.identifier.pages131-138
dc.identifier.urihttps://doi.org/10.1111/cgf.13251
dc.identifier.urihttps://diglib.eg.org:443/handle/10.1111/cgf13251
dc.publisherThe Eurographics Association and John Wiley & Sons Ltd.en_US
dc.subjectG.3 [Mathematics of Computing]
dc.subjectProbability and Statistics
dc.subjectProbabilistic Algorithms
dc.subject
dc.subjectI.3.5 [Computer Graphics]
dc.subjectComputational Geometry and Object Modelling
dc.subjectGeometric Algorithms
dc.subjectLanguages
dc.subjectand Systems
dc.titleStochastic Heat Kernel Estimation on Sampled Manifoldsen_US
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