Low rank matrix approximation (LRMA) has drawn increasing attention in recent years, due to its wide range of applications in computer vision and machine learning. However, LRMA, achieved by nuclear norm minimization (NNM), tends to over-shrink the rank components with the same threshold and ignore the differences between rank components. To address this problem, we propose a flexible and precise model named multi-band weighted $l_p$ norm minimization (MBWPNM). The proposed MBWPNM not only gives more accurate approximation with a Schatten $p$-norm, but also considers the prior knowledge where different rank components have different importance. We analyze the solution of MBWPNM and prove that MBWPNM is equivalent to a non-convex $l_p$ norm subproblems under certain weight condition, whose global optimum can be solved by a generalized soft-thresholding algorithm. We then adopt the MBWPNM algorithm to color and multispectral image denoising. Extensive experiments on additive white Gaussian noise removal and realistic noise removal demonstrate that the proposed MBWPNM achieves a better performance than several state-of-art algorithms.
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 …
Nonlocal image representation or group sparsity has attracted considerable interest in various low-level vision tasks and has led to several state-of-the-art image denoising techniques, such as BM3D, LSSC. In the past, convex optimization with sparsity-promoting convex regularization was usually regarded as a standard scheme for estimating sparse signals in noise. However, using convex regularization can not still obtain the correct sparsity solution under some practical problems including image inverse problems. In this paper we propose a non-convex weighted $\ell_p$ minimization based group …
Gachon University（嘉泉大学）；Hanyang University (ERICA)（汉阳大学（ERICA））
Recently, the application of low rank minimization to image denoising has shown remarkable denoising results which are equivalent or better than those of the existing state-of-the-art algorithms. However, due to iterative nature of low rank optimization, estimation of residual noise is an essential requirement after each iteration. Currently, this noise is estimated by using the filtered noise in the previous iteration without considering the geometric structure of the given image. This estimate may be affected in the presence of moderate and severe levels of noise. To obtain a more reliable e…
The Hong Kong Polytechnic University（香港理工大学）；Xidian University（西安电子科技大学）
Most of the existing denoising algorithms are developed for grayscale images, while it is not a trivial work to extend them for color image denoising because the noise statistics in R, G, B channels can be very different for real noisy images. In this paper, we propose a multi-channel (MC) optimization model for real color image denoising under the weighted nuclear norm minimization (WNNM) framework. We concatenate the RGB patches to make use of the channel redundancy, and introduce a weight matrix to balance the data fidelity of the three channels in consideration of their different noise sta…
The Hong Kong Polytechnic University（香港理工大学）；Institute of Automation, Chinese Academy of Sciences（中国科学院自动化研究所）；Harbin Institute of Technology（哈尔滨工业大学）
Low rank matrix approximation (LRMA), which aims to recover the underlying low rank matrix from its degraded observation, has a wide range of applications in computer vision. The latest LRMA methods resort to using the nuclear norm minimization (NNM) as a convex relaxation of the nonconvex rank minimization. However, NNM tends to over-shrink the rank components and treats the different rank components equally, limiting its flexibility in practical applications. We propose a more flexible model, namely the Weighted Schatten $p$-Norm Minimization (WSNM), to generalize the NNM to the Schatten $p$…
This paper presents a patch-wise low-rank based image denoising method with constrained variational model involving local and nonlocal regularization. On one hand, recent patch-wise methods can be represented as a low-rank matrix approximation problem whose convex relaxation usually depends on nuclear norm minimization (NNM). Here, we extend the NNM to the nonconvex schatten p-norm minimization with additional weights assigned to different singular values, which is referred to as the Weighted Schatten p-Norm Minimization (WSNM). An efficient algorithm is also proposed to solve the WSNM problem…
École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）
The development of neural networks has greatly improved the performance in various computer vision tasks. In the filed of image denoising, convolutional neural network based methods such as DnCNN break through the limits of classical methods, achieving better quantitative results. However, the epistemic uncertainty existing in neural networks limits further improvements in their performance over denoising tasks. Therefore, we develop and study different solutions to minimize uncertainty and further improve the removal of noise. From the perspective of ensemble learning, we implement manipulati…
Western Norway University of Applied Sciences（挪威西部应用科技大学）；University of Bergen（卑尔根大学）
The paper presents a fully coupled TV-Stokes model, and propose an algorithm based on alternating minimization of the objective functional whose first iteration is exactly the modified TV-Stokes model proposed earlier. The model is a generalization of the second order Total Generalized Variation model. A convergence analysis is given.
In image denoising (IDN) processing, the low-rank property is usually considered as an important image prior. As a convex relaxation approximation of low rank, nuclear norm based algorithms and their variants have attracted significant attention. These algorithms can be collectively called image domain based methods, whose common drawback is the requirement of great number of iterations for some acceptable solution. Meanwhile, the sparsity of images in a certain transform domain has also been exploited in image denoising problems. Sparsity transform learning algorithms can achieve extremely fa…
Color image denoising is frequently encountered in various image processing and computer vision tasks. One traditional strategy is to convert the RGB image to a less correlated color space and denoise each channel of the new space separately. However, such a strategy can not fully exploit the correlated information between channels and is inadequate to obtain satisfactory results. To address this issue, this paper proposes a new multi-channel optimization model for color image denoising under the nuclear norm minus Frobenius norm minimization framework. Specifically, based on the block-matchin…
University of Electronic Science and Technology of China（电子科技大学）；Nanyang Technological University（南洋理工大学）；Nanjing Institute of Technology（南京工程学院）；University of Macau（澳门大学）
The nuclear norm minimization (NNM) is commonly used to approximate the matrix rank by shrinking all singular values equally. However, the singular values have clear physical meanings in many practical problems, and NNM may not be able to faithfully approximate the matrix rank. To alleviate the above-mentioned limitation of NNM, recent studies have suggested that the weighted nuclear norm minimization (WNNM) can achieve a better rank estimation than NNM, which heuristically set the weight being inverse to the singular values. However, it still lacks a rigorous explanation why WNNM is more effe…
Hyperspectral images (HSIs) usually suffer from different types of pollution. This severely reduces the quality of HSIs and limits the accuracy of subsequent processing tasks. HSI denoising can be modeled as a low-rank tensor denoising problem. Tensor nuclear norm (TNN) induced by tensor singular value decomposition plays an important role in this problem. In this letter, we first reconsider three inconspicuous but crucial phenomenons in TNN. In the Fourier transform domain of HSIs, different frequency slices (FS) contain different information; different singular values (SVs) of each FS also r…
Yonsei University（延世大学）；Chungnam National University（忠南国立大学）
Regularization-based image restoration has remained an active research topic in computer vision and image processing. It often leverages a guidance signal captured in different fields as an additional cue. In this work, we present a general framework for image restoration, called deeply aggregated alternating minimization (DeepAM). We propose to train deep neural network to advance two of the steps in the conventional AM algorithm: proximal mapping and ?- continuation. Both steps are learned from a large dataset in an end-to-end manner. The proposed framework enables the convolutional neural n…
Hangzhou Dianzi University（杭州电子科技大学）；Tsinghua Shenzhen International Graduate School, Tsinghua University（清华大学深圳国际研究生院）；Tsinghua University（清华大学）；University of Science and Technology of China（中国科学技术大学）
The depth images denoising are increasingly becoming the hot research topic nowadays because they reflect the three-dimensional (3D) scene and can be applied in various fields of computer vision. But the depth images obtained from depth camera usually contain stains such as noise, which greatly impairs the performance of depth related applications. In this paper, considering that group-based image restoration methods are more effective in gathering the similarity among patches, a group based nuclear norm and learning graph (GNNLG) model was proposed. For each patch, we find and group the most …
In this paper, we propose a novel convolutional neural network (CNN) for image denoising, which uses exponential linear unit (ELU) as the activation function. We investigate the suitability by analyzing ELU's connection with trainable nonlinear reaction diffusion model (TNRD) and residual denoising. On the other hand, batch normalization (BN) is indispensable for residual denoising and convergence purpose. However, direct stacking of BN and ELU degrades the performance of CNN. To mitigate this issue, we design an innovative combination of activation layer and normalization layer to exploit and…
University of Goettingen（哥廷根大学）；Statistical and Applied Mathematical Science Institute（统计与应用数学科学研究所）
We consider the very challenging task of restoring images (i) which have a large number of missing pixels, (ii) whose existing pixels are corrupted by noise and (iii) the ideal image to be restored contains both cartoon and texture elements. The combination of these three properties makes this inverse problem a very difficult one. The solution proposed in this manuscript is based on directional global three-part decomposition (DG3PD) [ThaiGottschlich2016] with directional total variation norm, directional G-norm and $\ell_\infty$-norm in curvelet domain as key ingredients of the model. Image d…
Quaternion Nuclear Norm minus Frobenius Norm Minimization for color image reconstruction
School of Mathematical Science, Inner Mongolia University（内蒙古大学数学科学学院）；The Chinese University of Hong Kong（香港中文大学）；Hong Kong Baptist University（香港浸会大学）
The Technion - Israel Institute of Technology（以色列理工学院）
Image denoising is a well-known and well studied problem, commonly targeting a minimization of the mean squared error (MSE) between the outcome and the original image. Unfortunately, especially for severe noise levels, such Minimum MSE (MMSE) solutions may lead to blurry output images. In this work we propose a novel stochastic denoising approach that produces viable and high perceptual quality results, while maintaining a small MSE. Our method employs Langevin dynamics that relies on a repeated application of any given MMSE denoiser, obtaining the reconstructed image by effectively sampling f…
University of California at San Diego（加州大学圣地亚哥分校）；Purdue University（普渡大学）
We propose a data-dependent denoising procedure to restore noisy images. Different from existing denoising algorithms which search for patches from either the noisy image or a generic database, the new algorithm finds patches from a database that contains only relevant patches. We formulate the denoising problem as an optimal filter design problem and make two contributions. First, we determine the basis function of the denoising filter by solving a group sparsity minimization problem. The optimization formulation generalizes existing denoising algorithms and offers systematic analysis of the …
Low-rankness is important in the hyperspectral image (HSI) denoising tasks. The tensor nuclear norm (TNN), defined based on the tensor singular value decomposition, is a state-of-the-art method to describe the low-rankness of HSI. However, TNN ignores some physical meanings of HSI in tackling denoising tasks, leading to suboptimal denoising performance. In this paper, we propose the multi-modal and frequency-weighted tensor nuclear norm (MFWTNN) and the non-convex MFWTNN for HSI denoising tasks. Firstly, we investigate the physical meaning of frequency slices and reconsider their weights to im…
Total Variation, Adaptive Total Variation and Nonconvex Smoothly Clipped Absolute Deviation Penalty for Denoising Blocky Images
School of Computing Sciences, VIT University, Vellore, TN, India（计算机科学学院，VIT大学，Vellore，印度）；Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, 637371（数学科学系，南洋理工大学，新加坡）
University of Electronic Science and Technology of China（中国电子科技大学）
Recovering a large matrix from limited measurements is a challenging task arising in many real applications, such as image inpainting, compressive sensing and medical imaging, and this kind of problems are mostly formulated as low-rank matrix approximation problems. Due to the rank operator being non-convex and discontinuous, most of the recent theoretical studies use the nuclear norm as a convex relaxation and the low-rank matrix recovery problem is solved through minimization of the nuclear norm regularized problem. However, a major limitation of nuclear norm minimization is that all the sin…
The Hong Kong Polytechnic University（香港理工大学）；Harbin Institute of Technology（哈尔滨工业大学）；Xidian University（西安电子科技大学）
In recent years, the nuclear norm minimization (NNM) problem has been attracting much attention in computer vision and machine learning. The NNM problem is capitalized on its convexity and it can be solved efficiently. The standard nuclear norm regularizes all singular values equally, which is however not flexible enough to fit real scenarios. Weighted nuclear norm minimization (WNNM) is a natural extension and generalization of NNM. By assigning properly different weights to different singular values, WNNM can lead to state-of-the-art results in applications such as image denoising. Neverthel…
Centrum Wiskunde & Informatica（荷兰国家数学与计算机科学研究中心）；Leiden Institute of Advanced Computer Science, Leiden Universiteit（莱顿大学高级计算机科学研究所）
Recovering a high-quality image from noisy indirect measurements is an important problem with many applications. For such inverse problems, supervised deep convolutional neural network (CNN)-based denoising methods have shown strong results, but the success of these supervised methods critically depends on the availability of a high-quality training dataset of similar measurements. For image denoising, methods are available that enable training without a separate training dataset by assuming that the noise in two different pixels is uncorrelated. However, this assumption does not hold for inve…
Non-local low-rank tensor approximation has been developed as a state-of-the-art method for hyperspectral image (HSI) denoising. Unfortunately, with more spectral bands for HSI, while the running time of these methods significantly increases, their denoising performance benefits little. In this paper, we claim that the HSI underlines a global spectral low-rank subspace, and the spectral subspaces of each full band patch groups should underlie this global low-rank subspace. This motivates us to propose a unified spatial-spectral paradigm for HSI denoising. As the new model is hard to optimize, …
Tensor completion is a technique of filling missing elements of the incomplete data tensors. It being actively studied based on the convex optimization scheme such as nuclear-norm minimization. When given data tensors include some noises, the nuclear-norm minimization problem is usually converted to the nuclear-norm `regularization' problem which simultaneously minimize penalty and error terms with some trade-off parameter. However, the good value of trade-off is not easily determined because of the difference of two units and the data dependence. In the sense of trade-off tuning, the noisy te…
Friedrich-Alexander-Universität Erlangen-Nürnberg（埃尔朗根-纽伦堡大学）；Massachusetts Institute of Technology（麻省理工学院）
Optical coherence tomography (OCT) enables high-resolution and non-invasive 3D imaging of the human retina but is inherently impaired by speckle noise. This paper introduces a spatio-temporal denoising algorithm for OCT data on a B-scan level using a novel quantile sparse image (QuaSI) prior. To remove speckle noise while preserving image structures of diagnostic relevance, we implement our QuaSI prior via median filter regularization coupled with a Huber data fidelity model in a variational approach. For efficient energy minimization, we develop an alternating direction method of multipliers …
Nanjing University（南京大学）；Nokia Bell Labs（诺基亚贝尔实验室）；University of Oulu（奥卢大学）；University of Electronic Science and Technology of China（电子科技大学）
Rank minimization methods have attracted considerable interest in various areas, such as computer vision and machine learning. The most representative work is nuclear norm minimization (NNM), which can recover the matrix rank exactly under some restricted and theoretical guarantee conditions. However, for many real applications, NNM is not able to approximate the matrix rank accurately, since it often tends to over-shrink the rank components. To rectify the weakness of NNM, recent advances have shown that weighted nuclear norm minimization (WNNM) can achieve a better matrix rank approximation …
INRIA（法国国家信息与自动化研究所）；Instituto de Ingeniería del Conocimiento（知识工程研究所）；Universidad Autónoma de Madrid（马德里自治大学）；Massachusetts Institute of Technology（麻省理工学院）
Energy minimization has been an intensely studied core problem in computer vision. With growing image sizes (2D and 3D), it is now highly desirable to run energy minimization algorithms in parallel. But many existing algorithms, in particular, some efficient combinatorial algorithms, are difficult to par-allelize. By exploiting results from convex and submodular theory, we reformulate the quadratic energy minimization problem as a total variation denoising problem, which, when viewed geometrically, enables the use of projection and reflection based convex methods. The resulting min-cut algorit…
The robust principal component analysis (RPCA), which aims to estimate underlying low-rank and sparse structures from the degraded observation data, has found wide applications in computer vision. It is usually replaced by the principal component pursuit (PCP) model in order to pursue the convex property, leading to the undesirable overshrink problem. In this paper, we propose a dual weighted lp-norm (DWLP) model with a more reasonable weighting rule and weaker powers, which greatly generalizes the previous work and provides a better approximation to the rank minimization problem for original …
DiffusionAD: Norm-guided One-step Denoising Diffusion for Anomaly Detection
Institute of Trustworthy Embodied AI, Fudan University（复旦大学可信具身人工智能研究院）；School of Computer Science, Zhejiang University of Technology（浙江工业大学计算机学院）；School of Communication & Information Engineering, Shanghai University（上海大学通信与信息工程学院）
Institute of Computing Technology, Chinese Academy of Sciences（中国科学院计算技术研究所）；University of Chinese Academy of Sciences（中国科学院大学）；Peng Cheng Laboratory（鹏城实验室）；Huawei Technologies（华为技术有限公司）
Due to the domain discrepancy in visual domain adaptation, the performance of source model degrades when bumping into the high data density near decision boundary in target domain. A common solution is to minimize the Shannon Entropy to push the decision boundary away from the high density area. However, entropy minimization also leads to severe reduction of prediction diversity, and unfortunately brings harm to the domain adaptation. In this paper, we investigate the prediction discriminability and diversity by studying the structure of the classification output matrix of a randomly selected …
Polytechnic Institute of New York University（纽约大学理工学院）
This paper addresses signal denoising when large-amplitude coefficients form clusters (groups). The L1-norm and other separable sparsity models do not capture the tendency of coefficients to cluster (group sparsity). This work develops an algorithm, called 'overlapping group shrinkage' (OGS), based on the minimization of a convex cost function involving a group-sparsity promoting penalty function. The groups are fully overlapping so the denoising method is translation-invariant and blocking artifacts are avoided. Based on the principle of majorization-minimization (MM), we derive a simple iter…
This paper presents a tensor multiplication based smoothing algorithm that follows a two step denoising method. Unlike other traditional averaging approaches, our approach uses an element based normal voting tensor to compute smooth surfaces. By introducing a binary optimization on the proposed tensor together with a local binary neighborhood concept, our algorithm better retains sharp features and produces smoother umbilical regions than previous approaches. On top of that, we provide a stochastic analysis on the different kinds of noise based on the average edge length. The quantitative and …
Eindhoven University of Technology（埃因霍温理工大学）；Philips Research（飞利浦研究院）
Ultrasonography offers an inexpensive, widely-accessible and compact medical imaging solution. However, compared to other imaging modalities such as CT and MRI, ultrasound images notoriously suffer from strong speckle noise, which originates from the random interference of sub-wavelength scattering. This deteriorates ultrasound image quality and makes interpretation challenging. We here propose a new unsupervised ultrasound speckle reduction and image denoising method based on maximum-a-posteriori estimation with deep generative priors that are learned from high-quality MRI images. To model th…
It is an efficient and effective strategy to utilize the nuclear norm approximation to learn low-rank matrices, which arise frequently in machine learning and computer vision. So the exploration of nuclear norm minimization problems is gaining much attention recently. In this paper we shall prove that the following Low-Rank Representation (LRR) \cite{icml_2010_lrr,lrr_extention} problem: {eqnarray*} \min_{Z} \norm{Z}_*, & {s.t.,} & X=AZ, {eqnarray*} has a unique and closed-form solution, where $X$ and $A$ are given matrices. The proof is based on proving a lemma that allows us to get closed-fo…
The University of Adelaide（阿德莱德大学）；Nanjing University of Science and Technology（南京理工大学）
Minimization of the $L_\infty$ norm, which can be viewed as approximately solving the non-convex least median estimation problem, is a powerful method for outlier removal and hence robust regression. However, current techniques for solving the problem at the heart of $L_\infty$ norm minimization are slow, and therefore cannot scale to large problems. A new method for the minimization of the $L_\infty$ norm is presented here, which provides a speedup of multiple orders of magnitude for data with high dimension. This method, termed Fast $L_\infty$ Minimization, allows robust regression to be app…
The explosive rise of the use of Computer tomography (CT) imaging in medical practice has heightened public concern over the patient's associated radiation dose. However, reducing the radiation dose leads to increased noise and artifacts, which adversely degrades the scan's interpretability. Consequently, an advanced image reconstruction algorithm to improve the diagnostic performance of low dose ct arose as the primary concern among the researchers, which is challenging due to the ill-posedness of the problem. In recent times, the deep learning-based technique has emerged as a dominant method…
The Chinese University of Hong Kong（香港中文大学）；Shenzhen Research Institute of Big Data, the Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳）深圳大数据研究院）；University of Minnesota（明尼苏达大学）；Peking University（北京大学）；Shanghai Jiao Tong University（上海交通大学）
The heavy-tailed distributions of corrupted outliers and singular values of all channels in low-level vision have proven effective priors for many applications such as background modeling, photometric stereo and image alignment. And they can be well modeled by a hyper-Laplacian. However, the use of such distributions generally leads to challenging non-convex, non-smooth and non-Lipschitz problems, and makes existing algorithms very slow for large-scale applications. Together with the analytic solutions to lp-norm minimization with two specific values of p, i.e., p=1/2 and p=2/3, we propose two…
