The concept of ridges has appeared numerous times in the image processing liter ature. Sometimes the term is used in an intuitive sense. Other times a concrete definition is provided. In almost all cases the concept is used for very specific ap plications. When analyzing images or data sets, it is very natural for a scientist to measure critical behavior by considering maxima or minima of the data. These critical points are relatively easy to compute. Numerical packages always provide support for root finding or optimization, whether it be through bisection, Newton's method, conjugate gradient method, or other standard methods. It has not been natural for scientists to consider critical behavior in a higher-order sense. The con cept of ridge as a manifold of critical points is a natural extension of the concept of local maximum as an isolated critical point. However, almost no attention has been given to formalizing the concept. There is a need for a formal development. There is a need for understanding the computation issues that arise in the imple mentations. The purpose of this book is to address both needs by providing a formal mathematical foundation and a computational framework for ridges. The intended audience for this book includes anyone interested in exploring the use fulness of ridges in data analysis.
Inhaltsverzeichnis
1 Introduction. - 1. 1 A History of Ridges. - 1. 2 Reading Strategies. - 2 Mathematical Preliminaries. - 2. 1 Linear Algebra. - 2. 2 Differential Calculus. - 2. 3 Tensors. - 3 Ridges in Euclidean Geometry. - 3. 1 Generalized Local Extrema. - 3. 2 Height Ridge Definition. - 3. 3 1-Dimensional Ridges in ? 2. - 3. 4 1-Dimensional Ridges in ? 3. - 3. 5 1-Dimensional Ridges in ? n. - 3. 6 2-Dimensional Ridges in ? 3. - 3. 7 2-Dimensional Ridges in ? 4. - 3. 8 d-Dimensional Ridges in ? n. - 4 Ridges in Riemannian Geometry. - 4. 1 Generalized Local Extrema. - 4. 2 Height Ridge Definition. - 4. 3 1-Dimensional Ridges in ? 2. - 4. 4 1-Dimensional Ridges in ? 3. - 4. 5 1-Dimensional Ridges in ? n. - 4. 6 2-Dimensional Ridges in ? 3. - 4. 7 2-Dimensional Ridges in ? 4. - 4. 8 d-Dimensional Ridges in ? n. - 5 Ridges of Functions Defined on Manifolds. - 5. 1 Height Ridge Definition. - 5. 2 Maximal Curvature Ridge Definitions. - 6 Applications to Image and Data Analysis. - 6. 1 Medical Image Analysis. - 6. 2 Molecular Modeling. - 6. 3 Fluid Flow. - 7 Implementation Issues. - 7. 1 Bridging the Gap Between Theory and Practice. - 7. 2 B-spline Interpolation. - 7. 3 Eigensystem Solvers. - 7. 4 Ridge Construction.
