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2018 | Buch

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Nonlinear Eigenproblems in Image Processing and Computer Vision

verfasst von: Prof. Guy Gilboa

Verlag: Springer International Publishing

Buchreihe : Advances in Computer Vision and Pattern Recognition

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This unique text/reference presents a fresh look at nonlinear processing through nonlinear eigenvalue analysis, highlighting how one-homogeneous convex functionals can induce nonlinear operators that can be analyzed within an eigenvalue framework. The text opens with an introduction to the mathematical background, together with a summary of classical variational algorithms for vision. This is followed by a focus on the foundations and applications of the new multi-scale representation based on non-linear eigenproblems. The book then concludes with a discussion of new numerical techniques for finding nonlinear eigenfunctions, and promising research directions beyond the convex case.

Topics and features: introduces the classical Fourier transform and its associated operator and energy, and asks how these concepts can be generalized in the nonlinear case; reviews the basic mathematical notion, briefly outlining the use of variational and flow-based methods to solve image-processing and computer vision algorithms; describes the properties of the total variation (TV) functional, and how the concept of nonlinear eigenfunctions relate to convex functionals; provides a spectral framework for one-homogeneous functionals, and applies this framework for denoising, texture processing and image fusion; proposes novel ways to solve the nonlinear eigenvalue problem using special flows that converge to eigenfunctions; examines graph-based and nonlocal methods, for which a TV eigenvalue analysis gives rise to strong segmentation, clustering and classification algorithms; presents an approach to generalizing the nonlinear spectral concept beyond the convex case, based on pixel decay analysis; discusses relations to other branches of image processing, such as wavelets and dictionary based methods.

This original work offers fascinating new insights into established signal processing techniques, integrating deep mathematical concepts from a range of different fields, which will be of great interest to all researchers involved with image processing and computer vision applications, as well as computations for more general scientific problems.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Mathematical Preliminaries

Abstract

The book tries to address a relatively broad audience, from a variety of disciplines. Therefore, we make minimal assumptions on previous mathematical knowledge and attempt to have a self-contained book. In addition, to increase clarity and readability we sometimes avoid getting into some less crucial mathematical details. In these cases, we refer the reader to appropriate references with the complete formal definitions and settings.

Guy Gilboa

Chapter 2. Variational Methods in Image Processing

Abstract

A short review is given on the rationale for using cost functions and optimization methods for modeling image processing and computer vision problems. Classical examples of various costs and functionals are given, illustrating this highly effective algorithmic approach. We examine different mathematical models Sects. (2.1, 2.2 and 2.5), as well as image processing tasks Sects. (2.3, 2.4).

Guy Gilboa

Chapter 3. Total Variation and Its Properties

Abstract

In this chapter, we present in some more details the total variation functional, which is the most popular and well-studied one-homogeneous regularizer. Total variation (TV) is a fundamental regularizer which has been used extensively in image processing and computer vision. It is very simple to formalize and is parameter free, which makes it convenient for various applications. Deep theoretical research has been conducted investigating the properties of TV in the continuous and in the discrete settings. Today TV is understood quite well. TV is essentially the \(L^1\) norm of the gradient of a function. However, there are several flavors and generalizations of TV to be anisotropic, adaptive, nonlocal, and graph-based. In this chapter, we will give a general overview of the motivation for using TV and its main properties. Reference to additional theory, extensions and applications will be supplied for the interested reader.

Guy Gilboa

Chapter 4. Eigenfunctions of One-Homogeneous Functionals

Abstract

The motivation and interpretation of classical linear filtering strategies are closely linked to eigendecomposition of positive semidefinite linear operators (derivatives of quadratic functionals). In the following chapters, we show that one can define a nonlinear spectral decomposition framework based on eigenfunctions of convex one-homogeneous functionals and obtain a remarkable number of analogies to linear filtering techniques. In this chapter, we give an introduction of previous studies on the topic and give preliminary settings and properties of one-homogeneous functionals. We then explain in more detail the derivation of eigenfunctions of total variation.

Guy Gilboa

Chapter 5. Spectral One-Homogeneous Framework

Abstract

This chapter introduces a main topic of this book, of viewing variational methods through a nonlinear spectral perspective. It is shown how all regularization methods—gradient flow, variational methods, and inverse scale space can be used to decompose the image in a new way that is similar in some sense to linear spectral or Fourier decompositions. We use the gradient descent as the canonical scale space and show how a nonlinear transform can be defined, based on its solution. This transform takes any nonlinear eigenfunction to appear in a singular time (scale) which is inverse proportional to its eigenvalue. Moreover, effective, contrast-preserving filtering can be applied by simple amplification, preservation, or attenuation of the different spectral components. We begin with a more informal presentation of the topic, where in the later part of this chapter more rigorous results are shown in the finite-dimensional case. A fundamental result is that in some settings we can show a precise decomposition of the input signal into eigenfunctions. In addition, the spectral components turn to be orthogonal to each other.

Guy Gilboa

Chapter 6. Applications Using Nonlinear Spectral Processing

Abstract

In this chapter, we show some image processing applications that use the spectral framework. These are related to denoising, texture processing, enhancement, and image fusion. This area is currently investigated and developed. A main theme is that following the image decomposition one can use very basic operations of attenuating, enhancing, and mixing certain spectral bands. Thus, a single framework with a solid theory can have very diverse applications, similar to classical linear transforms. The aim here is to give several interesting examples and to show the potential of using the nonlinear spectral formulations.

Guy Gilboa

Chapter 7. Numerical Methods for Finding Eigenfunctions

Abstract

The solution of nonlinear eigenvalue problems is an active research topic. It appears also in far-related fields of optics, dynamical flows, and more. In the variational context, the research is quite preliminary. We outline the method of Hein and Buhler, based on the Rayleigh quotient. We present in more detail a recent work by Raz Nossek and the author where a flow is used to solve the problem. This can be generalized in various ways. A generalization of Aujol et al., which is well supported theoretically, is outlined at the end of this chapter.

Guy Gilboa

Chapter 8. Graph and Nonlocal Framework

Abstract

Graph-based methods have been extensively studied and are used for many applications...

Guy Gilboa

Chapter 9. Beyond Convex Analysis—Decompositions with Nonlinear Flows

Abstract

In this chapter, we generalize the notion of nonlinear spectral decomposition beyond the convex case. It is shown how general denoisers can be viewed as nonlinear operators, and the coarsening scale space they induce can be analyzed in a spectral manner. We provide some basic assumptions that help us solve the new spectral decomposition problem. Essentially, a common decay profile is sought in order to decompose the image. It is shown that this generalizes both Fourier transform and the TV (or 1-homogeneous) transform. This work is still in research progress. The ideas presented here are a result of a collaboration of Oren Katzir, Tomer Michaeli, and the author, where the essential ideas are presented in Oren’s thesis (On the scale space of filters and their applications, 2017, [1]). The major implication of these initial findings is that nonlinear eigenfunctions are fundamental in the understanding and analysis of a broad class of nonlinear systems.

Guy Gilboa

Chapter 10. Relations to Other Decomposition Methods

Abstract

We discuss here the spectral nonlinear framework through different perspectives, related to well-known signal processing disciplines. The relations to wavelets are given, showing one can recover wavelet processing within this framework. In the specific case of Haar wavelet, which is actually a small subset of the eigenfunction of TV, it is shown how the spectral TV can adapt better to the signal. A numerical example shows that fewer elements are needed to encode the signal. We further discuss the relation to generalized Rayleigh quotients and to sparse representations, where nonlinear eigenfunctions can be viewed as an overcomplete dictionary.

Guy Gilboa

Chapter 11. Future Directions

Abstract

This chapter reviews current research in progress in the field and suggests future interesting directions. We present some emerging research efforts in the field. We show some promising preliminary results with respect to both applications and theory.

Guy Gilboa

Backmatter

Titel: Nonlinear Eigenproblems in Image Processing and Computer Vision
verfasst von: Prof. Guy Gilboa
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-75847-3
Print ISBN: 978-3-319-75846-6
DOI: https://doi.org/10.1007/978-3-319-75847-3