It is shown that a wave front, or in general any scalar two-dimensional function, can be reconstructed from its discrete differences by a simple multiplicative filtering operation in the spatial-frequency domain by using complex exponentials as basis functions in a modal expansion. Various difference-sampling geometries are analyzed. The difference data are assumed to be corrupted by random, additive noise of zero mean. The derived algorithms yield unbiased reconstructions for finite data arrays. The error propagation from the noise on the difference data to the reconstructed wave fronts is minimal in a least-squares sense. The spatial distribution of the reconstruction error over the array and the dependence of the mean reconstruction error on the array size are determined. The algorithms are computationally efficient, noniterative, and suitable for large arrays since the required number of mathematical operations for a reconstruction is approximately proportional to the number of data points if fast-Fourier-transform algorithms are employed.
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