Fingerprints Theorems

We prove that the scale map of the zero-crossings of almost all signals filtered by a gaussian of variable size determines the signal uniquely, up to a constant scaling. Exceptions are signals that are antisymmetric about all their zeros (for instance infinitely periodic gratings). Our proof provides a method for reconstructing almost all signals from knowledge of how the zero-crossing contours of the signal, filtered by a gaussian filter, change with the size of the filter. The proof assumes that the filtered signal can be represented as a polynomial of finite, albeit possibly very high, order. The result applies to zero- and level-crossings of signals filtered by gaussian filters. The theorem is extended to two dimensions, that is to images. These results imply that extrema (for instance of derivatives) at different scales are a complete representation of a signal.