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

In this talk, we present an efficient method for a convex optimization problem involving a large non-symmetric and non-Toeplitz matrix. The proposed method is an instantiation of ADMM (Alternating Direction Method of Multipliers) applied in Krylov subspaces. Our method shows a significant advantage in computational time for convex optimization problems involving general matrices of large size. We applied the proposed method to a removal of spatially variant blur. Spatially variant blur is not given in the form of a block circulant matrix with circulant blocks (BCCB) and an efficient implementation based on the diagonalization of BCCB matrices by the discrete Fourier transform (DFT) is not available. However, due to our method’s efficiency in dealing with general matrices, spatially variant blur can be efficiently removed. Experimental results for total variation restoration in the case of spatially varian blur support our method’s superiority. PROPOSED OPTIMIZATION PROBLEM Krylov subspace methods consider an inverse problem of the form g = Af , where A ∈ Rd×d and f , g ∈ R. A Krylov subspace of order n is generated by a matrix A and a residual vector r := g −Af0 as follows, Kn(A, r) = span{r,Ar, · · · ,An−1r}, (1) where f0 ∈ R is an initial solution. Consider an optimization problem in the form of minimize u∈u0+Kn(A,r) h1(Au) + J ∑