Energy Minimization Methods in Computer Vision and Pattern Recognition

Simplicity of linear representations makes them a popular tool in several imaging analysis, and indeed many other applications involving high-dimensional data. In image analysis, the two widely used linear representations are: (i) linear projections of images to lowdimensional Euclidean subspaces, and (ii) linear spectral filtering of images. In view of the orthogonality and other constraints imposed on these representations (the subspaces or the filters), they take values on nonlinear manifolds (Grassmann, Stiefel, or rotation group). We present a family of algorithms that exploit the geometry of the underlying manifolds to find optimal linear representations for specified tasks. We illustrate the effectiveness of algorithms by finding optimal subspaces and sparse filters both in the context of image-based object recognition.