Table of Contents

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.