Fingerprints theorems for zero crossings

We prove that the scale map of the zero crossings of almost all signals filtered by a Gaussian filter of variable size determines the signal uniquely, up to a constant scaling. The proof assumes that the filtered signal can be represented as a polynomial of finite, albeit possibly high, order. The result applies to zero and level crossings of linear differential operators of Gaussian filters. In this case the signal is determined uniquely, modulus the null space of the linear operator. The theorem can be extended to two-dimensional functions. These results are reminiscent of Logan’s theorem Bell Syst. Tech. J.56, 487 ( 1977). They imply that extrema of derivatives at different scales are a complete representation of a signal. They are especially relevant for computational vision in the case of the Laplacian operator acting on image intensities, and they suggest rigorous foundations for the primal sketch.