A General Framework for Regularized, Similarity-Based Image Restoration

Abstract

Any image can be represented as a function defined on a weighted graph, in which the underlying structure of the image is encoded in kernel similarity and associated Laplacian matrices. In this paper, we develop an iterative graph-based framework for image restoration based on a new definition of the normalized graph Laplacian. We propose a cost function, which consists of a new data fidelity term and regularization term derived from the specific definition of the normalized graph Laplacian. The normalizing coefficients used in the definition of the Laplacian and associated regularization term are obtained using fast symmetry preserving matrix balancing. This results in some desired spectral properties for the normalized Laplacian such as being symmetric, positive semidefinite, and returning zero vector when applied to a constant image. Our algorithm comprises of outer and inner iterations, where in each outer iteration, the similarity weights are recomputed using the previous estimate and the updated objective function is minimized using inner conjugate gradient iterations. This procedure improves the performance of the algorithm for image deblurring, where we do not have access to a good initial estimate of the underlying image. In addition, the specific form of the cost function allows us to render the spectral analysis for the solutions of the corresponding linear equations. In addition, the proposed approach is general in the sense that we have shown its effectiveness for different restoration problems, including deblurring, denoising, and sharpening. Experimental results verify the effectiveness of the proposed algorithm on both synthetic and real examples.

Collaborative filtering via graph signal processing

“…With this interpretation one can see (8) as a filter that eliminates the irrelevant features (middle frequencies), smoothes out the similar components (low frequencies) and preserves the discriminative features (high frequencies). This band-stop behavior where both high and low graph frequencies are preserved is not uncommon in image processing (image de-noising and image sharpening, respectively) [13], and was also observed in brain signal analytics [8], [14].…”

Image deblurring using a perturbation-based regularization approach

“…Gaussian white noise with zero mean and three different standard deviation values (σ = 0.0005, 0.005 and 0.05) was added to the blurred images. The following benchmark methods were used for comparison: The Augmented Lagrangian Method for Total Variation (ALMTV) [4], the General Framework for Regularized, Similarity-Based Image Restoration (GFRSB) [19], and the Generalized Iterated Shrinkage Algorithm for Non-convex Sparse Coding (GISA) [18]. All the parameters of the benchmark methods were set as suggested by the authors.…”

Full 3D Reconstruction of Non-Rigidly Deforming Objects

“…∇H t represents the discrete gradient of H t , |.| denotes the L1-norm and µ is the regularization parameter. BTV regularization/denoising has been a topic of interest for researchers but most of the research has been restricted to organized color and depth images [28,33,36,39,42], where the neighborhoods are well defined and the gradient, based on intensity or depth values, is easy to compute e.g., via shift operators [34,51]. In the current problem,Ĥ f t is a set of unorganized 3D points without any connectivity or neighborhood information, therefore the extension of BTV regularization to 3D point clouds is not a straightforward problem.…”