Product Description
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. Thanks to the
formalization, mathematical questions about the soundness 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 segmentation energies, "edge" terms compete with
"region" terms in a way which is intended to impose regularity on both
regions and boundaries. The first part of the book presents a unified
presentation of the evidence in favour of the conjecture. It is proven
that the competition of one-dimensional and two-dimensional energy
terms in a variational formulation cannot create fractal-like behaviour
for the edges. The proof of regularity for the edges of a segmentation
constantly involves concepts from geometric measure theory, which
proves to be central in image 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").
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