Removing multiplicative noise by Douglas-Rachford splitting methods

Multiplicative noise appears in various image processing applications, e.g., in synthetic aperture radar (SAR), ultrasound imaging or in connection with blur in electronic microscopy, single particle emission computed tomography (SPECT) and positron emission tomography (PET). In this paper, we consider a variational restoration model consisting of the I-divergence as data fitting term and the total variation semi-norm or nonlocal means as regularizer. Although the I-divergence is the typical data fitting term when dealing with Poisson noise we substantiate why it is also appropriate for cleaning Gamma noise. We propose to compute the minimizer of our restoration functional by applying Douglas-Rachford splitting techniques, resp. alternating split Bregman methods, combined with an efficient algorithm to solve the involved nonlinear systems of equations. We prove the Q-linear convergence of the latter algorithm. Finally, we demonstrate the performance of our whole scheme by numerical examples. It appears that the nonlocal means approach leads to very good qualitative results

Dual constrained TV-based regularization on graphs

26 pagesInternational audienceAlgorithms based on Total Variation (TV) minimization are prevalent in image processing. They play a key role in a variety of applications such as image denoising, compressive sensing and inverse problems in general. In this work, we extend the TV dual framework that includes Chambolle's and Gilboa-Osher's projection algorithms for TV minimization. We use a flexible graph data representation that allows us to generalize the constraint on the projection variable. We show how this new formulation of the TV problem may be solved by means of fast parallel proximal algorithms. On denoising and deblurring examples, the proposed approach is shown not only to perform better than recent TV-based approaches, but also to perform well on arbitrary graphs instead of regular grids. The proposed method consequently applies to a variety of other inverse problems including image fusion and mesh filtering

A Review on Deep Learning in Medical Image Reconstruction

Medical imaging is crucial in modern clinics to guide the diagnosis and treatment of diseases. Medical image reconstruction is one of the most fundamental and important components of medical imaging, whose major objective is to acquire high-quality medical images for clinical usage at the minimal cost and risk to the patients. Mathematical models in medical image reconstruction or, more generally, image restoration in computer vision, have been playing a prominent role. Earlier mathematical models are mostly designed by human knowledge or hypothesis on the image to be reconstructed, and we shall call these models handcrafted models. Later, handcrafted plus data-driven modeling started to emerge which still mostly relies on human designs, while part of the model is learned from the observed data. More recently, as more data and computation resources are made available, deep learning based models (or deep models) pushed the data-driven modeling to the extreme where the models are mostly based on learning with minimal human designs. Both handcrafted and data-driven modeling have their own advantages and disadvantages. One of the major research trends in medical imaging is to combine handcrafted modeling with deep modeling so that we can enjoy benefits from both approaches. The major part of this article is to provide a conceptual review of some recent works on deep modeling from the unrolling dynamics viewpoint. This viewpoint stimulates new designs of neural network architectures with inspirations from optimization algorithms and numerical differential equations. Given the popularity of deep modeling, there are still vast remaining challenges in the field, as well as opportunities which we shall discuss at the end of this article.Comment: 31 pages, 6 figures. Survey pape

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