Variational Methods in Image Processing

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.

Introduction and Book Overview Introduction Overview Mathematical Background Tikhonov Regularization of Ill-Posed Inverse Problems Maximum a Posteriori (MAP) Estimate Convolution 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 ModelingNumerical 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 ContoursPiecewise-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 Applications 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 Implementation Numerical Experiments Variational Image Registration Models Introduction A Variational Image Registration Algorithm Using Nonlinear Elasticity Regularization Experimental Results A Piecewise-Constant Binary Model for Electrical Impedance Tomography Introduction 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 Introduction The S1 case The S2 case Numerical Experiments Concluding Remarks and Discussions for More General Manifolds Exercises appear at the end of each chapter.

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.