Variational Methods in Imaging

By: Otmar Scherzer, Markus Grasmair, Harald Grossauer

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Hardcover | 9 October 2008

At a Glance

Hardcover
336 Pages

Dimensions(cm)
23.39 x 15.6 x 1.91

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This book is devoted to the study of variational methods in imaging. The presentation is mathematically rigorous and covers a detailed treatment of the approach from an inverse problems point of view.

Key Features:

- Introduces variational methods with motivation from the deterministic, geometric, and stochastic point of view

- Bridges the gap between regularization theory in image analysis and in inverse problems

- Presents case examples in imaging to illustrate the use of variational methods e.g. denoising, thermoacoustics, computerized tomography

- Discusses link between non-convex calculus of variations, morphological analysis, and level set methods

- Analyses variational methods containing classical analysis of variational methods, modern analysis such as G-norm properties, and non-convex calculus of variations

- Uses numerical examples to enhance the theory

This book is geared towards graduate students and researchers in applied mathematics. It can serve as a main text for graduate courses in image processing and inverse problems or as a supplemental text for courses on regularization. Researchers and computer scientists in the area of imaging science will also find this book useful.

Industry Reviews

From the reviews:

"Imaging is a wide area of applied mathematics which covers inverse problems, data filtering ... medical diagnosis, etc. ... The book is structured in a logical manner, starting with motivating examples and building on them. ... One of the strengths of this book is its real-life applications and analytical and numerical results presented at each step, keeping the content real ... . This is ... a book for the seasoned researchers or graduate students who look to deepen their understanding of the subject." (Bogdan G. Nita, Mathematical Reviews, Issue 2009 j)

"The book is mainly devoted to variational methods in imaging. It is divided into three parts. ... The book is interesting in particular for its rigorous presentation of many proved mathematical results, and is ... important for the image processing community." (Alessandro Duci, Zentralblatt MATH, Vol. 1177, 2010)

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Fundamentals of Imaging
Case Examples of Imaging	p. 3
Denoising	p. 3
Chopping and Nodding	p. 6
Image Inpainting	p. 8
X-ray-Based Computerized Tomography	p. 10
Thermoacoustic Computerized Tomography	p. 13
Schlieren Tomography	p. 24
Image and Noise Models	p. 27
Basic Concepts of Statistics	p. 27
Digitized (Discrete) Images	p. 31
Noise Models	p. 33
Priors for Images	p. 36
Maximum A Posteriori Estimation	p. 43
MAP Estimation for Noisy Images	p. 46
Regularization
Variational Regularization Methods for the Solution of Inverse Problems	p. 53
Quadratic Tikhonov Regularization in Hilbert Spaces	p. 54
Variational Regularization Methods in Banach Spaces	p. 60
Regularization with Sparsity Constraints	p. 79
Linear Inverse Problems with Convex Constraints	p. 89
Schlieren Tomography	p. 109
Further Literature on Regularization Methods for Inverse Problems	p. 112
Convex Regularization Methods for Denoising	p. 115
The *-Number	p. 120
Characterization of Minimizers	p. 125
One-dimensional Results	p. 131
Taut String Algorithm	p. 137
Mumford-Shah Regularization	p. 151
Recent Topics on Denoising with Variational Methods	p. 155
Variational Calculus for Non-convex Regularization	p. 159
Direct Methods	p. 160
Relaxation on Sobolev Spaces	p. 162
Relaxation on BV	p. 167
Applications in Non-convex Regularization	p. 172
One-dimensional Results	p. 178
Examples	p. 180
Semi-group Theory and Scale Spaces	p. 185
Linear Semi-group Theory	p. 186
Non-linear Semi-groups in Hilbert Spaces	p. 190
Non-linear Semi-groups in Banach Spaces	p. 193
Axiomatic Approach to Scale Spaces	p. 197
Evolution by Non-convex Energy Functionals	p. 200
Enhancing	p. 202
Inverse Scale Spaces	p. 205
Iterative Tikhonov Regularization	p. 206
Iterative Regularization with Bregman Distances	p. 209
Recent Topics on Evolutionary Equations for Inverse Problems	p. 217
Mathematical Foundations
Functional Analysis	p. 221
General Topology	p. 221
Locally Convex Spaces	p. 224
Bounded Linear Operators and Functionals	p. 227
Linear Operators in Hilbert Spaces	p. 231
Weak and Weak* Topologies	p. 234
Spaces of Differentiable Functions	p. 237
Weakly Differentiable Functions	p. 239
Measure and Integration Theory	p. 239
Distributions and Distributional Derivatives	p. 248
Geometrical Properties of Functions and Domains	p. 250
Sobolev Spaces	p. 254
Convolution	p. 261
Sobolev Spaces of Fractional Order	p. 262
Bochner Spaces	p. 263
Functions of Bounded Variation	p. 265
Convex Analysis and Calculus of Variations	p. 273
Convex and Lower Semi-continuous Functionals	p. 274
Fenchel Duality and Subdifferentiability	p. 276
Duality Mappings	p. 280
Differentiability of Functionals and Operators	p. 281
Derivatives of Integral Functionals on L[superscript p] ([Omega])	p. 284
References	p. 287
Nomenclature	p. 309
Index	p. 315
Table of Contents provided by Ingram. All Rights Reserved.

Variational Methods in Imaging

At a Glance

Hardcover

Industry Reviews

Shipping

How to return your order

Defective items

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