Intuitively,
we can recognize shapes
(silhouettes or
outlines) despite a vast number of interfering factors. To us, a human
hand still looks like a human hand even if it is moved and rotated,
near
or far, tilted or turned, blurry or noisy, or even if it is partially
obscured (occluded) or changes configuration (fingers spread or
together, some folded or extended).
We are
interested in a mathematical description of shape (represented as a
planar curve or a silhouette) that allows us to construct some notion
of
'distance' between shapes. Particularly, we want two shapes that are
truly different to be far apart (a hand should not be close to a fish,
no matter what kind of fish and how the hand is viewed/configured)
while
shapes that are similar up to the
deformations/perturbations/interfering
factors listed above should have small distance (two different breeds
of fish, even viewed
from a different angles, should be close to each other). Further, our
description should be robust to
measurement noise.




Shape

Signature
representation of an
integral invariant

Toward this
end, we have developed a framework for integral groupinvariant
descriptions of shape. These functions, computed with integral
operators on the shape (represented as a closed simple planar contour),
do not change despite the action of groups, such as translations and
rotations. Because they are based on integrals, they are far more
robust
to noise than similar descriptions computed with derivatives, such as
curvature.
(More information)
With these
integral invariants in hand, we can now use them to define a "shape
distance" between two shapes. Shape distance is defined as the
integral over the contour of the difference between integral invariant
values at corresponding points.
We pose the computation of shape distance
and optimal correspondence as the optimization of a
single functional.
Correspondences
between noisy, rotated, and reconfigured hands.
(Click on image for full size version)
This shape
distance can now be used for shape comparison or shape retrieval from a
database of shapes,
even if the shape is noisy, translated, rotated, articulated, or
occluded.
(More information)
We continue to
study the implications of this robust shape description, including its
role in multiscale analysis and alternate definitions of shape
distance.