By Jean-Michel Morel, Sergio Solimini
This e-book comprises either a synthesis and mathematical research of a large set of algorithms and theories whose goal is the automated segmen tation of electronic pictures in addition to the knowledge of visible notion. a standard formalism for those theories and algorithms is got in a variational shape. Thank to this formalization, mathematical questions on the stability of algorithms will be raised and replied. belief conception has to accommodate the advanced interplay among areas and "edges" (or barriers) in a picture: within the variational seg mentation energies, "edge" phrases compete with "region" phrases in a manner that is presupposed to impose regularity on either areas and bounds. This truth used to be an experimental bet in conception phenomenology and desktop imaginative and prescient until eventually it used to be proposed as a mathematical conjecture through Mumford and Shah. The 3rd a part of the booklet provides a unified presentation of the evi dences in favour of the conjecture. it really is proved that the contest of one-dimensional and two-dimensional power phrases in a variational for mulation can't create fractal-like behaviour for the perimeters. The evidence of regularity for the perimeters of a segmentation always consists of con cepts from geometric degree thought, which proves to be vital in im age processing concept. the second one a part of the publication offers a quick and self-contained presentation of the classical concept of rectifiable units (the "edges") and unrectifiable units ("fractals").