Mathematical problems in image processing. Partial differential equations and the calculus of variations. Foreword by Olivier Faugeras.
2nd ed.

*(English)*Zbl 1110.35001
Applied Mathematical Sciences 147. New York, NY: Springer (ISBN 0-387-32200-0/hbk; 0-387-21766-5/ebook). xxiii, 377 p. (2006).

The book is devoted to the mathematical treatment of several problems in image processing, with particular attention to the methods involving partial differential equations and variational analysis, which were introduced in the eighties and received a lot of attention since then. This is the second edition of the volume; the first one appeared in 2002 (see Zbl 1109.35002), and its main goal was to present the various mathematical models in image processing, giving them a solid and rigorous mathematical background. In this way, the volume turned out to be very useful to the mathematical community, for providing a variety of challenging problems, where new tools were necessary to include the image processing models into a rigorous and well established variational framework. At the same time the computer vision community also took advantage of the volume, for providing precise and rigorous presentations to several tools that otherwise would have remained simply at the level of numerical experiments.

This second edition has the same goals, and the spirit in which the volume is written will again let it be a reference for both the mathematical and the computer vision communities. Besides the natural updating of the existing chapters, this second edition presents some new material: Sections 5.1, 5.2, 5.3.4, 5.5 are devoted to new applications and in addition, Sections 3.2.4, 3.3.4, 4.3.4 contain some new mathematical topics. Finally a web site (http://www-sop.inria.fr/books/imath) is associated with this second edition; it includes some complementary material, several related links, and also some source codes to be used as easy tests in some variational and PDE models in image processing.

This second edition has the same goals, and the spirit in which the volume is written will again let it be a reference for both the mathematical and the computer vision communities. Besides the natural updating of the existing chapters, this second edition presents some new material: Sections 5.1, 5.2, 5.3.4, 5.5 are devoted to new applications and in addition, Sections 3.2.4, 3.3.4, 4.3.4 contain some new mathematical topics. Finally a web site (http://www-sop.inria.fr/books/imath) is associated with this second edition; it includes some complementary material, several related links, and also some source codes to be used as easy tests in some variational and PDE models in image processing.

Reviewer: Giuseppe Buttazzo (Pisa)

##### MSC:

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

49-02 | Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control |

94A08 | Image processing (compression, reconstruction, etc.) in information and communication theory |

68U10 | Computing methodologies for image processing |

35J20 | Variational methods for second-order elliptic equations |

49J45 | Methods involving semicontinuity and convergence; relaxation |

49N99 | Miscellaneous topics in calculus of variations and optimal control |