Relative critical sets in RR(N) and applications to image analysis
Jason Miller · 1998
Geometric structure of pixel intensity functions can be used to identify the “shape” features of objects and figures in three dimensional greyscale images, such as MRI and CAT images. Such geometric descriptions provide tools for tackling medical imaging problems, including the segmentation and registration problems. Pizer and Eberly defined a shape skeleton, called the core, for greyscale images by applying a ridge construction to a medialness function that is directly derived from the greyscale image. The d-dimensional (height) ridges of a function f∈C∞U , for U⊂Rn , generalize the idea of local maxima of f. They are a special type of d-dimensional relative critical set, and fit into a ridge-valley-connector set that conveys important geometric information about f. The objective of this dissertation is to present a classification of the local generic geometric properties of d-dimensional ridges, relative critical sets, and ridge-valley-connector sets. These results will then be used to classify the local generic properties of Pizer and Eberly's 1- and 2-dimensional shape skeleton for three dimensional greyscale images. The Thom transversality theorem (and modified versions) will yield our genericity results once we identify submanifolds of jet space that define properties of d-dimensional relative critical sets. These submanifolds are defined using three auxiliary maps on an open dense subset U0⊂U , whose complement contains points at which H(f), the Hessian of f, has a repeated eigenvalue. To understand how repeated eigenvalues affect the geometric structure of the d-dimensional relative critical sets, we stratify the space of n × n symmetric matrices according to corank and eigenvalue multiplicity. This along with universal properties of the d-dimensional relative critical sets on U0 allow us to determine how a d-dimensional relative critical set ends and is continued by another d-dimensional relative critical set. Extending the classification to the space of medial functions involves expressing the medial construction as a local differential operator on the space of solutions to the heat equation. By showing the image of this operator is transverse to the submanifolds identified in the previous work, a modified version of the Thom transversality theorem yields the classification of local generic properties of Pizer and Eberly's shape skeleta.