On the Circular Area Signature for Graphs

The representation of curves by integral invariant signatures is an important step inshape recognition and classification. Integral invariants are preferred over their differential counter-parts due to their robustness with respect to noise. However, in contrast to differential invariants ofcurves, it is currently unknown whether integral signatures offer unique representations of curves. Inthis article, we prove some results on the uniqueness of the circular area signature. In particular, westudy the case for graphs of periodic functions. We show that the circular area signature is uniqueif taken with respect to parameterization by the x-axis. Furthermore, we prove that the true circu-lar area signature (parameterized 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 ofwidth 2r.