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Progress in Nonlinear Differential Equations and Their Applications

Variational Methods in Image Segmentation

with seven image processing experiments

Authors: Morel, Jean-Michel, Solimini, Sergio

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About this book

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").

Reviews

"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


Table of contents (16 chapters)

Table of contents (16 chapters)

Buy this book

eBook $84.99
price for USA in USD (gross)
  • ISBN 978-1-4684-0567-5
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $109.99
price for USA in USD
  • ISBN 978-1-4684-0569-9
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
Variational Methods in Image Segmentation
Book Subtitle
with seven image processing experiments
Authors
Series Title
Progress in Nonlinear Differential Equations and Their Applications
Series Volume
14
Copyright
1995
Publisher
Birkhäuser Basel
Copyright Holder
Birkhäuser Boston
eBook ISBN
978-1-4684-0567-5
DOI
10.1007/978-1-4684-0567-5
Softcover ISBN
978-1-4684-0569-9
Series ISSN
1421-1750
Edition Number
1
Number of Pages
XVI, 248
Topics