Affine-invariant, elastic shape analysis of planar contours

We present a Riemannian framework for analyzing shapes of planar contours in which metrics and other analyses are invariant to affine transformations and re-parameterizations of contours. Current methods that are affine invariant are restricted to point sets and do not handle full curves, while methods that analyze parameterized curves are restricted to equivalence under similarity transformation (rigid motion and scale). We construct a pre-shape manifold of standardized curves - curves whose centroid is at the origin, are of unit length, and their x and y coordinates are uncorrelated - and develop a path-straightening technique for computing geodesics on this nonlinear manifold under the elastic Riemannian metric. The removal of the rotation and the re-parameterization groups results in a quotient space, termed affine elastic shape space, and the resulting geodesic paths exhibit an improved matching of features across curves. These geodesics are used for shape comparison, retrieval, and statistical modeling of given curves. Experimental results using both simulated and real data, and an application involving pose-invariant activity recognition, demonstrate the success of this framework.