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"). This book contains both a synthesis and mathematical analysis of a wide set of algorithms and theories whose aim is the automatic segmenƯ 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 segƯ 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 eviƯ dences in favour of the conjecture. It is proved that the competition of one-dimensional and two-dimensional energy terms in a variational forƯ mulation cannot create fractal-like behaviour for the edges. The proof of regularity for the edges of a segmentation constantly involves conƯ cepts from geometric measure theory, which proves to be central in imƯ 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") 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 1. Edge Detection And Segmentation -- 2. Linear And Nonlinear Multiscale Filtering -- 3. Region And Edge Growing Methods -- 4. Variational Theories Of Segmentation -- 5. The Piecewise Constant Mumford-shah Model: Mathematical Analysis -- 6. Hausdorff Measure -- 7. Covering Lemmas In A Metric Space -- 8. Density Properties -- 9. Tangency Properties Of Regular Subsets Of R[superscript N] -- 10. Semicontinuity Properties Of Hausdorff Measure -- 11. Rectifiable Sets -- 12. Properties Of Regular And Rectifiable Sets -- 13. Properties Of The Approximating Image In The Mumford-shah Model -- 14. Small Oscillation Coverings And The Excision Method -- 15. Density Properties And Existence Theory For The Mumford-shah Minimizers -- 16. Further Properties Of The Minimizers: Covering The Edge Set With A Single Curve -- Index Of Segmentation Algorithms. Jean-michel Morel, Sergio Solimini. Includes Bibliographical References (p. [215]-237) And Index. This text contains a synthesis and a mathematical analysis of a wide set of algorithms and theories whose aim is the automatic segmentation of digital images as well as the understanding of visual perception. A common formalism for these theories and algorithms is obtained in variational form.
