^December January – 2012

The ease of seeing conceals many complexities. A fundamental one is the problem of fragmentation – we are able to recognize objects although they are optically incomplete, e.g., due to occlusions. To overcome this difficulty, biological and artificial visual systems use a mechanism for contour completion, which has been studied by the many disciplines of vision science, mostly in an intra-disciplinary fashion. Recent computational, neurophysiological, and psychophysical studies suggest that completed contours emerge from activation patterns of orientation selective cells in the primary visual cortex, or V1. In this work we suggest modeling these patterns as 3D curves in the mathematical continuous space R^2 × S^1, a.k.a. the unit tangent bundle associated with the image plane R^2, that abstracts V1. Then, we propose that the completed shape may follow physical/biological principles which are conveniently abstracted and analyzed in this space. We implement our theories by numerical algorithms to show ample experimental results of visually completed curves in natural and synthetic scenes.

A popular approach to deal with the "curse of dimensionality" in relation with high-dimensional data analysis is to assume that points in these datasets lie on a low-dimensional manifold immersed in a high-dimensional ambient space. Kernel methods operate on this assumption and introduce the notion of local affinities between data points via the construction of a suitable kernel. Spectral analysis of this kernel provides a global, preferably low-dimensional, coordinate system that preserves the qualities of the manifold. In this presentation, the scalar relations used in this framework will be extended to matrix relations, which can encompass multidimensional similarities between local neighborhoods of points on the manifold. We utilize the diffusion maps methodology together with linear-projection operators between tangent spaces of the manifold to construct a super-kernel that represents these relations. The properties of the presented super- kernels are explored and their spectral decompositions are utilized to embed the patches of the manifold into a tensor space in which the relations between them are revealed. Two applications of the patch- to-tensor embedding framework for data clustering and classification will be presented.