# Sobolev Active Contours

that

H

^{0}metrics on the space of curves lead to vanishing distances

between

curves, Yezzi and Mennucci proposed conformal variants of

H

^{0}

using

conformal factors dependent upon the total length of a given

curve.

The resulting metric was shown to yield non-vanishing distance

at

least when the conformal factor was greater than or equal to

the

curve length. The motivation for the conformal structure, was

to

preserve the directionality of the gradient of any functional

defined over the space of curves when compared to its

H

^{0}

gradient.

This desire came in part due to the fact that the H

^{0}metric

was

the consistent choice of metric in all variational active

contour

methods proposed since the early 90's. Even the well studied

geometric

heat flow is often referred to as the curve shrinking flow as

it

arises as the gradient descent of arclength with respect to the

H

^{0}

metric.

Changing strategies, we have decided to consider adapting

contour

optimization methods to a choice of metric on the space of

curves

rather than trying to constrain our metric choice in order to

conform to previous optimization methods. As such, we

reformulate

the gradient descent approach used for variational active

contours

by utilizing gradients with respect to H^{1} metrics rather than

H^{0}

metrics. We refer to this class of active contours as Sobolev

Active Contours and discuss their strengths when compared to

more

classical active contours based on the same underlying energy

functionals. Not only due Sobolev active contours exhibit more

regularity, regardless of the choice of energy to minimize, but

they

are ideally suited for applications in computer vision such as

tracking, where it is common that a contour to be tracked

changes

primarily by simple translation from frame to frame (a motion

which

is almost free for many Sobolev metrics).

(Joint work with G. Sundaramoorthi and A. Mennucci.)