In the second half of the 80's E. De Giorgi focussed and studied a new class of variational problems, named free discontinuity problems. Here, among these new problems, we deal with two models that are bidimensional, continuous and deterministic. The first, proposed by Mumford & Shah, represents a continuous extrapolation of a previous discrete formulation suggested independently by Geman & Geman and by Blake & Zisserman. The second model we are interested in was also proposed by Blake & Zisserman in order to overcome some inaccuracies of the previous one. We discuss the motivation and the analysis of the Blake& Zisserman’s thin-plate model, which improves the Mumford & Shah model in terms of better singularity characterization; in fact the Blake & Zisserman functional avoids the inconvenient of over-segmentation and distinguishes between abrupt jumps of the grey level and jumps of its gradient. We illustrate the feasibility of these two models for the image segmentation: some numerical experiments on digitalized synthetic images are presented and obtained with discretization of the Euler systems (associated to the elliptic approximating functionals) and the use of the Gauss-Siedel and Picard iterative schemes.
Image segmentation in the framework of free discontinuity problems
CARRIERO, Michele;SGURA, Ivonne
2004-01-01
Abstract
In the second half of the 80's E. De Giorgi focussed and studied a new class of variational problems, named free discontinuity problems. Here, among these new problems, we deal with two models that are bidimensional, continuous and deterministic. The first, proposed by Mumford & Shah, represents a continuous extrapolation of a previous discrete formulation suggested independently by Geman & Geman and by Blake & Zisserman. The second model we are interested in was also proposed by Blake & Zisserman in order to overcome some inaccuracies of the previous one. We discuss the motivation and the analysis of the Blake& Zisserman’s thin-plate model, which improves the Mumford & Shah model in terms of better singularity characterization; in fact the Blake & Zisserman functional avoids the inconvenient of over-segmentation and distinguishes between abrupt jumps of the grey level and jumps of its gradient. We illustrate the feasibility of these two models for the image segmentation: some numerical experiments on digitalized synthetic images are presented and obtained with discretization of the Euler systems (associated to the elliptic approximating functionals) and the use of the Gauss-Siedel and Picard iterative schemes.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
