Recovering Low-Rank and Sparse Components of Matrices from Incomplete and Noisy Observations

The maximum a posteriori estimation model for signal recovery with mixed Gaussian and impulse noise

We present a novel approach to structure from motion that can deal with missing data and outliers with an affine camera. We model the corruptions as sparse error. Therefore the structure from motion problem is reduced to the problem of recovering a low-rank matrix from corrupted observations. We first decompose the matrix of trajectories of features into low-rank and sparse components by nuclear-norm and ℓ 1-norm minimization, and then obtain the motion and structure from the low-rank components by the classical factorization method. Unlike pervious methods, which have some drawbacks such as depending on the initial value selection and being sensitive to the large magnitude errors, our method uses a convex optimization technique that is guaranteed to recover the low-rank matrix from highly corrupted and incomplete observations. Experimental results demonstrate that the proposed approach is more efficient and robust to large-scale outliers.

The augmented Lagrangian method is a classical solution method for nonlinear optimization problems. At each iteration, it minimizes an augmented Lagrangian function that consists of the constraint functions and the corresponding Lagrange multipliers. If the Lagrange multipliers in the augmented Lagrangian function are close to the exact Lagrange multipliers at an optimal solution, the method converges steadily. Since the conventional augmented Lagrangian method uses inaccurate estimated Lagrange multipliers, it sometimes converges slowly. In this paper, we propose a novel augmented Lagrangian method that allows the augmented Lagrangian function and its minimization problem to have variable constraints at each iteration. This allowance enables the new method to get more accurate estimated Lagrange multipliers by exploiting Karush–Kuhn–Tucker points of the subproblems and consequently to converge more efficiently and steadily.

Hyperspectral images (HSIs) are inevitably corrupted by mixture noise during their acquisition process, in which various kinds of noise, e.g., Gaussian noise, impulse noise, dead lines, and stripes, may exist concurrently. In this paper, mixture noise removal is well illustrated by the task of recovering the low-rank and sparse components of a given matrix, which is constructed by stacking vectorized HSI patches from all the bands at the same position. Instead of applying a traditional nuclear norm, a nonconvex low-rank regularizer, i.e., weighted Schatten p -norm (WSN), is introduced to not only give better approximation to the original low-rank assumption but also to consider the importance of different rank components. The resulted nonconvex low-rank matrix approximation (LRMA) model falls into the applicable scope of an augmented Lagrangian method, and its WSN minimization subproblem can be efficiently solved by generalized iterated shrinkage algorithm. Moreover, the proposed model is integrated into an iterative regularization schema to produce final results, leading to a completed HSI restoration framework. Extensive experimental testing on simulated and real data shows, both qualitatively and quantitatively, that the proposed method has achieved highly competent objective performance compared with several state-of-the-art HSI restoration methods.

The central goal of this dissertation is to design and model a smoothing filter based on the random single and mixed noise distribution that would attenuate the effect of noise while preserving edge details. Only then could robust, integrated and resilient edge detection methods be deployed to overcome the ubiquitous presence of random noise in images. Random noise effects are modeled as those that could emanate from impulse noise, Gaussian noise and speckle noise. In the first step, evaluation of methods is performed based on an exhaustive review on the different types of denoising methods which focus on impulse noise, Gaussian noise and their related denoising filters. These include spatial filters (linear, non-linear and a combination of them), transform domain filters, neural network-based filters, numerical-based filters, fuzzy based filters, morphological filters, statistical filters, and supervised learning-based filters. In the second step, switching adaptive median and fixed weighted mean filter (SAMFWMF) which is a combination of linear and non-linear filters, is introduced in order to detect and remove impulse noise. Then, a robust edge detection method is applied which relies on an integrated process including non-maximum suppression, maximum sequence, thresholding and morphological operations. The results are obtained on MRI and natural images. In the third step, a combination of transform domain-based filter which is a combination of dual tree – complex wavelet transform (DT-CWT) and total variation, is introduced in order to detect and remove Gaussian noise as well as mixed Gaussian and Speckle noise. Then, a robust edge detection is applied in order to track the true edges. The results are obtained on medical ultrasound and natural images. In the fourth step, a smoothing filter, which is a feed-forward convolutional network (CNN) is introduced to assume a deep architecture, and supported through a specific learning algorithm, l2 loss function minimization, a regularization method, and batch normalization all integrated in order to detect and remove impulse noise as well as mixed impulse and Gaussian noise. Then, a robust edge detection is applied in order to track the true edges. The results are obtained on natural images for both specific and non-specific noise-level.

Computed tomography images can be corrupted by mixed noises such as Gaussian and impulsive noise during acquisition time which results in reduction of its quality. Hence removing the noise from the image is very significant in medical image processing. The existing filters such as mean and median filter are not that efficient in removing impulse and Gaussian noise by retaining the details of the image. In this paper, a new filter is proposed which removes mixed noise such as Gaussian and impulse noise. Initially, pixels of image are separated into non-corrupted pixels and corrupted pixels based on existence of noises in their small neighborhood. The greyscale value of non-corrupted pixels are taken as output directly and for the corrupted pixels, removing Gaussian noises and impulse noises respectively is done based on their characteristics. The results demonstrate that the proposed filter can eliminate mixed noise of different density in a better way by also preserving the details of image when compared with the mean filter or the median filter for mixed noise.

Publication in the conference proceedings of EUSIPCO, Florence, Italy, 2006

Combining with vague sets, a filtering algorithm was proposed to filter mixed noises in image, which was polluted by impulse and Gaussian noises. The proposed filtering algorithm was first to filter impulse noise, and then was to filter Gaussian noise. In impulse noise filter, impulse noise was detected accurately first by homogeneity histogram or homogram and the significant peak detection method of histogram. The homogram was constructed by fuzzy entropy of vague sets. Combining with an adaptive adjusting filtering window method, the detected impulse noise was removed by median filter. In Gaussian noise filter, a fuzzy weighted filter that the weight was determined by similarity measure of vague sets was adopted to reduce Gaussian noise. The experimental results show that the proposed filtering algorithm for mixed noises in images has higher resolution, and can remove mixed impulse and Gaussian noises efficiently while protecting image details.

This paper proposes a new two-stage method for restoring image corrupted by additive impulsive and Gaussian noise based on local polynomial regression (LPR) and edge preserving regularization. In LPR, the observations are modeled locally by a polynomial using least-squares criterion with a kernel controlled by a certain bandwidth matrix. A refined intersection confidence intervals (RICI) adaptive scale selector for symmetric kernel is applied in LPR to achieve a better bias-variance tradeoff. The method is further extended to steering kernel with local orientation to adapt better to local characteristics of images. The resulting steering-kernel-based LPR with RICI method (SK-LPR-RICI) is applied to smooth images contaminated with Gaussian noise. Furthermore, to remove the impulsive noise in images, an edge-preserving regularization method is employed prior to SK-LPR-RICI and it gives rise to a two-stage method for suppressing both additive impulsive and Gaussian noises. Simulation results show that the proposed method performs satisfactorily and the SK-LPR-RICI method significantly improves the performance after edge-preservation regularization in suppressing the impulsive noise.

Denoising images corrupted with impulse, Gaussian, or a mixture of impulse and Gaussian noise

The Wigner distribution (WD) produces highly concentrated time-frequency (TF) representation of nonstationary signals. It may be used as an efficient signal analysis tool, including the cases of frequency modulated signals corrupted with the Gaussian noise. In some applications, a significant amount of impulse noise is present. Then, the WD fails to produce satisfactory results. The robust periodogram has been introduced for spectral estimation of this kind of noisy signals. It can produce good concentration for pure harmonic signals. However, it is not so efficient in the cases of signals with rapidly varying frequency. This is the motivation for introducing the robust WD. It is a reliable TF representation tool for wide class of nonstationary signals corrupted with impulse noise. This distribution produces good accuracy of the instantaneous frequency (IF) estimation. Using the Huber (1981) loss function, a generalization of the WD is presented. It includes both the standard and the robust WD as special cases. This distribution can be used for TF analysis of signals corrupted with a mixture of impulse and Gaussian noise. The presented theory is illustrated on examples, including applications on the IF estimation and time-varying filtering of signals corrupted with a mixture of the Gaussian and impulse noise. The case study analysis of the IF estimators' accuracy, based on the standard and the robust WD forms, is performed. In order to improve the IF estimation, a median filter is applied on the obtained IF estimate.

In this paper a Poisson impulse noise model is utilized to calculate error probability characteristics of a matched filter receiver operating in an additive combination of impulsive and Gaussian noise. Comparisons of the theoretical predictions are made with experimental data obtained from a simulated communication system. The results indicate that only a small amount of Gaussian noise power has a considerable effect on the probability of error for small signal-to-noise ratios (SNRs). Also, it is found that each noise component may be looked upon as acting separately, the Gaussian noise component dominating the error probability for low SNRs and the impulsive component dominating for high SNRs. Thus, approximate error probability curves may be calculated by using a result applicable for Gaussian interference at low SNRs, and an asymptotic formula corresponding to impulsive background noise at high SNRs.

To construct a parallel approach for solving optimization problems with orthogonality constraints is usually regarded as an extremely difficult mission, due to the low scalability of the orthonormalization procedure. However, such demand is particularly huge in some application areas such as materials computation. In this paper, we propose a proximal linearized augmented Lagrangian algorithm (PLAM) for solving optimization problems with orthogonality constraints. Unlike the classical augmented Lagrangian methods, in our algorithm, the prime variables are updated by minimizing a proximal linearized approximation of the augmented Lagrangian function, meanwhile the dual variables are updated by a closed-form expression which holds at any first-order stationary point. The orthonormalization procedure is only invoked once at the last step of the above mentioned algorithm if high-precision feasibility is needed. Consequently, the main parts of the proposed algorithm can be parallelized naturally. We establish global subsequence convergence, worst-case complexity and local convergence rate for PLAM under some mild assumptions. To reduce the sensitivity of the penalty parameter, we put forward a modification of PLAM, which is called parallelizable column-wise block minimization of PLAM (PCAL). Numerical experiments in serial illustrate that the novel updating rule for the Lagrangian multipliers significantly accelerates the convergence of PLAM and makes it comparable with the existent feasible solvers for optimization problems with orthogonality constraints, and the performance of PCAL does not highly rely on the choice of the penalty parameter. Numerical experiments under parallel environment demonstrate that PCAL attains good performance and high scalability in solving discretized Kohn-Sham total energy minimization problems.

Elimination of combined Gaussian and impulse noises in digital image processing with preservation of image details and suppression of noise are challenging problem. For this purpose, a new filter which is median filters combined with convolutional neural network for Gaussian and salt & pepper noises. The previous methods are application dependents; some used for impulse noise and other employed only for Gaussian noise. The elimination of Gaussian and impulse noise completed into two steps. First the detection of impulse noise with the rejection of noise by employed of 3 × 3 and 5 × 5 window size median filters. In the second step removal of Gaussian noise performed by residual learning denoising convolutional neural network. It is very favorable and the ability of learning and denoising performance in the field of digital image processing. Denoising convolutional neural network also has active Gaussian noise with an unknown level of noise. Experimental work showed that the proposed method can achieve low loss and root mean square error during training, high peak signal to noise ratio, low mean square error, image quality assessment with good quality and mean absolute error for close prediction between denoised and original color images.

Consider two variations of the method of multipliers, or classical augmented Lagrangian method for convex programming. The proximal method of multipliers adjoins quadratic primal proximal terms to the augmented Lagrangian, and has a stronger primal convergence theory than the standard method. On the other hand, the alternating direction method of multipliers, which uses a special kind of partial minimization of the augmented Lagrangian, is conducive to the derivation of decomposition methods finding application in parallel computing. This note shows convergence a method combining the features of these two variations. The method is closely related to some algorithms of Gols'shtein. A comparison of the methods helps illustrate the close relationship between previously separate bodies of Western and Soviet literature.