BMVC 2004, Kingston, 7th-9th Sept, 2004

From Subspace to Submanifold Methods
M. Brand (MERL Research Lab, USA)

Twenty years ago it was not obvious that subspace approximations would
be such a successful representation for faces and other phenomena whose
measurement-space manifolds exhibit clear nonlinearities. Now PCA is ubiquitous
in computer vision and, although its globally linear view of the data
manifold can be a liability as systems scale up, it is not obvious that one can
reliably construct a better nonlinear data model.
This question motivates a rapidly growing literature of graph-theoretic
and tensorial approximations that view the data manifold as linear only in locales
and slices, respectively. These limited-linearity "submanifold methods"
offer much richer descriptions of the data manifold, and may ultimately replace
subspace methods in computer vision. First, problems with suboptimality,
solution instability, and sample complexity must be overcome. Some of
the obstacles to "industrial-strength" manifold modeling are discussed, and
some methods that offer significant improvements in robustness and accuracy
are introduced which eschew the common assumptions about the manifold
geometry - in particular, a generalization of PCA that can exactly recover
the intrinsic coordinate systems of a significant class of nowhere-linear
manifolds. Connecting these data models to image analysis and synthesis
algorithms is only slightly more complicated than using subspace methods.
Some applications to tracking, recognition, and editing of objects and people
in video are demonstrated.

(pdf article)