Current techniques for shape analysis tend to seek invariance to similarity transformations (rotation, translation, and scale), but certain imaging situations require invariance to larger groups, such as affine or projective groups. Here we present a general Riemannian framework for shape analysis of planar objects where metrics and related quantities are invariant to affine and projective groups. Highlighting two possibilities for representing object boundaries-ordered points (or landmarks) and parameterized curves-we study different combinations of these representations (points and curves) and transformations (affine and projective). Specifically, we provide solutions to three out of four situations and develop algorithms for computing geodesics and intrinsic sample statistics, leading up to Gaussian-type statistical models, and classifying test shapes using such models learned from training data. In the case of parameterized curves, we also achieve the desired goal of invariance to re-parameterizations. The geodesics are constructed by particularizing the path-straightening algorithm to geometries of current manifolds and are used, in turn, to compute shape statistics and Gaussian-type shape models. We demonstrate these ideas using a number of examples from shape and activity recognition.

Statistical shape analysis develops methods for comparisons, deformations, summarizations, and modeling of shapes in given data sets. These tasks require a fundamental tool called parallel transport of tangent vectors along arbitrary paths. This tool is essential for: (1) computation of geodesic paths using either shooting or path-straightening method, (2) transferring deformations across objects, and (3) modeling of statistical variability in shapes. Using the square-root normal field (SRNF) representation of parameterized surfaces, we present a method for transporting deformations along paths in the shape space. This is difficult despite the underlying space being a vector space because the chosen (elastic) Riemannian metric is non-standard. Using a finite-basis for representing SRNFs of shapes, we derive expressions for Christoffel symbols that enable parallel transports. We demonstrate this framework using examples from shape analysis of parameterized spherical surfaces, in the three contexts mentioned above.