Variational models for multiplicative noise removal

학위논문 (박사)-- 서울대학교 대학원 자연과학대학 수리과학부, 2017. 8. 강명주.This dissertation discusses a variational partial differential equation (PDE) models for restoration of images corrupted by multiplicative Gamma noise. The two proposed models are suitable for heavy multiplicative noise which is often seen in applications. First, we propose a total variation (TV) based model with local constraints. The local constraint involves multiple local windows which is related a spatially adaptive regularization parameter (SARP). In addition, convergence analysis such as the existence and uniqueness of a solution is also provided. Second model is an extension of the first one using nonconvex version of the total generalized variation (TGV). The nonconvex TGV regularization enables to efficiently denoise smooth regions, without staircasing artifacts that appear on total variation regularization based models, and to conserve edges and details.1. Introduction 1 2. Previous works 6 2.1 Variational models for image denoising 6 2.2.1 Convex and nonconvex regularizers 6 2.2.2 Variational models for multiplicative noise removal 8 2.2 Proximal linearized alternating direction method of multipliers 10 3. Proposed models 13 3.1 Proposed model 1 :exp TV model with SARP 13 3.1.1 Derivation of our model 13 3.1.2 Proposed TV model with local constraints 16 3.1.3 A SARP algorithm for solving model (3.1.16) 27 3.1.4 Numerical results 32 3.2 Proposed model 2 :exp NTGV model with SARP 51 3.2.1 Proposed NTGV model 51 3.2.2 Updating rule for $\lambda(x)$ in (3.2.1) 52 3.2.3 Algorithm for solving the proposed model (3.2.1) 55 3.2.4 Numerical results 62 3.2.5 Selection of parameters 63 3.2.6 Image denoising 65 4. Conclusion 79Docto

First order algorithms in variational image processing

Variational methods in imaging are nowadays developing towards a quite universal and flexible tool, allowing for highly successful approaches on tasks like denoising, deblurring, inpainting, segmentation, super-resolution, disparity, and optical flow estimation. The overall structure of such approaches is of the form ${\cal D}(Ku) + \alpha {\cal R} (u) \rightarrow \min_u$ ; where the functional ${\cal D}$ is a data fidelity term also depending on some input data $f$ and measuring the deviation of $Ku$ from such and ${\cal R}$ is a regularization functional. Moreover $K$ is a (often linear) forward operator modeling the dependence of data on an underlying image, and $\alpha$ is a positive regularization parameter. While ${\cal D}$ is often smooth and (strictly) convex, the current practice almost exclusively uses nonsmooth regularization functionals. The majority of successful techniques is using nonsmooth and convex functionals like the total variation and generalizations thereof or $\ell_1$-norms of coefficients arising from scalar products with some frame system. The efficient solution of such variational problems in imaging demands for appropriate algorithms. Taking into account the specific structure as a sum of two very different terms to be minimized, splitting algorithms are a quite canonical choice. Consequently this field has revived the interest in techniques like operator splittings or augmented Lagrangians. Here we shall provide an overview of methods currently developed and recent results as well as some computational studies providing a comparison of different methods and also illustrating their success in applications.Comment: 60 pages, 33 figure

Multiplicative Noise Removal: Nonlocal Low-Rank Model and Its Proximal Alternating Reweighted Minimization Algorithm

The goal of this paper is to develop a novel numerical method for efficient multiplicative noise removal. The nonlocal self-similarity of natural images implies that the matrices formed by their nonlocal similar patches are low-rank. By exploiting this low-rank prior with application to multiplicative noise removal, we propose a nonlocal low-rank model for this task and develop a proximal alternating reweighted minimization (PARM) algorithm to solve the optimization problem resulting from the model. Specifically, we utilize a generalized nonconvex surrogate of the rank function to regularize the patch matrices and develop a new nonlocal low-rank model, which is a nonconvex nonsmooth optimization problem having a patchwise data fidelity and a generalized nonlocal low-rank regularization term. To solve this optimization problem, we propose the PARM algorithm, which has a proximal alternating scheme with a reweighted approximation of its subproblem. A theoretical analysis of the proposed PARM algorithm is conducted to guarantee its global convergence to a critical point. Numerical experiments demonstrate that the proposed method for multiplicative noise removal significantly outperforms existing methods such as the benchmark SAR-BM3D method in terms of the visual quality of the denoised images, and the PSNR (the peak-signal-to-noise ratio) and SSIM (the structural similarity index measure) values

Accelerated algorithms for linearly constrained convex minimization

학위논문 (박사)-- 서울대학교 대학원 : 수리과학부, 2014. 2. 강명주.선형 제한 조건의 수학적 최적화는 다양한 영상 처리 문제의 모델로서 사 용되고 있다. 이 논문에서는 이 선형 제한 조건의 수학적 최적화 문제를 풀기위한 빠른 알고리듬들을 소개하고자 한다. 우리가 제안하는 방법들 은 공통적으로 Nesterov에 의해서 개발되었던 가속화한 프록시말 그레디 언트 방법에서 사용되었던 보외법을 기초로 하고 있다. 여기에서 우리는 크게보아서 두가지 알고리듬을 제안하고자 한다. 첫번째 방법은 가속화한 Bregman 방법이며, 압축센싱문제에 적용하여서 원래의 Bregman 방법보다 가속화한 방법이 더 빠름을 확인한다. 두번째 방법은 가속화한 어그먼티드 라그랑지안 방법을 확장한 것인데, 어그먼티드 라그랑지안 방법은 내부 문제를 가지고 있고, 이런 내부문제는 일반적으로 정확한 답을 계산할 수 없다. 그렇기 때문에 이런 내부문제를 적당한 조건을 만족하도록 부정확하 게 풀더라도 가속화한 어그먼티드 라그랑지 방법이 정확하게 내부문제를 풀때와 같은 수렴성을 갖는 조건을 제시한다. 우리는 또한 가속화한 얼터 네이팅 디렉션 방법데 대해서도 비슷한 내용을 전개한다.Abstract i 1 Introduction 1 2 Previous Methods 5 2.1 Mathematical Preliminary . . . . . . . . . . . . . . . . . . . . 5 2.2 The algorithms for solving the linearly constrained convex minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2.1 Augmented Lagrangian Method . . . . . . . . . . . . . 8 2.2.2 Bregman Methods . . . . . . . . . . . . . . . . . . . . 9 2.2.3 Alternating direction method of multipliers . . . . . . . 13 2.3 The accelerating algorithms for unconstrained convex minimization problem . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3.1 Fast inexact iterative shrinkage thresholding algorithm 16 2.3.2 Inexact accelerated proximal point method . . . . . . . 19 3 Proposed Algorithms 23 3.1 Proposed Algorithm 1 : Accelerated Bregman method . . . . . 23 3.1.1 Equivalence to the accelerated augmented Lagrangian method . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.1.2 Complexity of the accelerated Bregman method . . . . 27 3.2 Proposed Algorithm 2 : I-AALM . . . . . . . . . . . . . . . . 35 3.3 Proposed Algorithm 3 : I-AADMM . . . . . . . . . . . . . . . 43 3.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.4.1 Comparison to Bregman method with accelerated Bregman method . . . . . . . . . . . . . . . . . . . . . . . . 54 3.4.2 Numerical results of inexact accelerated augmented Lagrangian method using various subproblem solvers . . . 60 3.4.3 Comparison to the inexact accelerated augmented Lagrangian method with other methods . . . . . . . . . . 63 3.4.4 Inexact accelerated alternating direction method of multipliers for Multiplicative Noise Removal . . . . . . . . 69 4 Conclusion 86 Abstract (in Korean) 94Docto