Jean Michel Morel, Sergio Solimini (auth.)9780817637200, 9783764337209, 0817637206, 3764337206
This book contains both a synthesis and mathematical analysis of a wide set of algorithms and theories whose aim is the automatic segmen tation of digital images as well as the understanding of visual perception. A common formalism for these theories and algorithms is obtained in a variational form. Thank to this formalization, mathematical questions about the soundness of algorithms can be raised and answered. Perception theory has to deal with the complex interaction between regions and “edges” (or boundaries) in an image: in the variational seg mentation energies, “edge” terms compete with “region” terms in a way which is supposed to impose regularity on both regions and boundaries. This fact was an experimental guess in perception phenomenology and computer vision until it was proposed as a mathematical conjecture by Mumford and Shah. The third part of the book presents a unified presentation of the evi dences in favour of the conjecture. It is proved that the competition of one-dimensional and two-dimensional energy terms in a variational for mulation cannot create fractal-like behaviour for the edges. The proof of regularity for the edges of a segmentation constantly involves con cepts from geometric measure theory, which proves to be central in im age processing theory. The second part of the book provides a fast and self-contained presentation of the classical theory of rectifiable sets (the “edges”) and unrectifiable sets (“fractals”). |
Table of contents :
Front Matter….Pages i-xvi
Front Matter….Pages 1-1
Edge Detection and Segmentation….Pages 3-7
Linear and Nonlinear Multiscale Filtering….Pages 8-20
Region and Edge Growing….Pages 21-34
Variational Theories of Segmentation….Pages 35-45
The Piecewise Constant Mumford and Shah Model: Mathematical Analysis….Pages 46-62
Front Matter….Pages 63-63
Hausdorff Measure….Pages 65-78
Covering Lemmas in a Metric Space….Pages 79-85
Density Properties….Pages 86-93
Tangency Properties of Regular Subsets of ℝ N ….Pages 94-117
Semicontinuity Properties of the Hausdorff Measure….Pages 118-126
Rectifiable Sets….Pages 127-135
Properties of Regular and Rectifiable Sets….Pages 136-147
Front Matter….Pages 149-149
Properties of the Approximating Image in the Mumford-Shah Model….Pages 151-164
Small Oscillation Coverings and the Excision Method….Pages 165-181
Density Properties and Existence Theory for the Mumford-Shah Minimizers….Pages 182-198
Further Properties of Minimizers: Covering the Edge Set with a Single Curve….Pages 199-208
Back Matter….Pages 209-248 |
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