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Intrinsic Volumes of Random Cubical Complexes

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Published on May 5, 2014

This video is similar to a presentation I gave at the Institute for Mathematics and its Applications on April 1, 2014. The abstract for the presentation was the following:

The intrinsic volumes generalize both Euler characteristic and Lebesgue volume, quantifying the size of a set in various ways. A random cubical complex is a union of (possibly high-dimensional) unit cubes selected from a lattice according to some probability model. I will describe a simple model of random cubical complex and derive exact polynomial formulae, dependent on a probability, for the expected value and variance of the intrinsic volumes of the complex. I will also give a central limit theorem and an interleaving theorem about the roots of the expected intrinsic volumes -- that is, the values of the probability parameter at which an expected value is zero. Lastly, I will discuss connections to random fields and applications, especially image recognition and the study of noise in digital images. This work is in collaboration with Michael Werman of The Hebrew University of Jerusalem.

A paper that contains all the math in this presentation and more is available at http://arxiv.org/abs/1402.5367 .

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