Optimal Linear Representations of Images Under Diverse Criteria

Image analysis often requires dimension reduction §before statistical analysis, in order to apply §sophisticated procedures. Motivated by eventual §applications, a variety of criteria have been §proposed: reconstruction error, class separation, §non-Gaussianity using kurtosis, sparseness, mutual §information, recognition of objects, and their §combinations. Although some criteria have analytical §solutions, the remaining ones require numerical §approaches. We present geometric tools for finding §linear projections that optimize a given criterion §for a given data set. The main idea is to formulate §a problem of optimization on a Grassmann or a §Stiefel manifold, and to use differential geometry §of the underlying space to construct optimization §algorithms. Purely deterministic updates lead to §local solutions, and addition of random components §allows for stochastic gradient searches that §eventually lead to global solutions. We demonstrate §these results using several image datasets, §including natural images and facial images. This §book should be useful for professionals, researches §and graduate students in Image Analysis field.