An important problem in the analysis of images and videos is the study of the persistence of objects in sequences of video frames and photographs that record surface shape changes in a visual scene. Surface shapes can appear, undergo a change because of varying light conditions, and finally they can disappear. Thus, to understand a visual scene we need to study three different aspects, namely geometry, topology of the scene and the physics of light, the character and energy of the photons colliding with curved surfaces. The book introduces different concepts from computational geometry, topology and physics which together form a very effective tools in the analysis of images and video sequences. Computational geometry captures the fine-grained structures embedded in image object shapes. Computational topology makes it possible to capture and analyse the proximities found in cellular complexes embedded in the geometry of triangulated visual scenes. And finally, computational physics is used in cases where we take into account the description of surface shapes and the light reflected from surface shapes stored in photos and video frames. The book not only presents the basics of the three subjects, but also gives a number of practical applications, e.g., cellular division trails, tracking changes in video frame shapes, comparison of collections of nesting, non-concentric vortex feature vectors, descriptive proximity in classifying physical object shapes etc. The material for the book grew out of a course on computer vision that the author has conducted over the past several years and of the author’s discussions with a number of researchers, graduate students and post-doctoral fellows.
Chapter 1 gives an introduction to the three main subjects, i.e., computational geometry, topology and physics. In all three cases, the mathematics and algorithms which provide a basis for the analysis, comparison and classification of physical shapes are presented. In Chapter 2 we find information on cell complexes, nerve structures and shapes. Cell complexes are used to represent shapes. Using this approach, we can introduce shape fingerprints that are very simple collections of things called filament skeletons with lucid measurable and comparable properties. In this way we are able to compare shapes lurking in the point-clouds in digital images. Chapter 3 takes another look at filament skeletons, skeletal vortexes and skeletal nerves in cell complexes. The author focuses on the group theory underlying the computational topology of digital images. In Chapter 4 the author explores what nerve structures in cell complexes tell us about approximating shapes revealed by light reflected from curved surfaces. The attention is focused on two forms of nerve complexes, namely Alexandroff nerve and Alexandroff star nerve, which were introduced by Paul Alexandroff. To study the relationships between sub-complexes in cell complexes, in Chapter 5, two basic types of proximities are introduced. These two types include spatial and descriptive proximities. In Chapter 6 a number of basic types of shape classes commonly found in closure weak topology complexes are presented. The proximities (from Chapter 5) and shape classes (from Chapter 6) are useful in clustering and separating subcomplexes in triangulated finite, bounded surface regions such as those found in visual scenes. Chapter 7 introduces an approximation approach to determining the closeness of descriptions of nerve shapes, which is highly application-oriented. This approximation approach is a relaxed form of descriptive proximity. In the final chapter, Chapter 8, the author takes a look at the Alexandroff version of the Brouwer-Lebesgue tiling theorem and introduces systems of nerve complexes that have proximity to each other and which are known shapes that cover all or part of the interior of unknown surface shapes in visual scenes.
The book is presented in a very accessible fashion. The author gives many examples presenting the notations and problems that are considered. Every chapter ends with sources, further reading and some problems to solve. The book is suitable for graduate students and researchers interested in computational geometry and computer vision. Moreover, it can be used as a professional reference.