Weight decay (WD) is a fundamental and practical regularization technique in improving generalization of current deep learning models. However, it is observed that the WD does not work effectively for an adaptive optimization algorithm (such as Adam), as it works for SGD. Specifically, the solution found by Adam with the WD often generalizes unsatisfactorily. Though efforts have been made to mitigate this issue, the reason for such deficiency is still vague. In this article, we first show that when using the Adam optimizer, the weight norm increases very fast along with the training procedure, which is in contrast to SGD where the weight norm increases relatively slower and tends to converge. The fast increase of weight norm is adverse to WD; in consequence, the Adam optimizer will lose efficacy in finding solution that generalizes well. To resolve this problem, we propose to tailor Adam by introducing a regularization term on the adaptive learning rate, such that it is friendly to WD. Meanwhile, we introduce first moment on the WD to further enhance the regularization effect. We show that the proposed method is able to find solution with small norm and generalizes better than SGD. We test the proposed method on general image classification and fine-grained image classification tasks with different networks. Experimental results on all these cases substantiate the effectiveness of the proposed method in help improving the generalization. Specifically, the proposed method improves the test accuracy of Adam by a large margin and even improves the performance of SGD by 0.84% on CIFAR 10 and 1.03 % on CIFAR 100 with ResNet-50. The code of this article is public available at xxx.

Low-rank matrix factorization (LRMF) is a canonical problem in non-convex optimization, the objective function to be minimized is non-convex and even non-smooth, which makes the global convergence guarantee of gradient-based algorithm quite challenging. Recent work made a breakthrough on proving that standard gradient descent converges to the $\varepsilon$-global minima after $O( \frac{d \kappa^2}{\tau^2} {\rm ln} \frac{d \sigma_d}{\tau} + \frac{d \kappa^2}{\tau^2} {\rm ln} \frac{\sigma_d}{\varepsilon})$ iterations from small initialization with a very small learning rate (both are related to the small constant $\tau$). While the dependence of the convergence on the \textit{condition number} $\kappa$ and \textit{small learning rate} makes it not practical especially for ill-conditioned LRMF problem. In this paper, we show that precondition helps in accelerating the convergence and prove that the scaled gradient descent (ScaledGD) and its variant, alternating scaled gradient descent (AltScaledGD) converge to an $\varepsilon$-global minima after $O( {\rm ln} \frac{d}{\delta} + {\rm ln} \frac{d}{\varepsilon})$ iterations from general random initialization. Meanwhile, for small initialization as in gradient descent, both ScaledGD and AltScaledGD converge to $\varepsilon$-global minima after only $O({\rm ln} \frac{d}{\varepsilon})$ iterations. Furthermore, we prove that as a proximity to the alternating minimization, AltScaledGD converges faster than ScaledGD, its global convergence does not rely on small learning rate and small initialization, which certificates the advantages of AltScaledGD in LRMF.

Nonlocal low-rank (LR) hyperspectral image (HSI) denoising approaches have gained a lot of attention because of their capacity to fully take advantage of spectral correlation and nonlocal self-similarity (NLSS). Most of the existing LR-tensor-based approaches use tensor singular value decomposition (t-SVD). However, fixed discrete Fourier transform-based t-SVD may compromise the low rank structure. Additionally, these approaches restricts flexibility in dealing with HSI data because it treats the singular values of each frontal slice equally. To overcome these issues, we propose a method using weighted nuclear norms of transformed tensors (WNNTT) for nonlocal HSI denoising. Our approach exploits the low-rankness in both the spectral and NLSS dimensions. We also compare our proposed method with other t-SVD based LR-tensor regularization methods under the same framework. Our experiments show that our WNNTT approach outperforms several state-of-the-art nonlocal t-SVD based methods on open HSI data.

The deep unfolding network (DUN) provides an efficient framework for image restoration. It consists of a regularization module and a data fitting module. In existing DUN models, it is common to directly use a deep convolution neural network (DCNN) as the regularization module, and perform data fitting before regularization in each iteration/stage. In this work, we present a DUN by incorporating a new regularization module, and putting the regularization module before the data fitting module. The proposed regularization model is deducted by using the regularization by denoing (RED) and plugging in it a newly designed DCNN. For the data fitting module, we use the closed-form solution with Faster Fourier Transform (FFT). The resulted DRED-DUN model has some major advantages. First, the regularization model inherits the flexibility of learned image-adaptive and interpretability of RED. Second, the DRED-DUN model is an end-to-end trainable DUN, which learns the regularization network and other parameters jointly, thus leads to better restoration performance than the plug-and-play framework. Third, extensive experiments show that, our proposed model significantly outperforms the-state-of-the-art model-based methods and learning based methods in terms of PSNR indexes as well as the visual effects. In particular, our method has much better capability in recovering salient image components such as edges and small scale textures.

Spectral regularization is a widely used approach for low-rank matrix recovery (LRMR) by regularizing matrix singular values. Most of the existing LRMR solvers iteratively compute the singular values via applying singular value decomposition (SVD) on a dense matrix, which is computationally expensive and severely limits their applications to large-scale problems. To address this issue, we present a generalized unitarily invariant gauge (GUIG) function for LRMR. The proposed GUIG function does not act on the singular values; however, we show that it generalizes the well-known spectral functions, including the rank function, the Schatten- p quasi-norm, and logsum of singular values. The proposed GUIG regularization model can be formulated as a bilinear variational problem, which can be efficiently solved without computing SVD. Such a property makes it well suited for large-scale LRMR problems. We apply the proposed GUIG model to matrix completion and robust principal component analysis and prove the convergence of the algorithms. Experimental results demonstrate that the proposed GUIG method is not only more accurate but also much faster than the state-of-the-art algorithms, especially on large-scale problems.

In this study, the authors propose a fractional derivative‐based image decomposition and denoising model which decomposes the image into the cartoon component (the component formed by homogeneous regions and with sharp boundaries) and the texture (or noise) component. The cartoon component is modelled by a function of the fractional‐order total bounded variation, while the texture component is modelled by an oscillatory function, bounded in the negative Sobolev space norm. The authors give the corresponding minimisation functional, after some transformations, and then the resulting fractional‐order partial differential equation can be solved using the Fourier transform. By symmetry and asymmetry of the fractional‐order derivative, some generalisations and variants of the proposed model are also introduced. Finally, the authors implement the algorithm by the fractional‐order finite difference in the frequency‐domain. The experimental results demonstrate that the proposed models make objective and visual improvements compared with other standard approaches in the task of decomposition and denoising.

Deep convolutional neural networks (DCNN) have been successfully used in many low-level vision problems such as image denoising. Recent studies on the mathematical foundation of DCNN has revealed that the forward propagation of DCNN corresponds to a dynamic system, which can be described by an ordinary differential equation (ODE) and solved by the optimal control method. However, most of these methods employ integer-order differential equation, which has local connectivity in time space and cannot describe the long-term memory of the system. Inspired by the fact that the fractional-order differential equation has long-term memory, in this paper we develop an advanced image denoising network, namely FOCNet, by solving a fractional optimal control (FOC) problem. Specifically, the network structure is designed based on the discretization of a fractional-order differential equation, which enjoys long-term memory in both forward and backward passes. Besides, multi-scale feature interactions are introduced into the FOCNet to strengthen the control of the dynamic system. Extensive experiments demonstrate the leading performance of the proposed FOCNet on image denoising. Code will be made available.

In the fusion process of medical computed tomography (CT) and magnetic resonance image (MRI), traditional multiscale methods often reduce the contrast of fused images. Although sparse representation (SR) methods overcome this shortcoming, they are often too smooth along the strong edges of the fusion image. To overcome these shortcomings, CT and MRI image fusion based on multiscale decomposition method and hybrid approach is proposed. There are three main steps. First, the cartoon parts and texture parts of CT and MRI are obtained by the improved image decomposition method using global sparse gradients. Second, the large structure cartoon parts are fused using the specific cartoon dictionary and the ‘L1‐max norm’ principle. The textured parts are fused using non‐subsampled contourlet transformation (NSCT) and the maximum energy rule. Finally, the final result is obtained by superimposing the fused cartoon part and the fused texture part. The experimental results demonstrate that the proposed method outperforms the state‐of‐the‐art method SR and NSCT in terms of visual effect and objective quality.

Subspace segmentation aims to segment a given data set into clusters with each cluster corresponding to a subspace. Most recent works focus on subspace representation-based methods, which construct the affinity matrix based on the subspace representation of the data points. Ideally, the affinity matrix should be inter-cluster sparse and intra-cluster uniform. The inter-cluster sparsity guarantees segmenting data into different subspaces from which they are originally drawn and the intra-cluster uniformity encourages clustering highly correlated data together. Most previous methods partly satisfy these properties and cannot obtain ideal results. To satisfy both properties, we propose an explicit data correlation adaptive regression model for the subspace representation. The proposed model essentially uses l2-norm on the coefficients of highly correlated data points while l1-norm on that of less correlated data points. The l2-norm tends to enforce the coefficients corresponding to highly correlated data have the grouping effect, while the l1-norm tends to enforce the coefficients corresponding to uncorrelated data to be zero. So, the proposed model can ensure the affinity matrix have two attractive properties: inter-subspace sparsity and intra-cluster uniformity. Experimental results on several commonly used clustering data sets show that our method performs better than the state-of-the-art methods.

Under the umbrella of tensor algebra, this paper proposes a new sparse-coding-based classifier (SCC) for hyperspectral imagery classification (HIC). By utilizing the tensor forms of hyperspectral pixels, we advance a tensor sparse-coding model which preserves as many original spatial constraints of a pixel and its spatial neighbors as possible. Furthermore, to alleviate the classification uncertainty resulted from widely existing mixed pixels, this paper constructs a regularization term for maximizing the likelihood of sparse-coding tensor defined on the posterior class probability. By combining the tensor sparse coding with maximizing likelihood estimation, a hybrid probabilistic SCC with spatial neighbor tensor (HPSCC-SNT) is proposed, which makes the pixels be well represented by the training pixels belonging to the same class. The performance of HPSCC-SNT is evaluated on three real hyperspectral imagery data sets, and the results show that it can achieve accurate and robust HIC results, and outperforms the state-of-the-art methods.

The ubiquitous large, complex and high dimensional datasets in computer vision and machine learning generates the problem of subspace clustering, which aims to partition the data into several low dimensional subspaces. By utilizing relatively limited labeled data and sufficient unlabeled data, the semi-supervised subspace clustering is more effective, practical and become more popular. In this work, we present a new regularity combing the labels and the affinity to ensure the coherence of the affinity between data points from the same subspace as well as the discrimination of cluster labels for data points from different subspaces. We combine it with the manifold smoothing term of the existing methods and the Gaussian fields and harmonic functions method to give a new unified optimization framework for semi-supervised subspace clustering. Analysis shows the proposed model fully combines the affinity and the labels to guide each other so that both are discriminative between clusters and coherent within clusters. Extensive experiments show that our method outperforms the existing state-of-the-art methods, thus suggests that the property of discriminative between clusters and coherent within clusters of our method is advantageous to semi-supervised subspace clustering.

Most of the existing denoising algorithms are developed for grayscale images. It is not trivial to extend them for color image denoising since the noise statistics in R, G, and 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 statistics. The proposed MC-WNNM model does not have an analytical solution. We reformulate it into a linear equality-constrained problem and solve it via alternating direction method of multipliers. Each alternative updating step has a closed-form solution and the convergence can be guaranteed. Experiments on both synthetic and real noisy image datasets demonstrate the superiority of the proposed MC-WNNM over state-of-the-art denoising methods.

Cross-domain visual data matching is one of the fundamental problems in many real-world vision tasks, e.g., matching persons across ID photos and surveillance videos. Conventional approaches to this problem usually involves two steps: i) projecting samples from different domains into a common space, and ii) computing (dis-)similarity in this space based on a certain distance. In this paper, we present a novel pairwise similarity measure that advances existing models by i) expanding traditional linear projections into affine transformations and ii) fusing affine Mahalanobis distance and Cosine similarity by a data-driven combination. Moreover, we unify our similarity measure with feature representation learning via deep convolutional neural networks. Specifically, we incorporate the similarity measure matrix into the deep architecture, enabling an end-to-end way of model optimization. We extensively evaluate our generalized similarity model in several challenging cross-domain matching tasks: person re-identification under different views and face verification over different modalities (i.e., faces from still images and videos, older and younger faces, and sketch and photo portraits). The experimental results demonstrate superior performance of our model over other state-of-the-art methods.

Conventional representation based classifiers, ranging from the classical nearest neighbor classifier and nearest subspace classifier to the recently developed sparse representation based classifier (SRC) and collaborative representation based classifier (CRC), are essentially distance based classifiers. Though SRC and CRC have shown interesting classification results, their intrinsic classification mechanism remains unclear. In this paper we propose a probabilistic collaborative representation framework, where the probability that a test sample belongs to the collaborative subspace of all classes can be well defined and computed. Consequently, we present a probabilistic collaborative representation based classifier (ProCRC), which jointly maximizes the likelihood that a test sample belongs to each of the multiple classes. The final classification is performed by checking which class has the maximum likelihood. The proposed ProCRC has a clear probabilistic interpretation, and it shows superior performance to many popular classifiers, including SRC, CRC and SVM. Coupled with the CNN features, it also leads to state-of-the-art classification results on a variety of challenging visual datasets.

In this paper, we propose an adaptive regularizer learning method in the framework of MAP for low rank approximation (ARLLR). We assume that the prior distribution of the singular values is Laplacian with varying scale parameters. By using a full maximize a posterior (MAP) we learn the optimal scale parameters iteratively. We indicate that ARLLR is equivalent to low rank approximation regularized by Logarithm on singular values. In theory, we prove that ARLLR (Logarithm regularization) although being non-convex can be solved in closed form, and we further prove that local minimum can be easily obtained. Finally, ARLLR is applied to image de-noising. Experimental results show that the proposed method enhances image denoising compared with state-of-the-art image denoising algorithms(especially for BM3D, SAIST and WNNM) in both quantity value (PSNR) and visual quality.

In this study, the authors propose a new image representation model that fully exploits the similarity inherent in natural images. The idea is based on an observation of similar patches. For a clean image, when a cluster of similar patches are collected to form the similar patch matrix (SPM), there exists high correlation among columns/rows of the SPM. This implies that, when the column/row vectors are linearly expressed by a dictionary, their coefficients of columns/rows should have high correlation, which leads to the coefficient matrixes of column/row representation are low‐ranked. This observation inspires them to propose a novel image denoising model named bidirectional low rank representation (BiLRR) with cluster adaptive dictionary. Specifically, they use low rank penalties simultaneously on the coefficient matrixes of column and row representations to recover the correlation structure of the SPM. Meanwhile, a cluster adaptive dictionary is learned to represent each SPM so as to well preserve the fine structure of image. By applying variable splitting and penalty technique, they present an efficient alternative minimisation algorithm to solve the proposed BiLRR model. Experimental results indicate the authors’ method achieves a competitive denoising performance in comparison with state‐of‐the‐art algorithms in terms of subjective and objective qualities.

Most of the previous sparse coding (SC) based super resolution (SR) methods partition the image into overlapped patches, and process each patch separately. These methods, however, ignore the consistency of pixels in overlapped patches, which is a strong constraint for image reconstruction. In this paper, we propose a convolutional sparse coding (CSC) based SR (CSC-SR) method to address the consistency issue. Our CSC-SR involves three groups of parameters to be learned: (i) a set of filters to decompose the low resolution (LR) image into LR sparse feature maps, (ii) a mapping function to predict the high resolution (HR) feature maps from the LR ones, and (iii) a set of filters to reconstruct the HR images from the predicted HR feature maps via simple convolution operations. By working directly on the whole image, the proposed CSC-SR algorithm does not need to divide the image into overlapped patches, and can exploit the image global correlation to produce more robust reconstruction of image local structures. Experimental results clearly validate the advantages of CSC over patch based SC in SR application. Compared with state-of-the-art SR methods, the proposed CSC-SR method achieves highly competitive PSNR results, while demonstrating better edge and texture preservation performance.

Patch based image modeling has achieved a great success in low level vision such as image denoising. In particular, the use of image nonlocal self-similarity (NSS) prior, which refers to the fact that a local patch often has many nonlocal similar patches to it across the image, has significantly enhanced the denoising performance. However, in most existing methods only the NSS of input degraded image is exploited, while how to utilize the NSS of clean natural images is still an open problem. In this paper, we propose a patch group (PG) based NSS prior learning scheme to learn explicit NSS models from natural images for high performance denoising. PGs are extracted from training images by putting nonlocal similar patches into groups, and a PG based Gaussian Mixture Model (PG-GMM) learning algorithm is developed to learn the NSS prior. We demonstrate that, owe to the learned PG-GMM, a simple weighted sparse coding model, which has a closed-form solution, can be used to perform image denoising effectively, resulting in high PSNR measure, fast speed, and particularly the best visual quality among all competing methods.

As a convex relaxation of the low rank matrix factorization problem, the nuclear norm minimization has been attracting significant research interest in recent years. The standard nuclear norm minimization regularizes each singular value equally to pursue the convexity of the objective function. However, this greatly restricts its capability and flexibility in dealing with many practical problems (e.g., denoising), where the singular values have clear physical meanings and should be treated differently. In this paper we study the weighted nuclear norm minimization (WNNM) problem, where the singular values are assigned different weights. The solutions of the WNNM problem are analyzed under different weighting conditions. We then apply the proposed WNNM algorithm to image denoising by exploiting the image nonlocal self-similarity. Experimental results clearly show that the proposed WNNM algorithm outperforms many state-of-the-art denoising algorithms such as BM3D in terms of both quantitative measure and visual perception quality.

Due to the complexity of very high resolution (VHR) images and the inaccurate correspondence, change feature extraction is the key difficulty of VHR image change detection. In this letter, change field is proposed to represent the complex changes between VHR images. Change field measures the complex changes based on the displacements and the compensated distance. Based on change field, a novel change detection approach is proposed, where the inter-class variability is improved and the changed class and the unchanged class can be separated effectively. Experiments demonstrate the effectiveness of the proposed approach.

Blind image quality assessment (BIQA) aims to evaluate the perceptual quality of a distorted image without information regarding its reference image. Existing BIQA models usually predict the image quality by analyzing the image statistics in some transformed domain, e.g., in the discrete cosine transform domain or wavelet domain. Though great progress has been made in recent years, BIQA is still a very challenging task due to the lack of a reference image. Considering that image local contrast features convey important structural information that is closely related to image perceptual quality, we propose a novel BIQA model that utilizes the joint statistics of two types of commonly used local contrast features: 1) the gradient magnitude (GM) map and 2) the Laplacian of Gaussian (LOG) response. We employ an adaptive procedure to jointly normalize the GM and LOG features, and show that the joint statistics of normalized GM and LOG features have desirable properties for the BIQA task. The proposed model is extensively evaluated on three large-scale benchmark databases, and shown to deliver highly competitive performance with state-of-the-art BIQA models, as well as with some well-known full reference image quality assessment models.

How to measure the perceptual quality of natural images is an important problem in low level vision. It is known that the Mean Squared Error (MSE) is not an effective index to describe the perceptual fidelity of images. Numerous perceptual fidelity indices have been developed, while the representatives include the Structural SIMilarity (SSIM) index and its variants. However, most of those perceptual measures are nonlinear, and they cannot be easily dopted as an objective function to minimize in various low level vision tasks. Can MSE be perceptual fidelity aware after some minor adaptation? In this paper we propose a simple framework to enhance the perceptual fidelity awareness of MSE by introducing an l2-norm structural error term to it. Such a Structural MSE (SMSE) can lead to very competitive image quality assessment (IQA) results. More surprisingly, we show that by using certain structure extractors, SMSE can be further turned into a Gaussian smoothed MSE (i.e., the Euclidean distance between the original and distorted images after Gaussian smooth filtering), which is much simpler to calculate but achieves rather better IQA performance than SSIM. The so called Perceptual-fidelity Aware MSE (PAMSE) can have great potentials in applications such as perceptual image coding and perceptual image restoration.

In many sparse coding based image restoration and image classification problems, using non-convex Ip-norm minimization (0 ≤ p1-norm minimization. A number of algorithms, e.g., iteratively reweighted least squares (IRLS), iteratively thresholding method (ITM-Ip), and look-up table (LUT), have been proposed for non-convex Ip-norm sparse coding, while some analytic solutions have been suggested for some specific values of p. In this paper, by extending the popular soft-thresholding operator, we propose a generalized iterated shrinkage algorithm (GISA) for Ip-norm non-convex sparse coding. Unlike the analytic solutions, the proposed GISA algorithm is easy to implement, and can be adopted for solving non-convex sparse coding problems with arbitrary p values. Compared with LUT, GISA is more general and does not need to compute and store the look-up tables. Compared with IRLS and ITM-Ip, GISA is theoretically more solid and can achieve more accurate solutions. Experiments on image restoration and sparse coding based face recognition are conducted to validate the performance of GISA.

The objective image quality assessment aims to model the perceptual fidelity of semantic information between two images. In this letter, we assume that the semantic information of images is fully represented by edge-strength of each pixel and propose an edge-strength-similarity-based image quality metric (ESSIM). Through investigating the characteristics of the edge in images, we define the edge-strength to take both anisotropic regularity and irregularity of the edge into account. The proposed ESSIM is considerably simple, however, it can achieve slightly better performance than the state-of-the-art image quality metrics as evaluated on six subject-rated image databases.

Similarities inherent in natural images have been widely exploited for image denoising and other applications. In fact, if a cluster of similar image patches is rearranged into a matrix, similarities exist both between columns and rows. Using the similarities, we present a two-directional nonlocal (TDNL) variational model for image denoising. The solution of our model consists of three components: one component is a scaled version of the original observed image and the other two components are obtained by utilizing the similarities. Specifically, by using the similarity between columns, we get a nonlocal-means-like estimation of the patch with consideration to all similar patches, while the weights are not the pairwise similarities but a set of clusterwise coefficients. Moreover, by using the similarity between rows, we also get nonlocal-autoregression-like estimations for the center pixels of the similar patches. The TDNL model leads to an alternative minimization algorithm. Experiments indicate that the model can perform on par with or better than the state-of-the-art denoising methods.

Sparse representation based classification has led to interesting image recognition results, while the dictionary used for sparse coding plays a key role in it. This paper presents a novel dictionary learning (DL) method to improve the pattern classification performance. Based on the Fisher discrimination criterion, a structured dictionary, whose dictionary atoms have correspondence to the class labels, is learned so that the reconstruction error after sparse coding can be used for pattern classification. Meanwhile, the Fisher discrimination criterion is imposed on the coding coefficients so that they have small within-class scatter but big between-class scatter. A new classification scheme associated with the proposed Fisher discrimination DL (FDDL) method is then presented by using both the discriminative information in the reconstruction error and sparse coding coefficients. The proposed FDDL is extensively evaluated on benchmark image databases in comparison with existing sparse representation and DL based classification methods.

As a recently proposed technique, sparse representation based classification (SRC) has been widely used for face recognition (FR). SRC first codes a testing sample as a sparse linear combination of all the training samples, and then classifies the testing sample by evaluating which class leads to the minimum representation error. While the importance of sparsity is much emphasized in SRC and many related works, the use of collaborative representation (CR) in SRC is ignored by most literature. However, is it really the l1-norm sparsity that improves the FR accuracy? This paper devotes to analyze the working mechanism of SRC, and indicates that it is the CR but not the l1-norm sparsity that makes SRC powerful for face classification. Consequently, we propose a very simple yet much more efficient face classification scheme, namely CR based classification with regularized least square (CRC_RLS). The extensive experiments clearly show that CRC_RLS has very competitive classification results, while it has significantly less complexity than SRC.

In this letter, variational models in pixel domain and wavelet domain are presented for fusion and denoising of noisy multifocus images. In pixel domain, the problem is formulized as minimizing a weighted energy functional, where the total variation (TV) is used as regularity constraint for noise reduction. A new family of weight functions for fusion is proposed that are based on the local average modulus of gradients and the power transform. In wavelet domain, the problem is formulized as shrinkage of the weighted wavelet coefficients of source images, where weight functions are based on the local average modulus of intra- and inter-scale wavelet coefficients and the power transform. The experiments are made to verify the effectiveness of the proposed methods.

This paper introduces a new class of fractional-order anisotropic diffusion equations for noise removal. These equations are Euler-Lagrange equations of a cost functional which is an increasing function of the absolute value of the fractional derivative of the image intensity function, so the proposed equations can be seen as generalizations of second-order and fourth-order anisotropic diffusion equations. We use the discrete Fourier transform to implement the numerical algorithm and give an iterative scheme in the frequency domain. It is one important aspect of the algorithm that it considers the input image as a periodic image. To overcome this problem, we use a folded algorithm by extending the image symmetrically about its borders. Finally, we list various numerical results on denoising real images. Experiments show that the proposed fractional-order anisotropic diffusion equations yield good visual effects and better signal-to-noise ratio.