Convergence Analysis in Fourier Domain

Abstract

Having shown in previous chapters that joint MAP estimator does work for blind deconvolution, we proceed by selecting TV based priors for both the image as well as the PSF. Since both the image and the PSF are unknowns, alternate minimization (AM) is used to solve the blind deconvolution problem. In this chapter we provide a Fourier domain convergence analysis of the AM procedure. TV prior being non-linear, a non-quadratic cost function is obtained at each iteration of the AM. This makes the convergence analysis difficult. For making the analysis feasible, the TV prior is replaced by its quadratic upper bound. With this approximation, the cost function at each step of AM becomes quadratic in nature. Even with this approximation, the system obtained is not a linear shift invariant (LSI) system, which makes analysis in the spectral domain difficult. To overcome this, we use a further approximation, which makes the system LSI at each iteration – with the system changing with iteration. We note that the resulting system behaves like an adaptive Wiener filter. Once the fixed-point is reached the regularization factors remain constant, and a Fourier domain analysis shows that the fixed-points for image and PSF magnitudes are similar to those derived in an already existing analysis. Our analysis differs in that the image and the PSF regularization factors become signal (image/PSF) dependent. It is observed that the convergence points of the magnitudes of the image and the PSF are related to each other, which is due to the approximation to make the system LSI. We analyze the error term in the approximation, which provides an insight into the regularizing capability of the TV prior.