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Variational Methods in Image Segmentation

with seven image processing experiments

  • Jean Michel Morel
  • Sergio Solimini

Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 14)

Table of contents

  1. Front Matter
    Pages i-xvi
  2. Modelization

    1. Front Matter
      Pages 1-1
    2. Jean Michel Morel, Sergio Solimini
      Pages 3-7
    3. Jean Michel Morel, Sergio Solimini
      Pages 8-20
    4. Jean Michel Morel, Sergio Solimini
      Pages 21-34
    5. Jean Michel Morel, Sergio Solimini
      Pages 35-45
    6. Jean Michel Morel, Sergio Solimini
      Pages 46-62
  3. Elements of Geometric Measure Theory

    1. Front Matter
      Pages 63-63
    2. Jean Michel Morel, Sergio Solimini
      Pages 65-78
    3. Jean Michel Morel, Sergio Solimini
      Pages 79-85
    4. Jean Michel Morel, Sergio Solimini
      Pages 86-93
    5. Jean Michel Morel, Sergio Solimini
      Pages 94-117
    6. Jean Michel Morel, Sergio Solimini
      Pages 118-126
    7. Jean Michel Morel, Sergio Solimini
      Pages 127-135
    8. 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
    2. Jean Michel Morel, Sergio Solimini
      Pages 151-164
    3. Jean Michel Morel, Sergio Solimini
      Pages 165-181
    4. Jean Michel Morel, Sergio Solimini
      Pages 182-198
    5. Jean Michel Morel, Sergio Solimini
      Pages 199-208
  5. Back Matter
    Pages 209-248

About this book

Introduction

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").

Keywords

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

Authors and affiliations

  • Jean Michel Morel
    • 1
  • Sergio Solimini
    • 2
  1. 1.CEREMADEUniversité Paris-DauphineParis Cedex 16France
  2. 2.Srada provinciale Lecce-ArnesanoUniversità degli Studi di LecceLecceItaly

Bibliographic information