UCLA Vision Lab

Integral Invariants for Shape Description


S. Manay, B. Hong, A.Yezzi, S. Soatto

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 invariants problematic.

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 scalar function).

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 area invariant." 

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.

Noisy Shape


Local Area Invariant

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 nuisance-insensitive 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.