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1996 | Book

Introduction Read chapter Read first chapter

Ridges in Image and Data Analysis

Author: David Eberly

Publisher: Springer Netherlands

Book Series : Computational Imaging and Vision

Included in: Professional Book Archive

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About this book

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.

Table of Contents

Frontmatter
Chapter 1. Introduction
Abstract
Webster’s Dictionary [108] provides the following definitions for ridge.
David Eberly
Chapter 2. Mathematical Preliminaries
Abstract
Each section of this chapter provides some background material which is used throughout the book. As an aid in the somewhat tedious task of tracking down definitions of symbols and notation encountered in later chapters, we provide in the index the page numbers of symbols where they are first defined.
David Eberly
Chapter 3. Ridges in Euclidean Geometry
Abstract
We now take a closer look at the definition for local extrema of a function f ∈ C2(ℝ n , ℝ). Ridges will be a generalization of local maxima whereby the test for maximality of f(x) is made in a restricted neighborhood of x. A similar concept of courses generalizes local minima, but since local minima of f are local maxima of —f, it is sufficient to study only the concept of ridge.
David Eberly
Chapter 4. Ridges in Riemannian Geometry
Abstract
Chapter 3 discussed the fundamental concepts of generalized local extrema and height ridges. The concepts were applied to functions f: ℝ n → ℝ where ℝ n is the set of n-tuples of real numbers. An implicit assumption was made that ℝ n , as a geometric entity, is standard Euclidean space whose metric tensor is the identity. The same concepts are definable even if ℝ n is assigned an arbitrary positive definite metric tensor. The extension to Riemannian geometry requires tensor calculus which is discussed in Section 2.3. Most notably the constructions involve the ideas of covariant and contravariant tensors and of covariant differentiation.
David Eberly
Chapter 5. Ridges of Functions Defined on Manifolds
Abstract
In Section 5.1 we apply the definitions of Chapter 4 to finding height ridges of functions defined on manifolds. The first case is simplest where we construct 1dimensional ridges of a function defined on a 2—dimensional surface embedded in 1R3. The second case is the most general where we construct d—dimensional ridges of a function defined on an n—dimensional manifold embedded in IRP. Section 5.2 provides an alternative definition for ridges based on principal curvatures and principal directions. Section 5.3 discusses a ridge definition which is an application of the definition of Section 5.2 to level sets.
David Eberly
Chapter 6. Applications to Image and Data Analysis
Abstract
We describe an approach to the processing of an image by a front—end vision system. This system is a geometry engine [58] that converts the image intensity data into concise geometric information that can be interpreted by semantical systems in later stages of processing [31]. The basis for the approach is linear scale space [111, 11, 57, 4]. There has been much research in the area of both linear and nonlinear scale spaces. Two good research texts on the topic are [64] and [97]. Applications of these ideas to medical image analysis are also presented.
David Eberly
Chapter 7. Implementation Issues
Abstract
Ridge construction is not just a straightforward application of the mathematical models presented in Chapters 3 through 5. Care must be taken when attempting to implement these ideas. Like most computational problems involving a large amount of multidimensional data, there is a desire to obtain the results in a reasonable amount of time. This requirement is typically satisfied by optimizations which reduce the amount of data to be processed and/or which reduce the number of operations per datum in obtaining the results. There is also a desire to obtain the results in a robust fashion. Given any reasonable input to the algorithm, one would always like to obtain output, and the output should be accurate.
David Eberly
Backmatter
Metadata
Title
Ridges in Image and Data Analysis
Author
David Eberly
Copyright Year
1996
Publisher
Springer Netherlands
Electronic ISBN
978-94-015-8765-5
Print ISBN
978-90-481-4761-8
DOI
https://doi.org/10.1007/978-94-015-8765-5