Three-dimensional and four-dimensional convolutions by coherent optical filtering

Three-dimensional (3-D) and four-dimensional (4-D) signals can be replaced by two-dimensional (2-D) sequences of sectional images if sampling in one or two dimensions is practicable. The convolution of such sequences results in a sequence that under certain conditions is the sampled version of the result of 3-D or 4-D convolution. Thus higher dimensional convolution can be performed in two dimensions, e.g., by coherent-optical filtering. Because of the usually large space-bandwidth products (SBP) of such sequences, the parallel computing ability of optical filtering makes it very feasible. A relation between input signal size, SBP of the filter, and parameters of the coherent-optical setup allows one to estimate limits in pixel numbers for the input signal size, e.g., for comparison with digital techniques. Based on linear systems theory, the necessary filter for sequence convolution can be calculated from a given higher dimensional point response or transfer function. Experimental results for 3-D bandpass filtering and a 4-D correlation are discussed.