Reconstructing Gridded Data from Higher Autocorrelations

The higher-order autocorrelations of integer-valued or rational-valued gridded data sets appear naturally in X-ray crystallography, and have applications in computer vision systems, correlation tomography, correlation spectroscopy, and pattern recognition. In this paper, we consider the problem of reconstructing a gridded data set from its higher-order autocorrelations. We describe an explicit reconstruction algorithm, and prove that the autocorrelations up to order 3r + 3 are always sufficient to determine the data up to translation, where r is the dimension of the grid. We also provide examples of rational-valued gridded data sets which are not determined by their autocorrelations up to order 3r + 2.