  In this paper, we show that the affine, non-rigid structure-from-motion
problem can be solved by rank-one, thus degenerate, basis shapes. It is a
natural reformulation of the classic low-rank method by Bregler et al., where
it was assumed that the deformable 3D structure is generated by a linear
combination of rigid basis shapes. The non-rigid shape will be decomposed into
the mean shape and the degenerate shapes, constructed from the right singular
vectors of the low-rank decomposition. The right singular vectors are affinely
back-projected into the 3D space, and the affine back-projections will also be
solved as part of the factorisation. By construction, a direct interpretation
for the right singular vectors of the low-rank decomposition will also follow:
they can be seen as principal components, hence, the first variant of our
method is referred to as Rank-1-PCA. The second variant, referred to as
Rank-1-ICA, additionally estimates the orthogonal transform which maps the
deformation modes into as statistically independent modes as possible. It has
the advantage of pinpointing statistically dependent subspaces related to, for
instance, lip movements on human faces. Moreover, in contrast to prior works,
no predefined dimensionality for the subspaces is imposed. The experiments on
several datasets show that the method achieves better results than the
state-of-the-art, it can be computed faster, and it provides an intuitive
interpretation for the deformation modes.
