Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations
Partial differential equations and variational methods were introduced into image processing about 15 years ago, and intensive research has been carried out since then. The main goal of this work is to present the variety of image analysis applications and the precise mathematics involved. It is intended for two audiences. The first is the mathematical community, to show the contribution of mathematics to this domain and to highlight some unresolved theoretical questions. The second is the computer vision community, to present a clear, self-contained, and global overview of the mathematics involved in image processing problems. This book will be useful to researchers and graduate students in mathematics and computer vision.
Why Read This Book
You should read this book if you want a rigorous, unified presentation of PDE- and variational-based approaches to classical image problems (denoising, restoration, segmentation, motion). It gives the mathematical foundations and links them to practical formulations so you can move from modeling to provable properties and discretization strategies.
Who Will Benefit
Graduate students, researchers, and advanced engineers in image/signal processing or applied mathematics who need a solid theoretical foundation for PDE- and variational-based image algorithms.
Level: Advanced — Prerequisites: Undergraduate calculus and linear algebra, basic PDEs and functional analysis, familiarity with variational calculus and elementary image-processing concepts; some numerical analysis is helpful.
Key Takeaways
- Formulate image-processing tasks (denoising, restoration, segmentation, motion) as variational problems and associated PDEs.
- Derive Euler–Lagrange equations and understand their connection to diffusion and scale-space processes.
- Apply and analyze nonlinear diffusion and total variation (TV) regularization methods for denoising and edge preservation.
- Analyze existence, uniqueness, and regularity results for variational models used in imaging.
- Implement and evaluate basic numerical discretizations (finite differences, time-stepping) for PDE-based image algorithms.
- Relate variational/PDE methods to classical and modern approaches (Mumford–Shah, optical flow, inverse problems).
Topics Covered
- Introduction: image problems and motivations
- Mathematical preliminaries: function spaces and distributions
- Calculus of variations and Euler–Lagrange framework
- Linear diffusion, scale-space and Gaussian smoothing
- Nonlinear diffusion and anisotropic filtering
- Total variation methods and TV-based denoising
- Variational segmentation and the Mumford–Shah functional
- Optical flow and variational motion estimation
- Inverse problems and regularization techniques
- Numerical methods: discretization, stability, and implementation
- Existence, uniqueness and regularity theory
- Extensions, applications and open mathematical questions
- Appendices: background in functional analysis and PDE tools
How It Compares
More mathematically rigorous than Chan & Shen's Image Processing and Analysis (which is more algorithmic and user-oriented) and complementary to Osher & Sethian's Level Set/Geometric methods; Aubert focuses on variational/PDE theory and proofs rather than extensive code or heuristic algorithms.
