K-SVD algorithm has been successfully applied to image denoising tasks dozens of years but the big bottleneck in speed and accuracy still needs attention to break. For the sparse coding stage in K-SVD, which involves $\ell_{0}$ constraint, prevailing methods usually seek approximate solutions greedily but are less effective once the noise level is high. The alternative $\ell_{1}$ optimization is proved to be powerful than $\ell_{0}$, however, the time consumption prevents it from the implementation. In this paper, we propose a new K-SVD framework called K-SVD$_P$ by applying the Primal-dual active set (PDAS) algorithm to it. Different from the greedy algorithms based K-SVD, the K-SVD$_P$ algorithm develops a selection strategy motivated by KKT (Karush-Kuhn-Tucker) condition and yields to an efficient update in the sparse coding stage. Since the K-SVD$_P$ algorithm seeks for an equivalent solution to the dual problem iteratively with simple explicit expression in this denoising problem, speed and quality of denoising can be reached simultaneously. Experiments are carried out and demonstrate the comparable denoising performance of our K-SVD$_P$ with state-of-the-art methods.
Cloud K-SVD is a dictionary learning algorithm that can train at multiple nodes and hereby produce a mutual dictionary to represent low-dimensional geometric structures in image data. We present a novel application of the algorithm as we use it to recover both noiseless and noisy images from overlapping patches. We implement a node network in Kubernetes using Docker containers to facilitate Cloud K-SVD. Results show that Cloud K-SVD can recover images approximately and remove quantifiable amounts of noise from benchmark gray-scaled images without sacrificing accuracy in recovery; we achieve an…
A TRPCA-Inspired Deep Unfolding Network for Hyperspectral Image Denoising via Thresholded t-SVD and Top-K Sparse Transformer
College of Computer Science and Engineering, Shandong University of Science and Technology（计算机科学与工程学院，山东科技大学）；Gosci Information Technology Co.,Ltd.（Gosci信息技术有限公司）
Northeastern University at Qinhuangdao（东北大学秦皇岛分校）；Univ Bretagne-Sud（南布列塔尼大学）
Patch-based low-rank minimization for image processing attracts much attention in recent years. The minimization of the matrix rank coupled with the Frobenius norm data fidelity can be solved by the hard thresholding filter with principle component analysis (PCA) or singular value decomposition (SVD). Based on this idea, we propose a patch-based low-rank minimization method for image denoising. The main denoising process is stated in three equivalent way: PCA, SVD and low-rank minimization. Compared to recent patch-based sparse representation methods, experiments demonstrate that the proposed …
University of Isfahan（伊斯法罕大学）；Hunan University（湖南大学）；Isfahan University of Medical Sciences（伊斯法罕医科大学）；Sichuan University（四川大学）
The data-driven sparse methods such as synthesis dictionary learning (e.g., K-SVD) and sparsifying transform learning have been proven effective in image denoising. However, they are intrinsically single-scale which can lead to suboptimal results. We propose two methods developed based on wavelet subbands mixing to efficiently combine the merits of both single and multiscale methods. We show that an efficient multiscale method can be devised without the need for denoising detail subbands which substantially reduces the runtime. The proposed methods are initially derived within the framework of…
In this paper a new dictionary learning algorithm for multidimensional data is proposed. Unlike most conventional dictionary learning methods which are derived for dealing with vectors or matrices, our algorithm, named KTSVD, learns a multidimensional dictionary directly via a novel algebraic approach for tensor factorization as proposed in [3, 12, 13]. Using this approach one can define a tensor-SVD and we propose to extend K-SVD algorithm used for 1-D data to a K-TSVD algorithm for handling 2-D and 3-D data. Our algorithm, based on the idea of sparse coding (using group-sparsity over multidi…
In this paper, the problem of de-noising of an image contaminated with additive white Gaussian noise (AWGN) is studied. This subject has been continued to be an open problem in signal processing for more than 50 years. In the present paper, we suggest a method based on global clustering of image constructing blocks. Noting that the type of clustering plays an important role in clustering-based de-noising methods, we address two questions about the clustering. First, which parts of data should be considered for clustering? Second, what data clustering method is suitable for de-noising? Clusteri…
Imperial College London（帝国理工学院）；Middlesex University（密德萨斯大学）；University of Oulu（奥卢大学）
Dictionary learning and component analysis are part of one of the most well-studied and active research fields, at the intersection of signal and image processing, computer vision, and statistical machine learning. In dictionary learning, the current methods of choice are arguably K-SVD and its variants, which learn a dictionary (i.e., a decomposition) for sparse coding via Singular Value Decomposition. In robust component analysis, leading methods derive from Principal Component Pursuit (PCP), which recovers a low-rank matrix from sparse corruptions of unknown magnitude and support. However, …
DeepTensor is a computationally efficient framework for low-rank decomposition of matrices and tensors using deep generative networks. We decompose a tensor as the product of low-rank tensor factors (e.g., a matrix as the outer product of two vectors), where each low-rank tensor is generated by a deep network (DN) that is trained in a self-supervised manner to minimize the mean-squared approximation error. Our key observation is that the implicit regularization inherent in DNs enables them to capture nonlinear signal structures (e.g., manifolds) that are out of the reach of classical linear me…
University of Electronic Science and Technology of China（电子科技大学）
Robust tensor principal component analysis (RTPCA) can separate the low-rank component and sparse component from multidimensional data, which has been used successfully in several image applications. Its performance varies with different kinds of tensor decompositions, and the tensor singular value decomposition (t-SVD) is a popularly selected one. The standard t-SVD takes the discrete Fourier transform to exploit the residual in the 3rd mode in the decomposition. When minimizing the tensor nuclear norm related to t-SVD, all the frontal slices in frequency domain are optimized equally. In this…
Image Denoising Using Global and Local Circulant Representation
School of Software Engineering, South China University of Technology（华南理工大学软件学院）；Department of Clinical Laboratory Medicine, Guangdong Provincial People’s Hospital, Southern Medical University（南方医科大学广东省级人民医院临床检验医学科）
Efficient Image Denoising Using Global and Local Circulant Representation
School of Software Engineering, South China University of Technology（华南理工大学软件学院）；Department of Clinical Laboratory Medicine, Guangdong Provincial People’s Hospital, Southern Medical University（广东省人民医院临床检验医学科）
The paper proposes a new high spatial resolution hyperspectral (HR-HS) image estimation method based on convex optimization. The method assumes a low spatial resolution HS (LR-HS) image and a guide image as observations, where both observations are contaminated by noise. Our method simultaneously estimates an HR-HS image and a noiseless guide image, so the method can utilize spatial information in a guide image even if it is contaminated by heavy noise. The proposed estimation problem adopts hybrid spatio-spectral total variation as regularization and evaluates the edge similarity between HR-H…
Imperial College London（帝国理工学院）；Middlesex University London（伦敦米德尔塞克斯大学）；University of Oulu（奥卢大学）
Dictionary learning and component analysis models are fundamental for learning compact representations that are relevant to a given task (feature extraction, dimensionality reduction, denoising, etc.). The model complexity is encoded by means of specific structure, such as sparsity, low-rankness, or nonnegativity. Unfortunately, approaches like K-SVD - that learn dictionaries for sparse coding via Singular Value Decomposition (SVD) - are hard to scale to high-volume and high-dimensional visual data, and fragile in the presence of outliers. Conversely, robust component analysis methods such as …
The spatio-spectral total variation (SSTV) model has been widely used as an effective regularization of hyperspectral images (HSI) for various applications such as mixed noise removal. However, since SSTV computes local spatial differences uniformly, it is difficult to remove noise while preserving complex spatial structures with fine edges and textures, especially in situations of high noise intensity. To solve this problem, we propose a new TV-type regularization called Graph-SSTV (GSSTV), which generates a graph explicitly reflecting the spatial structure of the target HSI from noisy HSIs a…
The truncated singular value decomposition (SVD) of the measurement matrix is the optimal solution to the_representation_ problem of how to best approximate a noisy measurement matrix using a low-rank matrix. Here, we consider the (unobservable)_denoising_ problem of how to best approximate a low-rank signal matrix buried in noise by optimal (re)weighting of the singular vectors of the measurement matrix. We exploit recent results from random matrix theory to exactly characterize the large matrix limit of the optimal weighting coefficients and show that they can be computed directly from data …
University of Illinois, Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
Many applications in signal processing benefit from the sparsity of signals in a certain transform domain or dictionary. Synthesis sparsifying dictionaries that are directly adapted to data have been popular in applications such as image denoising, inpainting, and medical image reconstruction. In this work, we focus instead on the sparsifying transform model, and study the learning of well-conditioned square sparsifying transforms. The proposed algorithms alternate between a $\ell_0$ "norm"-based sparse coding step, and a non-convex transform update step. We derive the exact analytical solutio…
IBM T.J. Watson Research Center（IBM T.J. 沃森研究中心）；UBC（不列颠哥伦比亚大学）；Mitsubishi Electric Research Laboratories（三菱电机研究实验室）；UC Berkeley（加州大学伯克利分校）
Recent SVD-free matrix factorization formulations have enabled rank minimization for systems with millions of rows and columns, paving the way for matrix completion in extremely large-scale applications, such as seismic data interpolation. In this paper, we consider matrix completion formulations designed to hit a target data-fitting error level provided by the user, and propose an algorithm called LR-BPDN that is able to exploit factorized formulations to solve the corresponding optimization problem. Since practitioners typically have strong prior knowledge about target error level, this inno…
Adaptive double-phase Rudin--Osher--Fatemi denoising model
Faculty of Mathematics, Universität Wien（维也纳大学数学系）；Faculty of Mathematics, Informatics and Mechanics, University of Warsaw（华沙大学数学、信息学与力学系）；Institute of Mathematics of the Polish Academy of Sciences（波兰科学院数学研究所）；Department of Mathematics, School of Applied Mathematical and Physical Sciences, National Technical University of Athens（雅典技术大学应用数学与物理科学学院数学系）
HIR-Diff: Unsupervised Hyperspectral Image Restoration Via Improved Diffusion Models
School of Computer Science and Technology, Xi’an Jiaotong University（西安交通大学计算机科学与技术学院）；School of Mathematics and Statistics, Xi’an Jiaotong University（西安交通大学数学与统计学院）
We develop a dictionary learning algorithm by minimizing the $\ell_1$ distortion metric on the data term, which is known to be robust for non-Gaussian noise contamination. The proposed algorithm exploits the idea of iterative minimization of weighted $\ell_2$ error. We refer to this algorithm as $\ell_1$-K-SVD, where the dictionary atoms and the corresponding sparse coefficients are simultaneously updated to minimize the $\ell_1$ objective, resulting in noise-robustness. We demonstrate through experiments that the $\ell_1$-K-SVD algorithm results in higher atom recovery rate compared with the …
Analysis and Synthesis Denoisers for Forward-Backward Plug-and-Play Algorithms
Laboratoire Interdisciplinaire des Sciences du Numériques, Inria, Université Paris-Saclay, CNRS, Gif-sur-Yvette, France（跨学科数字科学实验室，法国国家信息与自动化研究所，巴黎-萨克雷大学，法国国家科学研究中心，法国吉夫-sur-伊夫特）；Inria, Université Paris-Saclay, CEA, Palaiseau, France（法国国家信息与自动化研究所，巴黎-萨克雷大学，法国原子能委员会，法国帕莱索）；School of Mathematics and Computer Sciences and School of Engineering and Physical Sciences, Heriot-Watt University, Edinburgh, UK（数学与计算机科学学院和工程与物理科学学院，赫里奥特-瓦特大学，英国爱丁堡）；Maxwell Institute for Mathematical Sciences, Edinburgh, UK（数学科学麦克斯韦研究所，英国爱丁堡）
As quotidian use of sophisticated cameras surges, people in modern society are more interested in capturing fine-quality images. However, the quality of the images might be inferior to people's expectations due to the noise contamination in the images. Thus, filtering out the noise while preserving vital image features is an essential requirement. Current existing denoising methods have their own assumptions on the probability distribution in which the contaminated noise is sampled for the method to attain its expected denoising performance. In this paper, we utilize our recent Gramian-based f…
Optical image data have been used by the Remote Sensing workforce to study land use and cover since such data is easily interpretable. Synthetic Aperture Radar (SAR) has the characteristic of obtaining images during all-day, all-weather and provides object information that is different from visible and infrared sensors. However, SAR images have more speckle noise and fewer dimensions. This paper presents a method for denoising, feature extraction and compares classifications of Optical and SAR images. The image was denoised using K-Singular Value Decomposition (K-SVD) algorithm. A method to ma…
University of Cambridge（剑桥大学）；University of Liverpool（利物浦大学）；Graz University of Technology（格拉茨理工大学）；AIT Austrian Institute of Technology GmbH（奥地利技术研究所股份公司）
We propose several variants of the primal-dual method due to Chambolle and Pock. Without requiring full strong convexity of the objective functions, our methods are accelerated on subspaces with strong convexity. This yields mixed rates, $O(1/N^2)$ with respect to initialisation and $O(1/N)$ with respect to the dual sequence, and the residual part of the primal sequence. We demonstrate the efficacy of the proposed methods on image processing problems lacking strong convexity, such as total generalised variation denoising and total variation deblurring.
Indian Institute of Information Technology Vadodara（瓦多达拉印度信息技术学院）
Medical image acquisition is often intervented by unwanted noise that corrupts the information content. This paper introduces an unsupervised medical image denoising technique that learns noise characteristics from the available images and constructs denoised images. It comprises of two blocks of data processing, viz., patch-based dictionaries that indirectly learn the noise and residual learning (RL) that directly learns the noise. The model is generalized to account for both 2D and 3D images considering different medical imaging instruments. The images are considered one-by-one from the stac…
