Generic transitions of relative critical sets in parametrized families with applications to image analysis
James Damon, Robert Scott Keller · 1999
Many techniques for describing shape in greyscale computer images utilize image geometry. One method introduced by Pizer, Fritsch, and Morse uses cores, which are sets of points extracted from the image by applying a height ridge construction to a medial function associated to the image. A medial function measures the degree to which a point in an image feature at a specific scale behaves like a middle point. The height ridge, developed in detail by Eberly, belongs to a larger class of relative critical sets that are higher-dimensional analogues of local extrema of smooth functions. In earlier work, Damon and Miller determined the generic local structure of such sets. They applied these results to study the generic properties of cores in 2- and 3-dimensional greyscale images. Understanding how shape features change in families of images is equally important for questions in image analysis. Such families arise naturally in medical imaging in sequences of time-dependent images and sequences obtained in registration, a common problem in which different image poses must be aligned. In this dissertation, we investigate the classification of the generic transitions which occur for d-dimensional relative critical sets in parametrized families of functions on Cn . Three transversality conditions capture the structure of these sets. Work by Damon showed that the key to studying transition behavior is to understand how the transversality conditions fail in parametrized families. Using methods of singularity theory, we will classify the ways in which transversality fails generically. From this classification, we obtain local models for the transitions we expect to see in parametrized families. We give the complete classification of the generic transitions exhibited by 1- and 2-dimensional relative critical sets in 1- and 2-parameter families of smooth functions. These genericity results follow from extensions of Thom's Transversality Theorem. We then apply these results to study the generic changes occurring for cores in 1-parameter families of 3-dimensional greyscale images. A relative version of Thom's Transversality Theorem is employed to derive the corresponding genericity results for cores.