Circulant Preconditioners for the Super-Resolution Reconstruction Problem

Nhat Nguyen
SCCM Program
Gates Bldg 2B
Stanford, CA 94305

Gene Golub
Stanford University

Peyman Milanfar
SRI International


Abstract

In the super-resolution reconstruction problem, we would like to reconstruct a high-resolution image from a given set of low-resolution images. These low-resolution images are translated, blurred, uniformly down-sampled, and noisy representations of the high-resolution image to be reconstructed. We assume the blurring to be linearly space invariant and the noise image independent. The translations are subpixel shifts describing relative scene motions of the low-resolution images.

We model the problem in a classic reconstruction form

Y_k = D_k*C_k*F_k*X + E_k, 1<=k<=p

where X is the original high-resolution image, F_k the translation matrix, C_k the linear space invariant blur matrix, D_k the decimation operator, E_k the additive noise in the kth low-resolution image.

In general, we will not have all possible relative vertical and horizontal subpixel shifts in our set of given low-resolution images. The resulting system is ill-conditioned and underdetermined. In this talk we propose efficient circulant-type preconditioners based on the blur matrices C_k for the regularized minimum norm solution. Experimental results show that satisfactory reconstruction of the original image can be obtained in a few preconditioned CG iterations.