Variational Methods in Image Segmentation

1. Front Matter

   Pages i-xvi

   PDF

2. Modelization

   1. Front Matter

      Pages 1-1

      PDF

   2. Edge Detection and Segmentation

      • Jean Michel Morel, Sergio Solimini

      Pages 3-7

   3. Linear and Nonlinear Multiscale Filtering

      • Jean Michel Morel, Sergio Solimini

      Pages 8-20

   4. Region and Edge Growing

      • Jean Michel Morel, Sergio Solimini

      Pages 21-34

   5. Variational Theories of Segmentation

      • Jean Michel Morel, Sergio Solimini

      Pages 35-45

   6. The Piecewise Constant Mumford and Shah Model: Mathematical Analysis

      • Jean Michel Morel, Sergio Solimini

      Pages 46-62

3. Elements of Geometric Measure Theory

   1. Front Matter

      Pages 63-63

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   2. Hausdorff Measure

      • Jean Michel Morel, Sergio Solimini

      Pages 65-78

   3. Covering Lemmas in a Metric Space

      • Jean Michel Morel, Sergio Solimini

      Pages 79-85

   4. Density Properties

      • Jean Michel Morel, Sergio Solimini

      Pages 86-93

   5. Tangency Properties of Regular Subsets of ℝ^N

      • Jean Michel Morel, Sergio Solimini

      Pages 94-117

   6. Semicontinuity Properties of the Hausdorff Measure

      • Jean Michel Morel, Sergio Solimini

      Pages 118-126

   7. Rectifiable Sets

      • Jean Michel Morel, Sergio Solimini

      Pages 127-135

   8. Properties of Regular and Rectifiable Sets

      • Jean Michel Morel, Sergio Solimini

      Pages 136-147

4. Existence and Structural Properties of the Minimal Segmentations for the Mumford-Shah Model

   1. Front Matter

      Pages 149-149

      PDF

   2. Properties of the Approximating Image in the Mumford-Shah Model

      • Jean Michel Morel, Sergio Solimini

      Pages 151-164

   3. Small Oscillation Coverings and the Excision Method

      • Jean Michel Morel, Sergio Solimini

      Pages 165-181

   4. Density Properties and Existence Theory for the Mumford-Shah Minimizers

      • Jean Michel Morel, Sergio Solimini

      Pages 182-198

   5. Further Properties of Minimizers: Covering the Edge Set with a Single Curve

      • Jean Michel Morel, Sergio Solimini

      Pages 199-208

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").

• Mathematica
• algorithms
• behavior
• computer vision
• constant
• form
• fractal
• image segmentation
• interaction
• interpolation
• mathematical analysis
• measure theory
• presentation
• proof
• wavelet

"Well-written and very user-friendly. Each chapter is introduced by a brief summary of its aims... A very readable introduction to the mathematical theory of image segmentation."

--ZAA

• CEREMADE, Université Paris-Dauphine, Paris Cedex 16, France

  Jean Michel Morel

• Srada provinciale Lecce-Arnesano, Università degli Studi di Lecce, Lecce, Italy

  Sergio Solimini

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