Sangwoon Yun

An Accelerated Proximal Gradient Algorithm for Frame-Based Image Restoration via the Balanced Approach

Frame-based image restoration by using the balanced approach has been developed over the last decade. Many recently developed algorithms for image restoration can be viewed as an acceleration of the proximal forward-backward splitting algorithm. Accelerated proximal gradient (APG) algorithms studied by Nesterov, Nemirovski, and others have been demonstrated to be efficient in solving various regularized convex optimization problems arising in compressed sensing, machine learning, and control. In this paper, we adapt the APG algorithm to solve the

SymNMF: nonnegative low-rank approximation of a similarity matrix for graph clustering

\ell_1

A New Multiplicative Denoising Variational Model Based on

-regularized linear least squares problem in the balanced approach in frame-based image restoration. This algorithm terminates in

m

O(1/\sqrt{\epsilon})

th Root Transformation

iterations with an

Alternating Minimization Algorithm for Speckle Reduction With a Shifting Technique

\epsilon

Linearized proximal alternating minimization algorithm for motion deblurring by nonlocal regularization

-optimal solution, and we demonstrate that this single algorithmic framework can universally handle several image restoration problems, such as image deblurring, denoising, inpainting, and cartoon-texture decomposition. Our numerical results suggest that the APG algorithms are efficient and robust in solving large-scale image restoration problems. The algorithms we implemented are able to restore

Proximal Linearized Alternating Direction Method for Multiplicative Denoising

512\times512

A block coordinate gradient descent method for regularized convex separable optimization and covariance selection

images in various image restoration problems in less than 50 seconds on a modest PC. We also compare the numerical performance of our proposed algorithms applied to image restoration problems by using one frame-based system with that by using cartoon and texture systems for image deblurring, denoising, and inpainting.

Two-Level Convex Relaxed Variational Model for Multiplicative Denoising

Nonnegative matrix factorization (NMF) provides a lower rank approximation of a matrix by a product of two nonnegative factors. NMF has been shown to produce clustering results that are often superior to those by other methods such as K-means. In this paper, we provide further interpretation of NMF as a clustering method and study an extended formulation for graph clustering called Symmetric NMF (SymNMF). In contrast to NMF that takes a data matrix as an input, SymNMF takes a nonnegative similarity matrix as an input, and a symmetric nonnegative lower rank approximation is computed. We show that SymNMF is related to spectral clustering, justify SymNMF as a general graph clustering method, and discuss the strengths and shortcomings of SymNMF and spectral clustering. We propose two optimization algorithms for SymNMF and discuss their convergence properties and computational efficiencies. Our experiments on document clustering, image clustering, and image segmentation support SymNMF as a graph clustering method that captures latent linear and nonlinear relationships in the data.