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

\begin{eqnarray*}

V(n,m,k)=\{&&v_1 \otimes
\cdots

\otimes v_m \in \vtm~:~ \\

&&\dim \Span \{v_1,\ldots,v_m\}
\leq

k~\}~.

\end{eqnarray*}

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.