On Representation Theory in Computer Vision Problems

We introduce the following general question:
Let $V$ be a complex $n$-dimensional space and for $m \geq k$ consider the $GL(V)$-module $V(n,m,k)\subset \vtm$ defined by
V(n,m,k)=\{&&v_1 \otimes \cdots
\otimes v_m \in \vtm~:~ \\
&&\dim \Span \{v_1,\ldots,v_m\} \leq
We would like to determine $\dim V(n,m,k)$ for any choice of $n,m\gek$. This question appears in various disguises in computer vision problems where the constraints of a multi-linear problem occupy a low-dimensional 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 ${\cal P}^n\rightarrow {\calP}^n$ 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.