  Transformation-invariant analysis of signals often requires the computation
of the distance from a test pattern to a transformation manifold. In
particular, the estimation of the distances between a transformed query signal
and several transformation manifolds representing different classes provides
essential information for the classification of the signal. In many
applications the computation of the exact distance to the manifold is costly,
whereas an efficient practical solution is the approximation of the manifold
distance with the aid of a manifold grid. In this paper, we consider a setting
with transformation manifolds of known parameterization. We first present an
algorithm for the selection of samples from a single manifold that permits to
minimize the average error in the manifold distance estimation. Then we propose
a method for the joint discretization of multiple manifolds that represent
different signal classes, where we optimize the transformation-invariant
classification accuracy yielded by the discrete manifold representation.
Experimental results show that sampling each manifold individually by
minimizing the manifold distance estimation error outperforms baseline sampling
solutions with respect to registration and classification accuracy. Performing
an additional joint optimization on all samples improves the classification
performance further. Moreover, given a fixed total number of samples to be
selected from all manifolds, an asymmetric distribution of samples to different
manifolds depending on their geometric structures may also increase the
classification accuracy in comparison with the equal distribution of samples.
