Variational Methods in Image Processing (Hardback) book cover

Variational Methods in Image Processing

By Luminita A. Vese, Carole Le Guyader

© 2015 – Chapman and Hall/CRC

386 pages | 136 B/W Illus.

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Variational Methods in Image Processing presents the principles, techniques, and applications of variational image processing. The text focuses on variational models, their corresponding Euler–Lagrange equations, and numerical implementations for image processing. It balances traditional computational models with more modern techniques that solve the latest challenges introduced by new image acquisition devices.

The book addresses the most important problems in image processing along with other related problems and applications. Each chapter presents the problem, discusses its mathematical formulation as a minimization problem, analyzes its mathematical well-posedness, derives the associated Euler–Lagrange equations, describes the numerical approximations and algorithms, explains several numerical results, and includes a list of exercises. MATLAB® codes are available online.

Filled with tables, illustrations, and algorithms, this self-contained textbook is primarily for advanced undergraduate and graduate students in applied mathematics, scientific computing, medical imaging, computer vision, computer science, and engineering. It also offers a detailed overview of the relevant variational models for engineers, professionals from academia, and those in the image processing industry.


"The book’s contents are very well prepared for graduate-level students or advanced undergraduates who work in the field of mathematical image processing and computer vision. The book is also an indispensable resource for engineers and professionals in the image processing industry looking to adopt innovative concepts. Compared to existing textbooks, this one offers a useful view as it covers the fundamentals and many specific applications together in one place, balancing the traditional computational models with the more modern techniques developed to answer new challenges introduced by the new image acquisition devices."

—Dr. Jalal Fadili, École Nationale Supérieure d'Ingénieurs de Caen

"… very educational … a useful source of reference and inspiration for advanced undergraduate and graduate students in applied mathematics and/or computer vision as well for academic researchers or engineers from the image processing industry."

—Gilles Aubert, Professor of Mathematics, University of Nice-Sophia Antipolis

"This book will be immensely useful both as a reference and textbook, as it presents the fundamentals of variational methods in image processing. It covers all aspects of variational methods in image processing, with essential applications. Homework problems are also given at the end of each chapter. This book could be used as a textbook for a graduate course on variational methods in image processing. It will also be a reference book to researchers in the field."

—Jean-François Aujol, Professor of Mathematics, University of Bordeaux

"This book is a must-have for students and researchers working in mathematical image analysis, in particular on segmentation problems. It covers in a pedagogical way the mathematical foundations, classical convex and non-convex segmentation methods, as well as more advanced subjects such as non-local regularizations. This book also features a lot of graphical illustrations and pseudo-codes of algorithms. It showcases several concrete applications to medical imaging, and the availability of the corresponding MATLAB code is a great feature."

—Gabriel Peyré, CNRS Senior Researcher, Université Paris-Dauphine

"Written by two world specialists of image segmentation, this book is the most complete account to date of the amazing applications of partial differential equations to image processing. Being provided with code and exercises, I found that it provides an excellent pedagogic introduction to the subject."

—Jean-Michel Morel, Professor, École Normale Supérieure de Cachan

Table of Contents

Introduction and Book Overview



Mathematical Background

Tikhonov Regularization of Ill-Posed Inverse Problems

Maximum a Posteriori (MAP) Estimate


Fourier Transform

Topologies on Banach Spaces

Sobolev and BV Spaces

Calculus of Variations

Geometric Curve Evolution

Variational Level Set Methods

Numerical Analysis

Image Restoration

Variational Image Restoration Models

Linear Degradation Model with Gaussian Noise and Total Variation Regularization

Numerical Results for Image Restoration

Compressive Sensing for Computerized Tomography Reconstruction

Nonlocal Variational Methods in Image Restoration

Introduction to Neighborhood Filters and NL Means

Variational Nonlocal Regularization for Image Restoration

Numerical Results for Image Restoration

Image Decomposition into Cartoon and Texture


Numerical Results for Image Decomposition into Cartoon and Texture

Image Segmentation and Boundary Detection

Mumford and Shah Functional for Image Segmentation

Description of the Mumford and Shah Model

Weak Formulation of the Mumford and Shah Functional: MSH1

Mumford and Shah TV Functional: MSTV

Phase-Field Approximations to the Mumford and Shah Problem

Ambrosio and Tortorelli Phase-Field Elliptic Approximations

Shah Approximation to the MSTV Functional

Applications to Image Restoration

Region-Based Variational Active Contours

Piecewise-Constant Mumford and Shah Segmentation Using Level Sets

Piecewise-Smooth Mumford and Shah Segmentation Using Level Sets

Applications to Variational Image Restoration with Segmentation-Based Regularization and Level Sets

Edge-Based Variational Snakes and Active Contours

Snake Model

Geodesic Active Contours

Alignment Term

Topology-Preserving Snakes Model


Nonlocal Mumford–Shah and Ambrosio–Tortorelli Variational Models

Characterization of Minimizers u

Gâteaux Derivative of Nonlocal M-S Regularizers

Image Restoration with NL/MS Regularizers

Numerical Discretizations

Experimental Results and Comparisons

A Combined Segmentation and Registration Variational Model

Description of the Model


Numerical Experiments

Variational Image Registration Models


A Variational Image Registration Algorithm Using Nonlinear Elasticity Regularization

Experimental Results

A Piecewise-Constant Binary Model for Electrical Impedance Tomography


Formulation of the Minimization

Numerical Details and Reconstruction Results

Additive and Multiplicative Piecewise-Smooth Segmentation Models

Piecewise-Smooth Model with Additive Noise (APS)

Piecewise-Smooth Model with Multiplicative Noise (MPS)

Numerical Methods for p−Harmonic Flows


The S1 case

The S2 case

Numerical Experiments

Concluding Remarks and Discussions for More General Manifolds

Exercises appear at the end of each chapter.

About the Authors

Luminita A. Vese is a professor in the Department of Mathematics at UCLA. She is the author or co-author of numerous papers and book chapters on the calculus of variations, PDEs, numerical analysis, image analysis, curve evolution, computer vision, and free boundary problems.

Carole Le Guyader is an associate professor in the mathematical and software engineering department at the National Institute of Applied Sciences of Rouen. She has authored or co-authored many papers on analysis and simulation, digital imaging mathematics and applications, and parallel computing.

About the Series

Chapman & Hall/CRC Mathematical and Computational Imaging Sciences Series

Learn more…

Subject Categories

BISAC Subject Codes/Headings:
COMPUTERS / Computer Graphics
MATHEMATICS / Graphic Methods