Variational methods in imaging.

*(English)*Zbl 1177.68245
Applied Mathematical Sciences 167. New York, NY: Springer (ISBN 978-0-387-30931-6/hbk). xiii, 320 p. (2009).

The book is mainly devoted to variational methods in imaging. It is divided into three parts.

The first one presents typical problems in imaging like: denoising, image inpainting and computerized tomography. Of particular interest is the description of the Schlieren tomography. An entire chapter is dedicated to the description of typical noise models and to the application of the maximum a posteriori estimation for image denoising.

The second part presents in a mathematically rigorous way different regularization methods. A chapter is dedicated to the general theory of regularization for the solution of inverse problems both in Hilbert and in Banach spaces. Proofs of existence, stability, and convergence are provided with reasonable hypotheses together with stability estimates and convergence. Then, the conclusions are reconsidered from the point of view of convex analysis to derive additional results in the context of convex regularization methods applied to denoising. The book dedicates a chapter to the problem of non-convex regularization; it presents results related to the functional \[ F(u) = \int_\Omega \frac{(u-u^\delta)^2}{2|\nabla u|^p}+\alpha |\nabla u|^p \] with \(u^\delta \in L^\infty(\Omega)\) and \(1\leq p < \infty\). The last two chapters of this part are then dedicated to semigroup theory, scale space and inverse scale space.

The third part is a presentation of well-known facts in functional analysis, convex analysis and the calculus of variations that are useful to understand the mathematical formalism used in the book.

The book is interesting in particular for its rigorous presentation of many proved mathematical results, and is therefore important for the image processing community.

The first one presents typical problems in imaging like: denoising, image inpainting and computerized tomography. Of particular interest is the description of the Schlieren tomography. An entire chapter is dedicated to the description of typical noise models and to the application of the maximum a posteriori estimation for image denoising.

The second part presents in a mathematically rigorous way different regularization methods. A chapter is dedicated to the general theory of regularization for the solution of inverse problems both in Hilbert and in Banach spaces. Proofs of existence, stability, and convergence are provided with reasonable hypotheses together with stability estimates and convergence. Then, the conclusions are reconsidered from the point of view of convex analysis to derive additional results in the context of convex regularization methods applied to denoising. The book dedicates a chapter to the problem of non-convex regularization; it presents results related to the functional \[ F(u) = \int_\Omega \frac{(u-u^\delta)^2}{2|\nabla u|^p}+\alpha |\nabla u|^p \] with \(u^\delta \in L^\infty(\Omega)\) and \(1\leq p < \infty\). The last two chapters of this part are then dedicated to semigroup theory, scale space and inverse scale space.

The third part is a presentation of well-known facts in functional analysis, convex analysis and the calculus of variations that are useful to understand the mathematical formalism used in the book.

The book is interesting in particular for its rigorous presentation of many proved mathematical results, and is therefore important for the image processing community.

Reviewer: Alessandro Duci (Bergamo)