### 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)