@book{a7267788c3d14bedaa6952590da5d1f0,
title = "Variational Methods in Image Segmentation: with seven image processing experiments",
abstract = "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{"}).",
author = "Jean-Michel MOREL and Sergio SOLIMINI",
year = "1995",
doi = "10.1007/978-1-4684-0567-5",
language = "English",
isbn = "9781468405699",
series = "Progress in Nonlinear Differential Equations and Their Applications",
publisher = "Birkhauser",
address = "Switzerland",
}