SCALING AND FINGERPRINT THEOREMS FOR ZERO-CROSSINGS
ALAN L. YUILLE TOMASO POGGIO Massachusetts Institute of Technology
We characterize some properties of the zero-crossings of the Laplacian of signals--in particular images--filtered with linear filters, as a function of the scale of the filter, extending recent work by A. Witkin ( 1983). We review our two main results. First, we have proven that in any dimension the only filter that does not create generic zero-crossings as the scale increases is the Gaussian. This result can be generalized to apply to level- crossings of any linear differential operator: It applies in particular to ridges and ravines in the image intensity. Second, we have proven 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 also valid in two dimensions, that is, it applies to images. Thus, extrema (for instance of derivatives) at different scales are a complete representation of a signal. Finally we discuss the new results proven by Curtis ( 1985) for two- and higher-dimensional functions.
In most physical phenomena, changes in spatial or temporal structure occur over a wide range of scales. Images are no exception: Changes in