We seek a description of a shape, represented as a closed, planar
contour, that allows us to analyze its intrinsic geometry. We don't
wish to be distracted by nuisances like the choice of coordinate frame
or scaling. Further, we'd like two additional properties in our
descriptions. 1) Similar shapes should have similar descriptions; a
hand should have a similar description whether the fingers are splayed
or not. 2) The description should only be different in regions where
the shape is different; given two hands, one with a finger folded or
occluded, the description should be different only around the occluded
finger. This property is called locality.
In the literature, many
shape descriptions and measures of differences between shapes embed
these properties. However, most are based on derivatives
taken along the curve, thus the name differential invariants .
While differential invariants have been used with great theoretical
success, in practice measurement or discretization noise renders these
Instead, we propose a
shape descriptor that relies on integrals on the curve (or the
domain the curve is embedded in). These integral invariants
are far less sensitive to noise than their differential cousins, while
(by construction) they share all the desirable properties discussed
above. Below we discuss two such integral invariants.
We are interested in
closed, simple, planar curves parametrized from 0 to 1. The
descriptions we seek are functions mapping each point on the curve to a
scalar value, vector, or function. For simplicity, our invariants are
scalar, but extend naturally to vector- or functional-valued
invariants. (Curvature, the canonical differential invariant, is a
One such integral
invariant is computed by centering an isotropic kernel with a parameter
r at a point on the contour p. This kernel could
be a 2D isotropic
Gaussian with variance r, or a ball of radius r. The
invariant value is the (normalized) area of the intersection of the
kernel and the interior of the curve. We call this invariant the "local
Local Area Integral Invariant
Compute the area inside the kernel B(p).
The following figure
demonstrates the robustness of this method to noise on a simple shape.
With these invariants in
hand, we can construct distance measures between shapes that are not
affected by rotation, translation, or scaling. (Shapes can be made
invariant to affine transforms by normalization.) Further, nuisances
like noise, occlusion, and changes in configuration will affect the
distance proportionally (unlike distances based on differential
invariants, where small noise perturbations can have a
disproportionately large effect on shape distance). These
distances are constructed on the invariant instead of on the shape.
Application of these
invariants to the problem of computing shape distance and optimal
correspondence is discussed here.
For more details, please see the publications listed here.