ADMM in Krylov Subspace and its Application to Total Variation Restoration of Spatially Variant Blur

Abstract

In this paper we proposed an efficient method for a convex optimization problem which involves a large non-symmetric and non-Toeplitz matrix. The proposed method is an instantiation of alternating direction method of multipliers (ADMM) applied in Krylov subspace. Our method takes significant advantages in computational speed for the convex optimization problems involved with those large and general matrices. We applied the proposed method to the restoration of spatially variant blur. One of the most popular approaches for image restoration is solving a convex optimization problem accelerated by fast Fourier transform (FFT) based implementation. However the acceleration using the FFT may not be allowed when the involved matrix is not block circulant with circulant blocks (BCCB). The matrix representing spatially variant blur is not BCCB in general. Since the proposed method can efficiently work with non-BCCB matrices, the restoration of spatially variant blur is a good application of our method. Experimental results for total variation (TV) restoration of spatially variant blur show that the proposed method provides meaningful solutions in a short time.