Computations on the grassmann manifold and affine-permutation invariance in shape representation

We present a novel approach to computations on the Grassmann manifold and define a locally Riemmanian affine-permutation invariant shape space for the task of shape representation. The definition of a measure of similarity between shapes cannot be based on the geometry of Euclidean space. One of the essential concepts here is defining a shape to be a set of all equivalent representations obtained by acting on a rigid object with a class of distortions and then mapping each shape as a point of a geometric manifold. The underlying metric of the non-Euclidean space intrinsically defines an unlabeled-view-independent similarity measure between the shapes. We consider affine distortions and distortions induced by permutation of the unlabeled feature points. We solve this challenging problem of finding a similarity measure and of finding a locally geometric manifold with an unlabeled-view-independent metric properly defined on it. We construct geometric shape spaces for 2D and 3D rigid objects and define on these spaces distances among shapes that are invariant to a class of distortions---affine transformations and permutations of unlabeled feature points. We define equivalency sets of configurations of p -dim objects. Each equivalency class collects the configurations of a p-dim object that are transformable into each other under the class of affine transformations and the class of permutations of the feature points. We give an analytical mapping to the neighborhood of the real Grassmann manifold as a solution to the distance minimization problem. As the true distance minimization is not easily solvable, we present an approximate solution using linear programming and quadratic programming. The main result is the development of a new modeling and computational framework, and related general-purpose calculation formulas and algorithms. We derive explicit analytical linear solutions for the exponential map and the logarithmic map. We give a linear and efficient algorithm for the computation of the approximate barycentric combinations of points on the Grassmann manifold. The derived original general-purpose explicit formulas for the real Grassmann manifold allow easy computation of the basic characteristics, such as measuring distances between points (shapes) and computing sample means and higher order statistics, for the task of shape representation. They can also be used in many other fields besides shape representation theory. In the thesis, these concepts are directly applied to the affine-invariant shape space, allowing computation of similarity between shapes as well as the sample mean and higher order moments of a constellation of shapes. This work is of great practical importance as it provides with efficient and robust ways of applying the theory to real world applications, such as multi-author handwritten digit recognition. Our broad contribution adds another dimension to the problem of representation of a geometric object and bridges a gap between image processing algorithms in Riemannian spaces and conventional algorithms embedded for simplicity in Euclidean spaces.