Noise Variance Estimation Research Articles

Estimation of Break and Noise Variance and the Maximum Distance of Climate Stations Allowed in Relative Homogenisation of Annual Temperature Anomalies

ABSTRACTRelative homogenisation uses the difference time series of neighbouring climate stations in order to reduce the dominating variance of the climate signal. Larger distances between the stations reduce the signal‐to‐noise ratio (SNR) because the noise variance increases while the break variance remains constant. For different continents, we derive the distance where noise and break variance become equal (SNR = 1). This is an estimate for the maximum allowed distance because SNRs below 1 bear the risk that the homogenisation produces nonsense results. Technically, the break variance is derived by extrapolating the autocovariance to zero temporal distance. An analogous treatment of the variance provides an estimate of the sum of break and noise variance, so that the SNR can be concluded. However, the unavoidable usage of anomalies and limited time series shifts and bends the commonly assumed simple functions (decreasing exponential for the covariance and constant for the variance) in a complex manner. The actually occurring functions are theoretically derived and then fitted to observational data. Above that, we show that also in North America, the inhomogeneity levels themselves and not the jumps between them can be described as a random variable. Consequently, no Brownian‐motion‐type inhomogeneities seem to exist, as it was suggested in earlier studies.

Enhanced Wavelet Scattering Network for Image Inpainting Detection

The rapid advancement of image inpainting tools, especially those aimed at removing artifacts, has made digital image manipulation alarmingly accessible. This paper proposes several innovative ideas for detecting inpainting forgeries based on a low-level noise analysis by combining Dual-Tree Complex Wavelet Transform (DT-CWT) for feature extraction with convolutional neural networks (CNN) for forged area detection and localization, and lastly by employing an innovative combination of texture segmentation with noise variance estimations. The DT-CWT offers significant advantages due to its shift-invariance, enhancing its robustness against subtle manipulations during the inpainting process. Furthermore, its directional selectivity allows for the detection of subtle artifacts introduced by inpainting within specific frequency bands and orientations. Various neural network architectures were evaluated and proposed. Lastly, we propose a fusion detection module that combines texture analysis with noise variance estimation to give the forged area. Also, to address the limitations of existing inpainting datasets, particularly their lack of clear separation between inpainted regions and removed objects—which can inadvertently favor detection—we introduced a new dataset named the Real Inpainting Detection Dataset. Our approach was benchmarked against state-of-the-art methods and demonstrated superior performance over all cited alternatives.

A Novel Noise Variance and SNR Estimator for OFDM-IM Systems over Nakagami-m Fading Channel

A Novel Noise Variance and SNR Estimator for OFDM-IM Systems over Nakagami-m Fading Channel

Adaptive Gaussian Wiener Filter for CT-Scan Images with Gaussian Noise Variance

Medical imaging plays an important role in modern healthcare, with Computed Tomography (CT) being essential for high-resolution cross-sectional imaging. However, Gaussian noise often occurs within the CT scan images and makes it difficult for image interpretation and reduces the diagnostic accuracy, creating a significant obstacle to fully utilizing CT scanning technology. Existing denoising techniques have a hard time balance between noise reduction and preserving the important image details, failing to enable the optimal diagnostic precision. This study introduces Adaptive Gaussian Wiener Filter (AGWF), a novel filter aims to denoise CT scan images that have been corrupted with various Gaussian noise variance without compromising the image details. The AGWF combines the Gaussian filter for initial noise reduction, followed by the implementation of Wiener filter, which can adaptively estimate noise variance and signal power in localized regions. This approach not only outperforms other existing techniques but also showcases a remarkable balance between noise reduction and image detail preservation. The experiment evaluates 300 images from the dataset and each image is corrupted with Gaussian noise variance to ensure a comprehensive evaluation of the AGWF’s performance. The evaluation indicated that AGWF can improve the Signal-to-Noise Ratio (SNR) value and reduce the Root Mean Square Error (RMSE) and Mean Square Error (MSE) value, showing a qualitative improvement in CT scan imagery. The proposed method holds promising potential for advancing medical imaging technology with the implementation of deep learning.

Deep Variation Prior: Joint Image Denoising and Noise Variance Estimation without Clean Data.

With recent deep learning based approaches showing promising results in removing noise from images, the best denoising performance has been reported in a supervised learning setup that requires a large set of paired noisy images and ground truth data for training. The strong data requirement can be mitigated by unsupervised learning techniques, however, accurate modelling of images or noise variances is still crucial for high-quality solutions. The learning problem is ill-posed for unknown noise distributions. This paper investigates the tasks of image denoising and noise variance estimation in a single, joint learning framework. To address the ill-posedness of the problem, we present deep variation prior (DVP), which states that the variation of a properly learnt denoiser with respect to the change of noise satisfies some smoothness properties, as a key criterion for good denoisers. Building upon DVP and under the assumption that the noise is zero mean and pixel-wise independent conditioned on the image, an unsupervised deep learning framework, that simultaneously learns a denoiser and estimates noise variances, is developed. Our method does not require any clean training images or an external step of noise estimation, and instead, approximates the minimum mean squared error denoisers using only a set of noisy images. With the two underlying tasks being considered in a single framework, we allow them to be optimised for each other. The experimental results show a denoising quality comparable to that of supervised learning and accurate noise variance estimates.

Efficient PCA denoising of spatially correlated redundant MRI data

Abstract Marčenko-Pastur PCA (MPPCA) denoising is emerging as an effective means for noise suppression in MR imaging (MRI) acquisitions with redundant dimensions. However, MPPCA performance can be severely compromised by spatially correlated noise—an issue typically affecting most modern MRI acquisitions—almost to the point of returning the original images with little or no noise removal. In this study, we explore different threshold criteria for principal component analysis (PCA) component classification that enable efficient and robust denoising of MRI data even when noise exhibits high spatial correlations, especially in cases where data are acquired with Partial Fourier and when only magnitude data are available. We show that efficient denoising can be achieved by incorporating a-priori information about the noise variance into PCA denoising thresholding. Based on this, two denoising strategies developed here are: 1) General PCA (GPCA) denoising that uses a-priori noise variance estimates without assuming specific noise distributions; and 2) Threshold PCA (TPCA) denoising which removes noise components with a threshold computed from a-priori estimated noise variance to determine the upper bound of the Marčenko-Pastur (MP) distribution. These strategies were tested in simulations with known ground truth and applied for denoising diffusion MRI data acquired using pre-clinical (16.4T) and clinical (3T) MRI scanners. In synthetic phantoms, MPPCA denoising failed to denoise spatially correlated data, while GPCA and TPCA better classified components as dominated by signal/noise. In cases where the noise variance was not accurately estimated (as can be the case in many practical scenarios), TPCA still provides excellent denoising performance. Our experiments in pre-clinical diffusion data with highly corrupted by spatial correlated noise revealed that both GPCA and TPCA robustly denoised the data while MPPCA denoising failed. In in vivo diffusion MRI data acquired on a clinical scanner in healthy subjects, MPPCA weakly removed noised, while TPCA was found to have the best performance, likely due to misestimations of the noise variance. Thus, our work shows that these novel denoising approaches can strongly benefit future pre-clinical and clinical MRI applications.

Lifting Wavelet-Assisted EM Joint Estimation and Detection in Cooperative Spectrum Sensing.

Spectrum sensing in Cognitive radio (CR) is a way to improve spectrum utilization by detecting spectral holes to achieve a dynamic allocation of spectrum resources. As it is often difficult to obtain accurate wireless environment information in real-world scenarios, the detection performance is limited. Signal-to-noise ratio (SNR), noise variance, and channel prior occupancy rate are critical parameters in wireless spectrum sensing. However, obtaining these parameter values in advance is challenging in practical scenarios. A lifting wavelet-assisted Expectation-Maximization (EM) joint estimation and detection method is proposed to estimate multiple parameters and achieve full-blind detection, which uses lifting wavelet in noise variance estimation to improve detection probability and convergence speed. Moreover, a stream learning strategy is used in estimating SNR and channel prior occupancy rate to fit the scenario where the SU has mobility. The simulation results demonstrate that the proposed method can achieve comparable detection performance to the semi-blind EM method.

A Tight Coupling Algorithm for Strapdown Inertial Navigation System (SINS)/Global Positioning System (GPS) Adaptive Integrated Navigation Based on Variational Bayesian

Multi-source nonlinear noise exists in the process of multi-source navigation information fusion in long-endurance positioning systems in complex environments. In such engineering applications, the classical Kalman filter (KF) and the extended Kalman filter (EKF) have the phenomena of noise instability and parameter drift, which lead to the divergence of filtering results and reductions in accuracy over long periods of time. Aiming at the above problems, this paper proposes a fusion algorithm of the variational Bayesian (VB) and the cubature Kalman filter (CKF). Firstly, the system is modeled through nonlinear filtering, and the CKF error equation is established by taking the position difference and velocity difference between SINS and GPS as observation variables. Then, to address the problem of poor self-adaptation of the CKF algorithm, the variational Bayesian adaptive estimation method is introduced into the CKF algorithm, and a measurement noise variance estimation model is introduced to the process of time and measurement updates of the CKF algorithm to finally obtain the adaptive VB–CKF algorithm. The simulation results from the experimental platform show that the proposed fusion algorithm improves the combined SINS/GPS system by about 30% in terms of attitude angle accuracy and reduces speed and position estimation errors (RMSE) by about 45%. At the same time, comprehensive experiments on multiple types of sites show that compared with the CKF algorithm, the VB–CKF algorithm improves the positioning accuracy by 10% when the GPS signal is stable and improves the accuracy by about 38% when the GPS measurement noise changes dramatically in complex terrain, which effectively suppresses the accuracy divergence of the CKF algorithm and has high value for engineering applications.

Denoising for airborne transient electromagnetic data using noise-whitening-based weighted nuclear norm minimization

Abstract Recovering anomalous information covered under noise in late gates can enhance airborne transient electromagnetic (ATEM) detection. Conventional denoising mainly comprises filtering and gate correlation-based decomposition algorithms; the former fails to extract anomalies contaminated by noise and the latter relies on the correlation between gates, which may yield false late gate anomalies caused by early large-amplitude anomalies in early gates. In ATEM profiles, the correlation between anomalies in adjacent gates makes the anomalies to be measured to have low-rank characteristics relative to the noise-contaminated profiles; the noise is uniformly distributed in the profiles, which have nonlocal self-similarity. Therefore, the low-rank matrix approximation algorithm is applicable to ATEM data denoising. In this study, an algorithm—noise-whitening-based weighted nuclear norm minimization (NW-WNNM)—is designed to remove ATEM profile noise. First, we analyze the influence of patch size in block matching on anomalous and noisy patches and estimate the profile patch size adaptively. Then, we combine the estimation of noise variance in weighted nuclear norm minimization (WNNM) with the noise whitening of similar patch matrices to reduce the noise interference on the nuclear norm and add a whitening factor in the weight vector to make the soft-thresholding function applicable to the low-rank reconstruction of the whitened matrix. By analyzing the reconstructed low-rank matrix and its feature distribution, compared with WNNM, NW-WNNM can detect the feature information more accurately and eliminate the influence of noise on the nuclear norm. Simulation and field profile results indicate that NW-WNNM is superior to comparison denoising methods.

Singular value decomposition based learning identification for linear time‐varying systems: From recursion to iteration

AbstractSystem identification is a critical task in various engineering applications such as motion control, signal processing and robotics. In this article, the identification of linear time‐varying (LTV) systems that perform tasks repetitively over a finite‐time interval is investigated. Traditional LTV system identification typically adopts recursive algorithms in the time domain, which are incapable of tracking drastic‐varying parameters and are subject to estimation lag and numerical instability. To address these issues, this article proposes the utilization of an iteration axis in addition to the time axis for estimating repetitive time‐varying parameters. Specifically, the proposed approach involves an estimation algorithm for the time‐varying parameters based on a recursive least squares (RLS) method along the iteration axis, as well as an update algorithm for the covariance matrix based on singular value decomposition (SVD) to enhance numerical stability. Additionally, a bias compensation method based on noise variance estimation is introduced for the sake of eliminating estimation error induced by measurement noise. Numerical comparisons with existing methods are conducted to demonstrate the effectiveness and superiority of the proposed method.

A technology for monitoring the onset of malfunctions in oil field reservoir pressure maintenance equipment

Existing control and management systems of oil field reservoir pressure maintenance equipment do not warn of the beginning of the latent period of accidents. At the same time, accidents are preceded by the emergence of certain defects, followed by their latent period, after which they develop. Only after that they begin to affect the readings of measuring instruments of control and management systems. The duration of the latent period depends on the dynamics of defect development. Because of the above, control systems detect the beginning of an emergency when it becomes explicit. This leads to unreasonable costs, since elimination of the defect at the moment of its origin requires much less funds and time than after-accident pump repairs. It is shown in the paper that as a result of continuous rotational motion under high pressure, a vibration process is inevitably formed in this equipment, and the beginning of the latent period of malfunctions makes a clear impact on their noisy vibration signals, with a correlation emerging between the noise and the useful component. To control the beginning of malfunctions in reservoir pressure maintenance equipment it is proposed to use the estimates of noise variance and cross-correlation functions between the useful signal and the noise, which allow forming informative attributes for warning and control of the beginning of the latent period of malfunctions. The proposed technologies can also be used to improve accident-free operation at compressor stations of main oil and gas pipelines, at drilling rigs, at artesian wells, in transport, etc.

Blind Curvelet-Based Denoising of Seismic Surveys in Coherent and Incoherent Noise Environments

Distributed acoustic sensing (DAS) is a new seismic monitoring technology. DAS generates a large amount of data, necessitating the development of new technologies to allow for cost-effective processing and handling. The raw seismic data is noisy and must be processed. The curvelet transform is an excellent choice for processing seismic data due to its localized nature, as well as its frequency and dip characteristics. However, its capabilities are limited in case of noise other than white. This paper proposes a denoising method based on a combination of the curvelet transform and a whitening filter, as well as a procedure for estimating noise variance. The whitening filter is included to improve the performance of the curvelet transform in both coherent and incoherent noise cases, as well as to simplify the noise variance estimation method and make it easier to use standard threshold methodology without delving into the curvelet domain. Two data sets are used to validate the suggested technique. Pseudo-synthetic data set created by adding noise to the actual noise-free data collection from the Netherlands offshore F3 block and the on-site data set (with ground roll noise) from east Texas, USA. Experimental results demonstrate that the proposed algorithm achieves the best results under various types of noise.

Estimation of process noise variances from the measured output sequence with application to the empirical model of type 1 diabetes

This paper presents a novel method for estimating a priori unknown variances of the process noise affecting a general discrete-time stochastic state-space model under the assumption that the elements of the process noise vector are not correlated. To evaluate the proposed method in practice, only the system model and a sufficiently long sequence of output measurements are required, so there is no need for the system state to be measured or the state observer to be involved. In detail, the design of the method is based on the autocorrelation function of the stochastic output component, which is used to form the equivalent linear regression system with respect to the unknown vector of the process noise variances. Since the data-based estimate of the autocorrelation function has to be treated as uncertain, the generalized least squares method is proposed to solve the derived linear regression system, yielding the estimate with minimal variance. To verify the theory, the method is first validated within a case study that considers an artificial linear stochastic system, while this experiment was carried out under controlled statistical conditions. However, the intended target application domain of the method is the empirical model of type 1 diabetes, so another experiment based on virtual diabetic data obtained by simulating a complex physiology-based model that was subjected to input uncertainties was carried out.

Identification of dual‐rate sampled errors‐in‐variables systems with time delays

AbstractFor identification of dual‐rate sampled errors‐in‐variables (EIV) systems with time delays, this paper utilizes the polynomial transformation technique and the redundant rule to derive the augmented dual‐rate model, and proposes a bias compensation‐based least squares (BC‐LS) algorithm. The basic idea is to obtain a biased parameter estimate by using the LS algorithm, then compensate the noise‐induced bias by the estimated noise variances, and finally estimate the time delay according to the unbiased parameter estimate and the given threshold. Considering that proper threshold selection has great influence on the performance of the BC‐LS identification algorithm for high noise level, the generalized orthogonal matching pursuit (GOMP) algorithm is introduced to simultaneously estimate the biased parameters and the time delay, and the bias compensation‐based GOMP (BC‐GOMP) algorithm is further proposed to identify the dual‐rate sampled EIV systems with time delays. The simulation examples demonstrate the effectiveness of the proposed two algorithms, and the BC‐GOMP algorithm performs better especially when the modeling dataset is small.

An Optimal Fusion Method of Multiple Inertial Measurement Units Based on Measurement Noise Variance Estimation

At present, most inertial systems generally only contain a single inertial measurement unit (IMU). Considering the low cost and low accuracy of the micro-electromechanical system (MEMS)-IMU, it has attracted much attention to fuse multiple IMUs to improve the accuracy and robustness of the system. In this article, two online noise variance estimators based on second-order-mutual-difference (SOMD) algorithm are proposed for two redundant measurements and multiple redundant measurements, respectively. In addition, the unbiasedness and consistency of the estimators are proved. Using the proposed noise variance estimators, measurement noise variances of each sensor can be estimated in real time when multiple IMUs exist. Based on the estimated noise variance of each sensor, the weighted least squares (WLS) estimation method is used to generate the optimal virtual IMU (VIMU) in the observation domain. Finally, comparative simulations and the real-world experiment were conducted to evaluate the proposed online noise estimation algorithm. The simulation results demonstrate its superiority compared with other noise variance estimation methods, and the real-world experiment results show the effectiveness of the IMU fusion method based on the proposed noise variance estimation algorithm.

A Reverberation Noise Suppression Method of Sonar Image Based on Shearlet Transform

A sonar image affected by reverberation noise would generate serious speckle noise in the image, which causes great trouble for the later sonar image target segmentation and detection. Therefore, noise suppression is a crucial preprocessing technique for sonar image applications. To suppress the reverberation noise effectively in the sonar image, a reverberation noise suppression method of sonar image based on shearlet transform is proposed. First, to effectively apply the noise suppression method to sonar images, a method of combining the transform from the reverberation noise model to the Gaussian additive noise model with the shearlet transform is presented. Second, in the process of the shearlet transform, constructing a shear wave filter based on the Meyer window function, which has an excellent performance in both time and frequency domains, is suggested. Third, a noise variance estimation based on weak texture blocks would be made to avoid the effect of gray overflow in sonar images. Then, the estimated variance is combined with the root mean square coefficient obtained from the shearlet transform of the simulated noise. For this reason, obtaining the adaptive filtering thresholds of each scale and each direction is essential. With this process, a better reverberation suppression effect could be achieved. Simulation and experimental results demonstrate that the method proposed in this article can suppress reverberation noise more effectively, and a better edge-preserving effect could be obtained. Compared with other comparison algorithms, the algorithm proposed in this article could achieve the best results on the peak signal-to-noise ratio (PSNR) and SSMI.

Adaptive multiscale sparse unmixing for hyperspectral remote sensing image

Sparse unmixing of hyperspectral images aims to separate the endmembers and estimate the abundances of mixed pixels. This approach is the essential step for many applications involving hyperspectral images. The multi-scale spatial sparse hyperspectral unmixing algorithm (MUA) could achieve higher accuracy than many state-of-the-art algorithms. The regularization parameters, whose combinations markedly influence the unmixing accuracy, are determined by manually searching in the broad parameter space, leading to time consuming. To settle this issue, the adaptive multi-scale spatial sparse hyperspectral unmixing algorithm (AMUA) is proposed. Firstly, the MUA model is converted into a new version by using of a maximum a posteriori (MAP) system. Secondly, the theories indicating that andnorms are equivalent to Laplacian and multivariate Gaussian functions, respectively, are applied to explore the strong connections among the regularization parameters, estimated abundances and estimated noise variances. Finally, the connections are applied to update the regularization parameters adaptively in the optimization process of unmixing. Experimental results on both simulated data and real hyperspectral images show that the AMUA can substantially improve the unmixing efficiency at the cost of negligible accuracy. And a series of sensitive experiments were undertook to verify the robustness of the AMUA algorithm.

When artificial parameter evolution gets real: particle filtering for time-varying parameter estimation in deterministic dynamical systems

Estimating and quantifying uncertainty in unknown system parameters from limited data remains a challenging inverse problem in a variety of real-world applications. While many approaches focus on estimating constant parameters, a subset of these problems includes time-varying parameters with unknown evolution models that often cannot be directly observed. This work develops a systematic particle filtering approach that reframes the idea behind artificial parameter evolution to estimate time-varying parameters in nonstationary inverse problems arising from deterministic dynamical systems. Focusing on systems modeled by ordinary differential equations, we present two particle filter algorithms for time-varying parameter estimation: one that relies on a fixed value for the noise variance of a parameter random walk; another that employs online estimation of the parameter evolution noise variance along with the time-varying parameter of interest. Several computed examples demonstrate the capability of the proposed algorithms in estimating time-varying parameters with different underlying functional forms and different relationships with the system states (i.e. additive vs. multiplicative).

Blind-noise image denoising with block-matching domain transformation filtering and improved guided filtering

The adaptive block size processing method in different image areas makes block-matching and 3D-filtering (BM3D) have a very good image denoising effect. Based on these observation, in this paper, we improve BM3D in three aspects: adaptive noise variance estimation, domain transformation filtering and nonlinear filtering. First, we improve the noise-variance estimation method of principle component analysis using multilayer wavelet decomposition. Second, we propose compressive sensing based Gaussian sequence Hartley domain transform filtering to reduce noise. Finally, we perform edge-preserving smoothing on the preprocessed image using the guided filtering based on total variation. Experimental results show that the proposed denoising method can be competitive with many representative denoising methods on the evaluation criteria of PSNR. However, it is worth further research on the visual quality of denoised images.

Speckle Noise Removal Model Based on Diffusion Equation and Convolutional Neural Network

The image denoising model based on convolutional neural network (CNN) can achieve a good denoising effect. However, its robustness is poor, and it is not suitable for direct noise removal tasks. Differently, the image denoising method based on the diffusion equation is more stable and has theoretical guarantees. In order to give full play to the advantages of CNN and diffusion equation in image denoising, this paper proposes a speckle noise denoising model via a combination of the two tools. Firstly, based on the mathematical model of speckle noise, a class of neural network speckle noise removal model which mixes residual learning and structure learning is proposed using image decomposition theory. Then, in order to solve the hyperparameter problem that the model depends on noise variance, a noise variance estimation algorithm based on a nonlinear diffusion equation is proposed. Finally, a speckle noise denoising model based on diffusion equation and CNN is obtained. Numerical simulation experiments verify the accuracy of the variance estimation algorithm and also the denoising effect and practical application value of the proposed method.