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
Introduction and Book Overview
Tikhonov Regularization of Ill-Posed Inverse Problems
Maximum a Posteriori (MAP) Estimate
Topologies on Banach Spaces
Sobolev and BV Spaces
Calculus of Variations
Geometric Curve Evolution
Variational Level Set Methods
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
Geodesic Active Contours
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
Experimental Results and Comparisons
A Combined Segmentation and Registration Variational Model
Description of the Model
Variational Image Registration Models
A Variational Image Registration Algorithm Using Nonlinear Elasticity Regularization
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
Concluding Remarks and Discussions for More General Manifolds
Exercises appear at the end of each chapter.