Abstract. In this paper we propose an efficient method for a convex optimization problem which involves a large nonsymmetric and non-Toeplitz matrix. The proposed method is an instantiation of the alternating direction method of multipliers applied in Krylov subspace. Our method offers significant advantages in computational speed for the convex optimization problems involved with general matrices of large size. We apply the proposed method to the restoration of spatially variant blur. The matrix representing spatially variant blur is not block circulant with circulant blocks (BCCB). Efficient implementation based on diagonalization of BCCB matrices by the discrete Fourier transform is not applicable for spatially variant blur. Since the proposed method can efficiently work with general matrices, the restoration of spatially variant blur is a good application of our method. Experimental results for total variation restoration of spatially variant blur show that the proposed method provides meaningful solutions in a short time.Key words. ADMM, Krylov subspace methods, image restoration, space variant blur, total variation AMS subject classifications. 65F10, 65F22, 68U10, 90C25DOI. 10.1137/16M10835051. Introduction. Many interesting problems in image processing involve finding a solution of a system of linear equations given by g = Af , where A ∈ R d×d and g, f ∈ R d . When the problem is ill-posed, a regularized solution can be found by reducing the solution space with constraints that reflect our prior knowledge about the solution. With convex constraints that reflect our prior knowledge, the problem can be written as a convex optimization. The alternating direction method of multipliers (ADMM) [11,26,27] is a convex optimization tool, which recently received considerable attention for its ease of incorporating diverse convex constraints to the problem, ease of implementation, and fast computational speed.The Krylov subspace method [44,45] is a projection method which restricts the solution space of the problem g = Af to the subspace spanned by the nth order Krylov sequence x, Ax, A 2 x, . . . , A n−1 x constructed from a vector x ∈ R d . By reducing the solution space to a subspace of small dimensionality, a solution can be found efficiently. The orthonormal basis of the Krylov subspace can be found by the Lanczos bidiagonalization when A is symmetric and by the Arnoldi process when A is nonsymmetric. Examples of the Krylov subspace method are minimum residual (MINRES) [41] and generalized minimum residual (GMRES)
Images acquired by a camera show lens blur due to imperfection in the optical system. Lens blur is non-stationary in a sense the amount of blur depends on pixel locations in a sensor. Lens blur is also asymmetric in a sense the amount of blur is different in the radial and tangential directions, and also in the inward and outward radial directions. This paper presents parametric blur kernel models based on the normal sinh-arcsinh distribution function. The proposed models can provide flexible shapes of blur kernels with different symmetry and skewness to model complicated lens blur accurately. Blur of single focal length lenses is estimated and the accuracy of the models is compared with existing parametric blur models. Advantage of the proposed models is demonstrated through deblurring experiments.6
Scanned laser pico projectors utilising laser light sources and microelectromechanical systems mirror devices introduce geometric distortion to the displayed images because of nonlinear mirror driving methods. A pre‐processing algorithm is proposed that corrects the nonlinear geometric distortion of scanned laser pico projectors. The distortion is modelled as blockwise homography transforms. The geometry of the images being displayed is pre‐distorted using the pixel indices obtained by the backward blockwise homography transforms. The pre‐distorted images are displayed by the pico projector whose geometrical distortion undoes the pre‐distortion such that distortion‐free images can be displayed. The proposed algorithm requires a small amount of information stored in the memory and three bilinear interpolations per pixel, thus providing an efficient way to correct nonlinear distortion without additional optical components, such as aspherical lenses.
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