Shift Variant Image Degradation and Restoration Using Singular Value Decomposition

Abstract: Shift-variant image degradation is frequently encountered in practical imaging systems where the point spread function (PSF) varies across the image field due to motion, optical aberrations, atmospheric turbulence, or sensor-related effects. Unlike shift-invariant, shift-variant degradation presents significant challenges for image restoration because the degradation process cannot be represented by a single convolution kernel. This paper proposes a singular value decomposition (SVD)-based framework for restoring images degraded by shift-variant motion blur. The proposed approach determines the contribution of small singular values using a singular-value energy retention criterion. Specifically, the number of small singular values is selected based on a specified percentage of cumulative singular-value energy, providing a systematic approach for controlling noise amplification while preserving useful image information. The degradation model is formulated using a position-dependent PSF represented by a shift-variant imaging operator. Three representative one dimensional shift-variant motion PSFs are considered: bidirectional linear motion, Gaussian motion, and simple harmonic motion. The image degradation process is modeled as a linear system, and SVD is employed to analyze and invert the corresponding degradation operator. The singular-value representation provides insight into the ill-conditioned nature of the restoration problem and enables the development of stable inversion techniques. The proposed SVD-based restoration algorithm is applied to three degraded images. Experimental results demonstrate the effectiveness of the proposed approach in recovering image details and reducing blur artifacts under different motion models.