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Sobolev Active Contours

Monday, April 3, 2006 - 2:00pm - 3:00pm
EE/CS 3-180
Anthony Yezzi (Georgia Institute of Technology)
Following the observation first noted by Michor and Mumford,
that
H0 metrics on the space of curves lead to vanishing distances
between
curves, Yezzi and Mennucci proposed conformal variants of
H0
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
H0
gradient.
This desire came in part due to the fact that the H0 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
H0
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 H1 metrics rather than
H0
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.)

MSC Code: 
46E39
Keywords: