We prove that the scale map of the zero-crossings of almost all signals filtered by the second derivative of a gaussian of variable size determines the signal uniquely, up to a constant scaling and a harmonic function. 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. An argument suggests that this restriction is not essential. Stability of the reconstruction scheme is briefly discussed. The result applies to zero- and level-crossings of linear differential operators of gaussian filters. The theorem is extended to two dimensions, that is to images. These results are reminiscent of Logan's theorem. They imply that extrema of derivatives at different scales are a complete representation of a signal.
