On the Circular Area Signature for Graphs

The representation of curves by integral invariant signatures is an important step in shape recognition and classification. Integral invariants are preferred over their differential counterparts due to their robustness with respect to noise. However, in contrast to differential invariants of curves, it is currently unknown whether integral signatures offer unique representations of curves. In this article, we prove some results on the uniqueness of the circular area signature. In particular, we study the case for graphs of periodic functions. We show that the circular area signature is unique if taken with respect to parametrization by the $x$-axis. Furthermore, we prove that the true circular area signature (parametrized by arclength) is unique in a neighborhood of constant functions. Finally, we show uniqueness in the special case that the functions of interest agree on an interval of width $2r$.