On Representation Theory in Computer Vision Problems
We introduce the following general question: Let V be a complex n-dimensional space and for m k consider the GL V -module V n m k V m defined by V n m k f v vm V m dimSpanfv vmg k g We would like to determine dimV n m k for any choice of n m k. This question appears in various disguises in computer vision problems where the constraints of a multi-linear problem occupy a lowdimensional subspace. We discuss two such problems: analysis of constraints in single view indexing functions (the 8-point shape tensor), and the analysis of the constraints in dynamic P n P alignments, i.e., where the point sets are allowed to move within a k-dimensional subspace while the n-dimensional space is being multiply projected (multiple views) onto copies of the m-dimensional space. We then derive the solution to the general problem using tools from representation theory.