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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.
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.