Three-dimensional Super-resolution with Nonuniform Cutoff Frequencies

Super-resolution theory provides a theoretical foundation to numerous applications in, e.g., signal processing and conummications. Many applications have multi-dimensional measurements. We investigate the three-dimensional super-resolution problem that aims to recover the locations and complex amplitudes of point sources in [0, 1](3) from the low-pass Fourier measurements. In literature, sufficient minimum separations of the point sources for exact recovery have been found for one and two dimensional cases with equal cutoff frequency for each dimension. A cutoff frequency of a dimension is the bandwidth of the low pass measurement. We report the minimum separation for the three dimensional case, where we also allow the cutoff frequencies to be different for different dimensions. We introduce a new minimum separation factor and show that, when the measurements arc band-limited to the integer cuboid [-f(c,1), f(c,1)] x [-f(c,2), f(c,2)] x [f(c,3), f(c,3)] where f(c,1), f(c,2) and f(c,3) are not necessarily equal, the locations and complex amplitudes can be recovered by minimizing total variation norm if the minimum separation factor is at least 3.36.