Analysis of projective shapes of curves using projective frames

In classical projective shape analysis one works with a finite number of landmarks. In many practical problems, however, planar curves are really of interest. Examples include identification of shapes of large areas of land, shape changes in contours of seas, forests, deserts, etc, as well as in the detection of silhouettes in human surveillance, medical imaging, etc. Our example is a “toy model” of analysis involving “footprint” like images. To compare the projective shape of two configurations of curves plus a projective frame, one uses a statistical testing method for functional data. One approach to Projective Shape Analysis (Patrangenaru, 1999), is based on the idea of a projective frame selected from the points of a finite generic k-ad in m dimensions. The resulting space of projective shapes of k-ads of is a product of k-m-2 copies of axial spaces (Mardia and Patrangenaru, 2005) . Such a representation has the advantage that it associates to the full set of projective invariants (Goodall and Mardia, 1999) of such a configuration one point on a projective shape manifold. The projective frame approach can be used to identify the projective shape of a planar curve in the context of a scene that contains four points in general position, that are not necessarily on the curve. The projective frame determined by such control points is used for registration. Ideally two registered images of the same scene should be identical; nevertheless due to registration errors and departure from a planar scene, they are different, and this raises questions about identification of the mean projective shape of a curve, testing for equality of such mean projective shapes of curves, etc. There are a few problems that arise in such a testing problem from curves in digital images. One is of image processing and of automatic detection of the actual curve, even from less noisy images. Secondly one has to address the curve registration problem. Thirdly, the classical simple null hypothesis of equality of two mean curves, even when they arise form the same scene, is very likely to be rejected because of inherent registration errors; therefore a neighborhood null hypothesis for functional data is preferred (Dette and Munk, 1998). Finally once the test is established one has to dwell with intensive computational algorithms involved in functional data analysis. In this preliminary report, we discuss the first three steps in such a testing problem, for the “Bigfoot” data set, a toy example of pattern recognition of contours from aerial images.