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Parametric Morse theory and sequential image anaysis
by
Neza Mramor
University of Ljubljana, Faculty of Computer and Information Science, Slovenia
Coauthors: Jure Zabkar, Gregor Jerse
In this paper, we present three applications of parametrized discrete Morse theory to image analysis. Discrete Morse theory, introduced by Foreman, represents a discrete analogue of classical smooth Morse theory. It enables an analysis of the critical points of a smooth function f, sampled in a finite number of points, and a decomposition of the domain of f into descending and ascending regions which simulates the Morse-Smale decomposition.
Given a regular cellular decomposition K of the domain, the sampled values of f can be extended to a discrete Morse function on K, that is, a function associating a value to every cell of K which is monotone with respect to dimension in almost all cases. The exceptions to this rule determine a pairing of the cells, called a discrete vector field, defined on a subset of K. Cells which remain unpaired are the critical cells of the discrete Morse function and correspond to critical points of a smooth function in smooth Morse theory.
A parametric discrete Morse function, as considered by King, Knudson and Mramor, is a collection of discrete Morse functions fti, i=1, ..., m defined on K, and a pairing of the cells that occur in adjacent slices. It produces a bifurcation diagram for the births and deaths of critical cells of the functions fti with increasing t.
Parametric discrete Morse functions can be successfully applied to analyze of various types of image sequences. In our first application, we consider a sequence of meteorological radar images. Sequential radar images show the process of precipitation intensity over a fixed region which is viewed as a parametric discrete Morse function. Accompanied with additional information from other data sources (e.g. temperature profiles, wind speed and direction etc.) the bifurcation diagram could be used to study the development of thunderstorms. In the second example, we analyze a spatial sequence of CT scans of a human abdomen. Each image represents a slice of the abdomen at a different height.
The third application tackles a robotic domain in which the robot observes a red and a blue ball with an on-board camera. Its task is to learn the notion of object occlusion, that is, the robot learns from examples that objects should partially overlap before total occlusion. The model is represented in the form of a finite automaton.
Date received: May 22, 2008
Copyright © 2008 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caxd-29.