  We propose a metric learning framework for the construction of invariant
geometric functions of planar curves for the Eucledian and Similarity group of
transformations. We leverage on the representational power of convolutional
neural networks to compute these geometric quantities. In comparison with
axiomatic constructions, we show that the invariants approximated by the
learning architectures have better numerical qualities such as robustness to
noise, resiliency to sampling, as well as the ability to adapt to occlusion and
partiality. Finally, we develop a novel multi-scale representation in a
similarity metric learning paradigm.
