AVVISO DI SEMINARIO
Matematica
Lunedì 19 settembre 2005 ore 15.00
Scuola Normale Superiore
Pisa
(Aula Mancini)
Anthony Yezzi
Georgia Tech - U.S.A.
Terrà un seminario dal titolo:
"Conformal Metrics and True "Gradient Flows" for Curves"
Abstract
The problem of finding shapes in images is a long standing and far-fetched one. This problem is related and fundamental to such issues as image segmentation, shape analysis, shape optimization, etc. Following the introduction of snakes by Kass, Witkin, and Terzopoulos, a method known as active contours has played a prominent role. These active contours are closed planar curves that, driven by the minimization of suitable energies, move to achieve desireable segmentations of the image (foreground/background partitioning of the image domain). Early reasearch on active contours saw the transition from parameterization dependent models to geometric models independent of the parameterization of the evolving curve. Next, there were many efforts to incorporate region based image information to make the active contour depend upon global information about the image rather than just the traditional locally computed edge descriptors. In recent years, the latest trend in active contour research seems to be that of incorporating global shape priors into the active contour paradigm. This has brought up non-trivial questions such as how to define an "average shape" or how to characterize "variations in shape". All of these questions ultimately lead to a more basic and fundamental question of how to define a Riemannian Geometry in the space of curves.
Surprisingly, almost two decades of literature on variational approaches to active contours suggests a consistent metric on the space of curves. The rather unanimous suggestion of this underlying metric structure is made implicitly, however, through the widespread reference to a variety of evolution models as "gradient flows". Despite a large number of very different energy functionals proposed in the literature, a consistent underlying metric is utilized in each case (knowingly or unknownlingly) to derive a "gradient flow" for each energy. Perhaps just as surpringly, this consistent metric structure has not been used in the shape analysis literature to measure distances between curves, compute average curves, or to quantify variations in shape. We begin this talk by pointing out that (even more surprisingly) attempts to convert this implied metric into a distance between curves leads to a pathological and useless notion that the distance between any two curves is zero making it pointless to even consider geodesics between curves (optimal homotopies). We then show how to adjust the same metric through a conformal factor to correct its pathological properties and thereby allow us to compute geodesics between curves. A particularly nice property of this new class of metrics is that, due to their conformal structure, all variational active contour models that have been called "gradient flows" in the past will constitute gradient flows with respect to these new metrics after appropriate reparameterization in time
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