UCLA Vision Lab

Integral Invariants for Shape Description and Matching


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

Signature representation of an
 integral invariant

Toward this end, we have developed a framework for integral group-invariant 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.

Hands with point correspondences
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.


S. Manay, D. Cremers, B. Hong, A. Yezzi, S. Soatto
Integral Invariants and Shape Matching
PAMI, Vol. 28, no. 10, Oct. 2006

S. Manay, D. Cremers, A. Yezzi, S. Soatto
One-shot Integral Invariant Shape Priors for Variational Segmentation
Proc. EMMCVPR 2005, pp. 414-426

S. Manay, B. Hong, A. Yezzi, S. Soatto
Integral Invariant Signatures
Proc. ECCV 2004

Page (c) 2006 by S. Manay.  Updated 14 Sept. 2006.