Aliasing is Good for You: Joint Registration and Reconstruction for Super-Resolution
In many applications, the sampling frequency is limited by the physical characteristics of the components: the pixel pitch, the rate of the A/D converter, etc. A low-pass filter is then often applied before the sampling operation to avoid aliasing. However, when multiple copies are available, it is possible to use the information that is inherently present in the aliasing to reconstruct a higher resolution signal. If the different copies have unknown relative offsets, this is a non-linear problem in the offsets and the signal coefficients. They are not easily separable in the set of equations describing the super-resolution problem. Thus, we perform joint registration and reconstruction from multiple unregistered sets of samples. We give a mathematical formulation for the problem when there are M sets of N samples of a signal that is described by L expansion coefficients. We prove that the solution of the registration and reconstruction problem is generically unique if MN >= L+M-1. We describe two subspace-based methods to compute this solution. Their complexity is analyzed, and some heuristic methods are proposed. Finally, some numerical simulation results on one and two-dimensional signals are given to show the performance of these methods.