Motion-free superresolution and the role of relative blur

Motion-free superresolution refers to the process of generating a high-resolution image from a set of defocused and downsampled observations blurred to different extents. We first examine the theory of motion-free superresolution and derive analytical expressions in the discrete Fourier transform domain for obtaining a superresolved image from its blurred and downsampled versions. The analysis leads to the construction of an appropriate system matrix whose inverse yields the desired reconstruction filters. Next, we address the effect of relative blurring among the defocused observations on the quality of the reconstructed image. To get a quantitative perspective of this effect, we derive the Cramér-Rao lower bound (CRLB) for the covariance of the error in the estimate of the superresolved image. The CRLB depends on the system matrix, and the condition number of this matrix is studied as a function of the blur kernels. It is shown that the stability of the problem and the estimate of the superresolved image become better as the relative blur increases. Adding more observations without enriching the blur span does not improve the quality of reconstruction. Several simulation results are given for the purpose of validation.