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Heat kernel smoothing, hot spots conjecture and Fiedler vector

Banff International Research Station for Mathematical Innovation and Discovery

The second eigenfunction of the Laplace-Beltrami operator (often called Fiedler vector in discrete settings) follows the pattern of the overall shape of an object. This geometric property is well known and used for various applications including mesh processing, feature extraction, manifold learning, spectral embedding and the minimum linear arrangement problem. Surprisingly, this geometric property has not been precisely formulated yet. This problem is directly related to the somewhat obscure hot spots conjecture in differential geometry that postulates the behavior of heat diffusion near boundary. The aim of the talk is to discuss and raise the awareness of the problem. As an application of the hot spots conjecture, we show how the second eigenfunction alone can be used for shape modeling of elongated anatomical structures such as hippocampus and mandible, and determining the diameter of large-scale brain networks. This talk is based on Chung et al. 2015 Medical Image Analysis 22:63076.

Mathematics; Statistics; Partial differential equations

video/mp4

Author affiliation: University of Wisconsin-Madison

2019-03-10

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/68570

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/