Variance Of Gaussian Noise Research Articles

Approximate support recovery of atomic line spectral estimation: A tale of resolution and precision

This work investigates the parameter estimation performance of super-resolution line spectral estimation using atomic norm minimization. The focus is on analyzing the algorithm's accuracy of inferring the frequencies and complex magnitudes from noisy observations. When the Signal-to-Noise Ratio is reasonably high and the true frequencies are separated by $O(\frac{1}{n})$, the atomic norm estimator is shown to localize the correct number of frequencies, each within a neighborhood of size $O(\sqrt{{\log n}/{n^3}} \sigma)$ of one of the true frequencies. Here $n$ is half the number of temporal samples and $\sigma^2$ is the Gaussian noise variance. The analysis is based on a primal-dual witness construction procedure. The obtained error bound matches the Cram\'er-Rao lower bound up to a logarithmic factor. The relationship between resolution (separation of frequencies) and precision or accuracy of the estimator is highlighted. Our analysis also reveals that the atomic norm minimization can be viewed as a convex way to solve a $\ell_1$-norm regularized, nonlinear and nonconvex least-squares problem to global optimality.

A design for a dynamic biomimetic sonarhead inspired by horseshoe bats

The noseleaf and pinnae of horseshoe bats (Rhinolophus ferrumequinum) have both been shown to actively deform during biosonar operation. Since these baffle structures directly affect the properties of the animals biosonar system, this work mimics horseshoe bat sonar system with the goal of developing a platform to study the dynamic sensing principles horseshoe bats employ. Consequently, two robotic devices were developed to mimic the dynamic emission and reception characteristics of horseshoe bats. The noseleaf and pinnae shapes were modeled as smooth blanks matched to digital representations of a horseshoe bat specimens noseleaf and pinnae. Local shape features mimicking structures on the pinnae and noseleaf were added digitally. Flexible baffles with local shape feature combinations were manufactured and paired with actuation mechanisms to mimic pinnae and noseleaf deformations in vivo. Two noseleaves with and without local shape features were considered. Each noseleaf baffle was mounted to a platform called the dynamic emission head to actuate three surface elements of the baffle. Similarly, 12 pinna realizations composed of combinations of three local shape features were mounted to a platform called the dynamic reception head to deform the left and right pinnae independently. Motion of the noseleaf and pinnae were synchronized to the incoming and outgoing sonar waveform, and the joint time-frequency properties of the noseleaf and pinnae local feature combinations and pairs of pinnae and noseleaf thereof were characterized across spatial direction. Amplitude modulations to the outgoing and incoming sonar pulse information across spatial direction were observed for all pinnae and noseleaf local shape feature combinations. Peak modulation variance generated by motion of the pinnae and combinations of the noseleaf and pinnae approached a white Gaussian noise variance bound. It was found the dynamic emitter generated less modulation than either the combined or reception scenarios.

Blind estimation of white Gaussian noise variance in highly textured images

In the paper, a new method of blind estimation of noise variance in a single highly textured image is proposed. An input image is divided into 8x8 blocks and discrete cosine transform (DCT) is performed for each block. A part of 64 DCT coefficients with lowest energy calculated through all blocks is selected for further analysis. For the DCT coefficients, a robust estimate of noise variance is calculated. Corresponding to the obtained estimate, a part of blocks having very large values of local variance calculated only for the selected DCT coefficients are excluded from the further analysis. These two steps (estimation of noise variance and exclusion of blocks) are iteratively repeated three times. For the verification of the proposed method, a new noise-free test image database TAMPERE17 consisting of many highly textured images is designed. It is shown for this database and different values of noise variance from the set {25, 49, 100, 225}, that the proposed method provides approximately two times lower estimation root mean square error than other methods.

Two-Dimensional CS Adaptive FIR Wiener Filtering Algorithm for the Denoising of Satellite Images

In the recent years, researchers are quite much attracted in designing two-dimensional (2-D) adaptive finite-impulse response (FIR) filters driven by an optimization algorithm to self-adjust the filter coefficients, with applications in different domains of research. For signal processing applications, FIR Wiener filters are commonly used for noisy signal restorations by computing the statistical estimates of the unknown signal. In this paper, a novel 2-D Cuckoo search adaptive Wiener filtering algorithm (2D-CSAWF) is proposed for the denoising of satellite images contaminated with Gaussian noise. Till date, study based on 2-D adaptive Wiener filtering driven by metaheuristic algorithms was not found in the literature to the best of our knowledge. Comparisons are made with the most studied and recent 2-D adaptive noise filtering algorithms, so as to analyze the performance and computational efficiency of the proposed algorithm. We have also included comparisons with recent adaptive metaheuristic algorithms used for satellite image denoising to ensure a fair comparison. All the algorithms are tested on the same satellite image dataset, for denoising images corrupted with three different Gaussian noise variance levels. The experimental results reveal that the proposed novel 2D-CSAWF algorithm outperforms others both quantitatively and qualitatively. Investigations were also carried out to examine the stability and computational efficiency of the proposed algorithm in denoising satellite images.

Measurement variance ignorant target motion analysis

The paper is devoted to Bayesian target motion analysis (TMA) for the case when the variance of additive white zero-mean Gaussian measurement noise is unknown. Two Rao–Blackwellised particle filters for TMA are developed, which jointly estimate the target state and the measurement variance. The error performance of the two particle filters is compared against the theoretical Cramer–Rao lower bound. The bound suggests the error in target state estimation is not affected by the ignorance of the measurement noise variance. Both developed TMA algorithms reach this theoretical bound, however, one is significantly faster.

Robust Facial Expression Recognition with Low-Rank Sparse Error Dictionary Based Probabilistic Collaborative Representation Classification

The performance of facial expression recognition (FER) would be degraded due to some factors such as individual differences, Gaussian random noise and so on. Prior feature extraction methods like Local Binary Patterns (LBP) and Gabor filters require explicit expression components, which are always unavailable and difficult to obtain. To make the facial expression recognition (FER) more robust, we propose a novel FER approach based on low-rank sparse error dictionary (LRSE) to remit the side-effect caused by the problems above. Then the query samples can be represented and classified by a probabilistic collaborative representation based classifier (ProCRC), which exploits the maximum likelihood that the query sample belonging to the collaborative subspace of all classes can be better computed. The final classification is performed by seeking which class has the maximum probability. The proposed approach which exploits ProCRC associated with the LRSE features (LRSE ProCRC) for robust FER reaches higher average accuracies on the different databases (i.e., 79.39% on KDEF database, 89.54% on CAS-PEAL database, 84.45% on CK+ database etc.). In addition, our method also leads to state-of-the-art classification results from the aspect of feature extraction methods, training samples, Gaussian noise variances and classification based methods on benchmark databases.

Dictionary learning feature space via sparse representation classification for facial expression recognition

Facial expression recognition (FER) plays a significant role in human-computer interaction. In this paper, adopting a dictionary learning feature space (DLFS) via sparse representation classification (SRC), we propose a method for FER. First, we obtain a difference dictionary (DD) from the feature space by indirectly using an auxiliary neutral training set. Next, we use a dictionary learning algorithm to train the DD; this algorithm considers the samples from the DD are approximately symmetrical structure. Finally, we use SRC to represent and determine the label of each query sample. We then verify out proposed method from the perspective of training samples, dimension reduction methods and Gaussian noise variances using a variety of public databases. In addition, we compare our DLFS_SRC approach with DLFS_CRC and DLFS_LRC approaches on the Extended Cohn-Kanade (CK+) database to analyze recognition results. Our simulation experiments show that our proposed method achieved satisfying performance levels for FER.

Adaptive joint block-weighted collaborative representation for facial expression recognition

Facial expression recognition (FER) plays a significant role in human-computer interactions. Recently, regularized linear representation-based classification has achieved satisfying results in FER. Considering that different blocks in a sample should contribute differently to the representation and classification, we propose an adaptive joint block-weighted collaborative representation-based classification (JBW_CRC) method to effectively exploit the similarity and distinctiveness of different blocks. In JBW_CRC, samples are divided into different blocks and each block of the query sample is represented as a feature vector. Each feature vector is coded on its related block dictionary, which considers the similarity among the feature vectors. Additionally, the distinctiveness of different feature vectors is obtained by weighting its distance to other features, which addresses the distinctiveness in the different feature vectors. The proposed method is verified from the aspect of training samples, time complexity, and Gaussian noise variances on benchmark databases and the extensive experiments show that the proposed method is very competitive with some similar pattern classification methods.

Multiple receivers in a high-resolution near-infrared heterodyne spectrometer.

The paper describes a modification of a high-resolution heterodyne NIR spectrometer described by Rodin, et al. in Opt. Express22, 13825 (2014), wherein the noise amplitude of the heterodyne signal squared was proportional to the power of the incoherent radiation source (the sun). The addition of a second receiver collecting radiation from the sun into a second single-mode fiber created up to a factor of two increase in detected source power spectrum. The ability to add uncorrelated heterodyne IF signals as Gaussian noise (variance) provides the means by which multiple heterodyne receivers of signal from an incoherent source can increase the detected source power by the same multiple, thus avoiding the limit imposed by the antenna theorem for a single receiver (A. E. Siegman, Appl. Opt.5, 1588 (1966)).

Superpixel-based image noise variance estimation with local statistical assessment

Noise estimation is fundamental and essential in a wide variety of computer vision, image, and video processing applications. It provides an adaptive mechanism for many restoration algorithms instead of using fixed values for the setting of noise levels. This paper proposes a new superpixel-based framework associated with statistical analysis for estimating the variance of additive Gaussian noise in digital images. The proposed approach consists of three major phases: superpixel classification, local variance computation, and statistical determination. The normalized cut algorithm is first adopted to effectively divide the image into a set of superpixel regions, from which the noise variance is computed and estimated. Subsequently, the Jarque–Bera test is used to exclude regions that are not normally distributed. The smallest standard deviation in the remaining regions is finally selected as the estimation result. A wide variety of noisy images with various scenarios were used to evaluate this new noise estimation algorithm. Experimental results indicated that the proposed framework provides accurate estimations across various noise levels. Comparing with many state-of-the-art methods, our algorithm strikes a good compromise between low-level and high-level noise estimations. It is suggested that the proposed method is of potential in many computer vision, image, and video processing applications that require automation.

Mixed noise removal based on a novel non-parametric Bayesian sparse outlier model

We develop a novel non-parametric Bayesian sparse outlier model for the problem of mixed noise removal. Based on the assumptions of sparse data and isolated outliers, the proposed model is considered for decomposing the observed data into three components of ideal data, Gaussian noise and outlier noise. Then the spike-slab prior is employed for outlier noise and sparse coefficients of ideal data. The proposed method can automatically infer noise statistics (e.g., Gaussian noise variance) from the training data without changing model hyper-parameter settings. It is also robust to initialization without using adaptive median filter as in other denoising methods. Experimental results demonstrate proposed model can achieve better objective and subjective performances on mixed noise removal than other state-of-the-art methods.

The augmented lagrange multipliers method for matrix completion from corrupted samplings with application to mixed Gaussian-impulse noise removal.

This paper studies the problem of the restoration of images corrupted by mixed Gaussian-impulse noise. In recent years, low-rank matrix reconstruction has become a research hotspot in many scientific and engineering domains such as machine learning, image processing, computer vision and bioinformatics, which mainly involves the problem of matrix completion and robust principal component analysis, namely recovering a low-rank matrix from an incomplete but accurate sampling subset of its entries and from an observed data matrix with an unknown fraction of its entries being arbitrarily corrupted, respectively. Inspired by these ideas, we consider the problem of recovering a low-rank matrix from an incomplete sampling subset of its entries with an unknown fraction of the samplings contaminated by arbitrary errors, which is defined as the problem of matrix completion from corrupted samplings and modeled as a convex optimization problem that minimizes a combination of the nuclear norm and the -norm in this paper. Meanwhile, we put forward a novel and effective algorithm called augmented Lagrange multipliers to exactly solve the problem. For mixed Gaussian-impulse noise removal, we regard it as the problem of matrix completion from corrupted samplings, and restore the noisy image following an impulse-detecting procedure. Compared with some existing methods for mixed noise removal, the recovery quality performance of our method is dominant if images possess low-rank features such as geometrically regular textures and similar structured contents; especially when the density of impulse noise is relatively high and the variance of Gaussian noise is small, our method can outperform the traditional methods significantly not only in the simultaneous removal of Gaussian noise and impulse noise, and the restoration ability for a low-rank image matrix, but also in the preservation of textures and details in the image.

Localization precision of stochastic optical localization nanoscopy using single frames

The Cramer-Rao lower bound (CRLB) on localization precision of unbiased estimators is analyzed for stochastic optical localization nanoscopy that localizes emitters frame by frame independently. It is found that the CRLB is a function of the mean number of detected photons per emitter, signal to Poisson noise ratio, signal to Gaussian noise ratio, point spread function (PSF), pixel size, and relationship of emitter locations. With a slight and practical approximation, effect of Gaussian noise is equivalent to increasing the mean photon count of Poisson noise by a number equal to the variance of Gaussian noise. Numerical examples demonstrate that the CRLB of emitters located on a curve increase fast as the distance of adjacent emitters increases. The mean CRLB of randomly uniformly distributed emitters in both two-dimensional and three-dimensional imaging increases exponentially fast as the emitter density increases. The effects of PSF, standard deviation of PSF, mean number of detected photons per emitter, signal to noise ratio, axial thickness, and pixel size on the CRLB are also numerically investigated. The analytical and numerical results provide a guideline for the design of location estimators and a benchmark for the achievable localization precision of stochastic optical localization nanoscopy.

An Improved Wavelet-Based Approach for Estimating the Variance of Noise in Images

In the paper an improved approach based on Donoho’s method is proposed to estimate the variance of Gaussian noise in images. The proposed approach exploits the interscale correlation in wavelet domain to estimate the coefficients of image signals, and then get the purer noise coefficients by deducting the estimated values from the coefficients of the noisy image, then does estimation once again using Donoho’s formula with the updated data. Simulation results are presented which indicate that the proposed approach outperforms other methods, especially in scenarios where there is less noise or much more image details.

Rateless codes aided coherent M‐ary PSK systems in the presence of receiver imperfections and Rician fading

ABSTRACTIn this paper, we study the bit error rate (BER) and the symbol error probability (SEP) performance for a rateless coded M‐ary phase‐shift keying (MPSK) system in a situation where stochastic phase error, quadrature error and in‐phase–quadrature (I–Q) gain mismatch take place all concurrently over a Rician fading channel. Similar scenario has been considered before by Djordjevic et al. for quasi‐cyclic low‐density parity‐check (QC‐LDPC) codes, but the effect of I–Q gain mismatch is not correctly modeled, because the Gaussian noise components in I and Q paths are assumed to have the same variance. By taking two different Gaussian noise variances into consideration, precise expressions are derived for the SEP of a coherent MPSK system under the aforementioned condition (with the stochastic phase error depicted by Tikhonov distribution), thus providing a baseline to evaluate the merits of the proposed rateless coded system and that of QC‐LDPC codes. The validity of the rateless coded system is substantiated by simulation results. It is observed that rateless codes outperform QC‐LDPC codes even the I–Q gain mismatch is modeled in the same way as Djordjevic's work when the target BER is lower than 10−2. Copyright © 2011 John Wiley & Sons, Ltd.

Homogeneity Localization Using Particle Filters With Application to Noise Estimation

This paper proposes a method for localizing homogeneity and estimating additive white Gaussian noise (AWGN) variance in images. The proposed method uses spatially and sparsely scattered initial seeds and utilizes particle filtering techniques to guide their spatial movement towards homogeneous locations. This way, the proposed method avoids the need to perform the full search associated with block-based noise estimation methods. To achieve this, the paper proposes for the particle filter a dynamic model and a homogeneity observation model based on Laplacian structure detectors. The variance of AWGN is robustly estimated from the variances of blocks in the detected homogeneous areas. A proposed adaptive trimmed-mean based robust estimator is used to account for the reduction in estimation samples from the full search approach. Our results show that the proposed method reduces the number of homogeneity measurements required by block-based methods while achieving more accuracy.

Hyper Chaotic Logistic Phase Coded Signal and Its Sidelobe Suppression

A novel hyperchaotic coding scheme and corresponding optimal selection method are proposed to obtain the hyper chaotic logistic phase code (HCLPC) signal, which exhibits great electronic counter-counter measure (ECCM) performance in the presence of digital radio frequency memory (DRFM) repeat jammers and netted radar systems. Similar to the random noise signal, the HCLPC signal has an ideal average ambiguity function and cross ambiguity function. However, the HCLPC signal can overcome the difficulties of reference signal storage and transmitter-receiver isolation for random signals. High range sidelobe would exist for finite length of transmit signal in practice. The Tikhonov regularization constrained total least squares algorithm (TRCTLSA) is first proposed to range sidelobe suppression of the HCLPC signal when data inaccuracy and white Gaussian noise is taken into account, and a global optimal solution to this algorithm is given. Simulation shows that it has a better sidelobe suppression effect than other ways. When Gaussian-noise variance is less than 0.01, the sidelobe level can decrease to the order of -55 dB, but the signal-to-noise ratio (SNR) loss is less than 0.05 dB. As a digital sidelobe suppression filter procedure, it can be widely used in noise reduction environments, image enhancements, and easily ported to sidelobe suppression applications of other signals as well.

Noise Parameter Estimation From Quantized Data

In this paper, the parametric estimation of the variance of white Gaussian noise is considered when available data are obtained from a quantized noisy stimulus. The Crameacuter-Rao lower bound is derived, and the statistical efficiency of a maximum-likelihood parametric estimator is discussed, along with the estimation algorithm proposed in IEEE Standard 1241

Matrix Factorization Algorithms for the Identification of Muscle Synergies: Evaluation on Simulated and Experimental Data Sets

Several recent studies have used matrix factorization algorithms to assess the hypothesis that behaviors might be produced through the combination of a small number of muscle synergies. Although generally agreeing in their basic conclusions, these studies have used a range of different algorithms, making their interpretation and integration difficult. We therefore compared the performance of these different algorithms on both simulated and experimental data sets. We focused on the ability of these algorithms to identify the set of synergies underlying a data set. All data sets consisted of nonnegative values, reflecting the nonnegative data of muscle activation patterns. We found that the performance of principal component analysis (PCA) was generally lower than that of the other algorithms in identifying muscle synergies. Factor analysis (FA) with varimax rotation was better than PCA, and was generally at the same levels as independent component analysis (ICA) and nonnegative matrix factorization (NMF). ICA performed very well on data sets corrupted by constant variance Gaussian noise, but was impaired on data sets with signal-dependent noise and when synergy activation coefficients were correlated. Nonnegative matrix factorization (NMF) performed similarly to ICA and FA on data sets with signal-dependent noise and was generally robust across data sets. The best algorithms were ICA applied to the subspace defined by PCA (ICAPCA) and a version of probabilistic ICA with nonnegativity constraints (pICA). We also evaluated some commonly used criteria to identify the number of synergies underlying a data set, finding that only likelihood ratios based on factor analysis identified the correct number of synergies for data sets with signal-dependent noise in some cases. We then proposed an ad hoc procedure, finding that it was able to identify the correct number in a larger number of cases. Finally, we applied these methods to an experimentally obtained data set. The best performing algorithms (FA, ICA, NMF, ICAPCA, pICA) identified synergies very similar to one another. Based on these results, we discuss guidelines for using factorization algorithms to analyze muscle activation patterns. More generally, the ability of several algorithms to identify the correct muscle synergies and activation coefficients in simulated data, combined with their consistency when applied to physiological data sets, suggests that the muscle synergies found by a particular algorithm are not an artifact of that algorithm, but reflect basic aspects of the organization of muscle activation patterns underlying behaviors.

Robust estimation of Gaussian-noise variance in the presence of pulsed interference

A robust estimator of the mean power of a Gaussian noise signal in the presence of pulsed interference is proposed for remote sensing. The technique is based on the following rationale: under certain conditions, the histogram of a signal contaminated by pulsed interference is close to a normal distribution in a certain range of snapshot values, i.e., the histogram is an estimate almost insensitive to contaminating pulsed interference. Correspondingly, we propose a modified minimum χ2 method, which is called the method of robust approximation, for estimating the mean power of the “useful” Gaussian component. It is shown that the proposed estimate yields a higher accuracy compared with, e.g., median estimates and is close to the Cramer-Rao lower bound. The results of experimental testing of the technique using hydroacoustic data are presented.