This book is concerned with digital image processing techniques that use partial differential equations (PDEs) for the task of image 'inpainting', an artistic term for virtual image restoration or interpolation, whereby missing or occluded parts in images are completed based on information provided by intact parts. Computer graphic designers, artists and photographers have long used manual inpainting to restore damaged paintings or manipulate photographs. Today, mathematicians apply powerful methods based on PDEs to automate this task. This book introduces the mathematical concept of PDEs for virtual image restoration. It gives the full picture, from the first modelling steps originating in Gestalt theory and arts restoration to the analysis of resulting PDE models, numerical realisation and real-world application. This broad approach also gives insight into functional analysis, variational calculus, optimisation and numerical analysis and will appeal to researchers and graduate students in mathematics with an interest in image processing and mathematical analysis.
Author(s): Carola-Bibiane Schönlieb
Series: Cambridge Monographs on Applied and Computational Mathematics 29
Publisher: Cambridge University Press
Year: 2015
Language: English
Pages: 265
Contents......Page 7
Preface......Page 10
1.1 Digital Image Restoration in Modern Society......Page 12
1.2 What is a Digital Image?......Page 14
1.3 Image Inpainting......Page 16
2 Overview of Mathematical Inpainting Methods......Page 19
2.1 Variational and PDE Methods......Page 21
2.2 Structure Versus Texture Inpainting......Page 32
2.3 Inpainting of Colour Images......Page 35
2.4 Video Inpainting......Page 36
3 The Principle of Good Continuation......Page 37
3.1 Gestalt Theory......Page 38
3.2 Kanizsa’s Amodal Completion......Page 40
4.1 An Axiomatic Approach to Image Inpainting......Page 43
4.2 Harmonic Image Inpainting......Page 52
4.3 Total Variation Inpainting......Page 56
4.4 Absolutely Minimising Lipschitz Extensions......Page 66
4.5 Further Reading and Some Extensions......Page 70
5.1 Second- Versus Higher-Order Approaches......Page 74
5.2 Curvature-Based Inpainting......Page 77
5.3 Cahn-Hilliard and TV-H−1 Inpainting......Page 96
5.4 Low Curvature Image Simplifiers......Page 130
5.5 Second-Order Total Variation Inpainting......Page 132
5.6 Further Reading and Some Extensions......Page 144
6.1 Inpainting by Transport Along Level Lines......Page 148
6.2 Inpainting with Coherence Transport......Page 155
6.3 GuideFill: Fast Artist-Guided Transport Inpainting......Page 161
7.1 Inpainting with Mumford-Shah......Page 172
7.2 Mumford-Shah-Euler Inpainting......Page 181
8 Inpainting Mechanisms of Transport and Diffusion......Page 185
9.1 Restoration of Medieval Frescoes......Page 191
9.2 Road Reconstruction......Page 200
9.3 Sinogram Inpainting for Limited Angle Tomography......Page 202
9.4 Inpainting for 3D Conversion......Page 215
Appendix A Exercises......Page 222
Appendix B Mathematical Preliminaries......Page 228
Appendix C MATLAB Implementation......Page 240
Appendix D Image Credits......Page 242
Glossaries......Page 244
References......Page 248
Index......Page 264
