Case Of Additive Noise Research Articles

The stochastic heat equation with fractional time and fractional time–space white noise

We give an introduction to the time-fractional stochastic heat equation driven by 1+d-parameter fractional time–space white noise, in the following two cases:(i) With additive fractional time–space white noise(ii) With multiplicative time–space Brownian noiseThe fractional time derivative is interpreted as the Caputo derivative of order α∈(0,2) and we assume that the Hurst coefficient H=(H0,H1,H2,…,Hd) of the time–space fractional white noise is in (12,1)1+d.We find an explicit expression for the unique solution in the sense of distribution of the equation in the additive noise case (i).In the multiplicative case (ii) we show that there is a unique solution in the Hida space (S)∗ of stochastic distributions and we show that the solution coincides with the solution of an associated fractional stochastic Volterra equation. Then we give an explicit expression for the solution of this Volterra equation. See Appendix below or Di Nunno et al. (2008), Holden et al. (2010) and Øksendal (2023) for more information about Hida spaces and white noise theory.A solution Y(t,x) is called mild if E[Y2(t,x)]<∞ for all t,x. For both the additive noise case and the multiplicative noise case we show that if α≥1 then the solution is mild if d=1 or d=2, while if α<1 the solution is not mild for any d.This paper is partly a survey paper, explaining the concepts and methods behind the results. It is also partly a research paper, in the sense that some results are new, to the best of our knowledge.

Lossy Compression of Single-channel Noisy Images by Modern Coders

Lossy compression of remote-sensing images is a typical stage in their processing chain. In design or selection of methods for lossy compression, it is commonly assumed that images are noise-free. Meanwhile, there are many practical situations where an image or a set of its components are noisy. This fact needs to be taken into account since noise presence leads to specific effects in lossy compressed data. The main effect is the possible existence of the optimal operation point (OOP) shown for JPEG, JPEG2000, some coders based on the discrete cosine transform (DCT), and the better portable graphics (BPG) encoder. However, the performance of such modern coders as AVIF and HEIF with application to noisy images has not been studied yet. In this paper, analysis is carried out for the case of additive white Gaussian noise. We demonstrate that OOP can exist for AVIF and HEIF and the performance characteristics in it are quite similar to those for the BPG encoder. OOP exists with a higher probability for images of simpler structure and/or high-intensity noise, and this takes place according to different metrics including visual quality ones. The problems of providing lossy compression by AVIF or HEIF are shown and an initial solution is proposed. Examples for test and real-life remote-sensing images are presented.

PECULIARITIES OF NOISY IMAGE LOSSY COMPRESSION

This paper deals with lossy compression applied to grayscale noisy images using several coders. The case of additive white Gaussian noise is considered as the first step in studying the problem. A special attention is paid to quite new coders AVIF and HEIF for which, to the best of our knowledge, lossy compression of noisy images of different complexity has not been thoroughly considered yet. It is shown that optimal operation point is possible for all coders under certain conditions according to different quality metrics including conventional peak signal-to-noise ratio and visual quality metrics such as PSNR-HVS-M and MS-SSIM. The compression parameters (metrics) in optimal operation point significantly depend on image complexity and noise intensity where the existence of optimal operation point is more probable for simple structure images and/or intensive noise. Comparison of metric values in optimal operation point for the considered coders shows that slightly better characteristics are provided by better portable graphics (BPG) and advanced discrete cosine transform (ADCT) coders. The results for JPEG are significantly worse. The AVIF and HEIF encoders provide similar results and they outperform JPEG significantly. Optimal operation point for all studied coders is observed more rarely for visual quality metrics than for peak signal-to-noise ratio. General tendencies concerning dependence of optimal compression parameters on noise intensity are presented. It is shown that compression ratio in optimal operation point is often larger than 20 and can be as large as 60. The problem with AVIF and HEIF is that it is currently unclear how to choose quality factor for them to carry out lossy image compression in the neighborhood of the corresponding optimal operation point. Meanwhile, there is the tendency to optimal quality factor reduction if intensity of additive white Gaussian noise increases. The directions of further research are discussed.

Poisson stable solutions for stochastic PDEs driven by Lévy noise

This paper is mainly concerned with the existence, uniqueness and Poisson stability (including stationarity, periodicity, almost periodicity and almost automorphy) of solutions for a class of stochastic partial differential equations driven by Lévy noise, where the involved coefficients are assumed to be strictly monotone. Based on the variational method, we establish the well-posedness of L2-bounded solution and then prove that it has the same characters of periodicity, almost periodicity and almost automorphy as the coefficients of the equation. Moreover, we also investigate the additive noise case under strong monotone condition. In particular, we illustrate our results by applying to concrete models such as stochastic reaction-diffusion equations, stochastic porous media equations and stochastic p-Laplace equations.

Resolution Limits of Non-Adaptive 20 Questions Search for a Moving Target

Using the 20 questions estimation framework with query-dependent noise, we study non-adaptive search strategies for a moving target over the unit cube with unknown initial location and velocities under a piecewise constant velocity model. In this search problem, there is an oracle who knows the instantaneous location of the target at any time. Our task is to query the oracle as few times as possible to accurately estimate the location of the target at any specified time. We first study the case where the oracle’s answer to each query is corrupted by discrete noise and then generalize our results to the case of additive white Gaussian noise. In our formulation, the performance criterion is the resolution, which is defined as the maximal L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> distance between the true locations and estimated locations. We characterize the minimal resolution of an optimal non-adaptive query procedure with a finite number of queries by deriving non-asymptotic and asymptotic bounds. Our bounds are tight in the first-order asymptotic sense when the number of queries satisfies a certain condition and our bounds are tight in the stronger second-order asymptotic sense when the target moves with a constant velocity. To prove our results, we relate the current problem to channel coding, borrow ideas from finite blocklength information theory and construct bounds on the number of possible quantized target trajectories.

Strong convergence of an adaptive time-stepping Milstein method for SDEs with monotone coefficients

We introduce an explicit adaptive Milstein method for stochastic differential equations with no commutativity condition. The drift and diffusion are separately locally Lipschitz and together satisfy a monotone condition. This method relies on a class of path-bounded time-stepping strategies which work by reducing the stepsize as solutions approach the boundary of a sphere, invoking a backstop method in the event that the timestep becomes too small. We prove that such schemes are strongly L_2 convergent of order one. This order is inherited by an explicit adaptive Euler–Maruyama scheme in the additive noise case. Moreover we show that the probability of using the backstop method at any step can be made arbitrarily small. We compare our method to other fixed-step Milstein variants on a range of test problems.

BPG-Based Lossy Compression of Three-Channel Noisy Images with Prediction of Optimal Operation Existence and Its Parameters

Nowadays, there is a clear trend toward increasing the number of remote-sensing images acquired and their average size. This leads to the need to compress the images for storage, dissemination, and transfer over communication lines where lossy compression techniques are more popular. The images to be compressed or some of their components are often noisy. They must therefore be compressed taking into account the properties of the noise. Due to the noise filtering effect obtained during lossy compression of noisy images, an optimal operating point (OOP) may exist. The OOP is a parameter that controls the compression for which the quality of the compressed image is closer (closest) to the corresponding noise-free image than the quality of the noisy (original, uncompressed) image according to some quantitative criterion (metric). In practice, it is important to know whether the OOP exists for a given image, because if the OOP exists, it is appropriate to perform the compression in the OOP or at least in its neighborhood. Since the real image is absent in practice, it is impossible to determine a priori whether the OOP exists or not. Here, we focus on three-channel-remote-sensing images and show that it is possible to easily predict the existence of the OOP. Furthermore, it is possible to predict the metric values or their improvements with appropriate accuracy for practical use. The BPG (better portable graphics) encoder is considered a special case of an efficient compression technique. As an initial design step, the case of additive white Gaussian noise with equal variance in the three components is considered. While previous research was mainly focused on predicting the improvement (reduction) of the PSNR and PSNR-HVS-M metrics, here we focus on the modern visual quality metrics, namely PSNR-HA and MDSI. We also discuss what to do if, according to the prediction, an OOP is absent. Examples of lossy compression of noisy three-channel remote sensing images are given. It is also shown that the use of three-dimensional compression provides a compression ratio increase by several times compared with component-wise compression in the OOP.

Ambient energy harvesting from colored multiplicative noise

The present study examines the effect of multiplicative Gaussian noise on the conversion of vibration energy into electrical energy. Nonlinear oscillators are considered in the present model. The results demonstrate that similar to additive noise case, the harvesting of the vibration energy to electricity could be good if the voltage can closely track the advancement of the amplitude of the oscillator. Another observation is that the power of the micro-electromechanical systems (MEMS) goes through a maximum as with inverse of noise correlation time. Further, the position of the maxima in power generation occurs for higher noise correlation time due to the space dependency associated with the multiplicative Gaussian noise. Along with these, we have investigated how the power generation is dependent on the environment, properties of the oscillating systems and the piezoelectric dynamics.

SPAER: Sparse Deep Convolutional Autoencoder Model to Extract Low Dimensional Imaging Biomarkers for Early Detection of Breast Cancer Using Dynamic Thermography

Early diagnosis of breast cancer unequivocally improves the survival rate of patients and is crucial for disease treatment. With the current developments in infrared imaging, breast screening using dynamic thermography seems to be a great complementary method for clinical breast examination (CBE) prior to mammography. In this study, we propose a sparse deep convolutional autoencoder model named SPAER to extract low-dimensional deep thermomics to aid breast cancer diagnosis. The model receives multichannel, low-rank, approximated thermal bases as input images. SPAER provides a solution for high-dimensional deep learning features and selects the predominant basis matrix using matrix factorization techniques. The model has been evaluated using five state-of-the-art matrix factorization methods and 208 thermal breast cancer screening cases. The best accuracy was for non-negative matrix factorization (NMF)-SPAER + Clinical and NMF-SPAER for maintaining thermal heterogeneity, leading to finding symptomatic cases with accuracies of 78.2% (74.3–82.5%) and 77.7% (70.9–82.1%), respectively. SPAER showed significant robustness when tested for additive Gaussian noise cases (3–20% noise), evaluated by the signal-to-noise ratio (SNR). The results suggest high performance of SPAER for preserveing thermal heterogeneity, and it can be used as a noninvasive in vivo tool aiding CBE in the early detection of breast cancer.

A Residual Dense U-Net Neural Network for Image Denoising

In recent years, convolutional neural networks have achieved considerable success in different computer vision tasks, including image denoising. In this work, we present a residual dense neural network (RDUNet) for image denoising based on the densely connected hierarchical network. The encoding and decoding layers of the RDUNet consist of densely connected convolutional layers to reuse the feature maps and local residual learning to avoid the vanishing gradient problem and speed up the learning process. Moreover, global residual learning is adopted such that, instead of directly predicting the denoised image, the model predicts the residual noise of the corrupted image. The algorithm was trained for the case of additive white Gaussian noise and using a wide range of noise levels. Hence, one advantage of the proposal is that the denoising process does not require prior knowledge about the noise level. In order to evaluate the model, we conducted several experiments with natural image databases available online, achieving competitive results compared with state-of-the-art networks for image denoising. For comparison purpose, we use additive Gaussian noise with levels 10, 30, 50. In the case of grayscale images, we achieved PSNR of 34.39, 29.11, 26.99, and SSIM of 0.9297, 0.8193, 0.7491. For color images we obtained PSNR of 36.68, 31.43, 29.12, and SSIM of 0.9600, 0.8961, 0.8465.

Likelihood Estimation and Wavelet Transformation Based Optimization for Minimization of Noisy Pixels

In recent times the statistical computation techniques are gaining a lot of interest for analyzing the behavior of various mathematical distributions. This paper derives the likelihood distribution of image priors which has been further used to denoise the image. This paper aims to maximize the likelihood estimation of the parameters of interest i.e. prior and posterior estimation. We have proposed a prior-based distribution model which has been applied to additive, multiplicative and mixed noise cases. The various estimation parameters such as statistical variance and mean parameters have been used to evaluate the maximum likelihood of image priors for these noise models. Later, we have used an optimization technique based on the likelihood to reconstruct noise-free images efficiently. This paper uses conditional likelihood and wavelet transformation-based minimization techniques to minimize the noise in the pixels and a final denoised image is recovered. The conditional likelihood of the image has been optimized using pixel-based minimization w.r.t. the wavelet transformation coefficients. The simulation and analytical results have also been presented for the different noise cases.

A discontinuous Galerkin method for systems of stochastic differential equations with applications to population biology, finance, and physics

In this paper, we propose a discontinuous Galerkin (DG) method for systems of stochastic differential equations (SDEs) driven by m-dimensional Brownian motion. We first construct a new approximate system of SDEs on each element using whose converges to the solution of the original system. The new system is then discretized using the standard DG method for deterministic ordinary differential equations (ODEs). For the case of additive noise, we prove that the proposed scheme is convergent in the mean-square sense. Our numerical experiments suggest that our results hold true for the case of multiplicative noise as well. Several linear and nonlinear test problems are presented to show the accuracy and effectiveness of the proposed method. In particular, the proposed scheme is illustrated by considering different examples arising in population biology, physics, and mathematical finance.

Learning Causal Structures Based on Divide and Conquer

This article addresses two important issues of causal inference in the high-dimensional situation. One is how to reduce redundant conditional independence (CI) tests, which heavily impact the efficiency and accuracy of existing constraint-based methods. Another is how to construct the true causal graph from a set of Markov equivalence classes returned by these methods. For the first issue, we design a recursive decomposition approach where the original data (a set of variables) are first decomposed into two small subsets, each of which is then recursively decomposed into two smaller subsets until none of these subsets can be decomposed further. Redundant CI tests can be reduced by inferring causalities from these subsets. The advantage of this decomposition scheme lies in two aspects: 1) it requires only low-order CI tests and 2) it does not violate d -separation. The complete causality can be reconstructed by merging all the partial results of the subsets. For the second issue, we employ regression-based CI tests to check CIs in linear non-Gaussian additive noise cases, which can identify more causal directions by [Formula: see text] (or [Formula: see text]). Consequently, causal direction learning is no longer limited by the number of returned V -structures and consistent propagation. Extensive experiments show that the proposed method can not only substantially reduce redundant CI tests but also effectively distinguish the equivalence classes.

Influence of temporal averaging in the performance of a rotating retarder imaging Stokes polarimeter

We study the optimum operating conditions for a rotating retarder fixed polarizer (RRFP) when the measurements are not quasi-instantaneous but time-averaged. We obtain the optimum retardance and retarder orientations as a function of the integrated angle interval. We also study how the increase in the number of time-averaged measurements leads to a better equally weighted variance (EWV) value, and thus, to a better performance of the polarimeter in terms of noise amplification for the case of additive noise. Two different analyzers configurations are studied in this work: uniformly spaced retarder angles and when measurements are taken at optimum angles (non-uniformly spaced angles). We also consider the case of polychromatic illumination. We discuss the best measurement conditions in terms of the signal-to-noise ratio depending on whether there is a fixed or a limited amount of photons per measurement.

Recursively Learning Causal Structures Using Regression-Based Conditional Independence Test

This paper addresses two important issues in causality inference. One is how to reduce redundant conditional independence (CI) tests, which heavily impact the efficiency and accuracy of existing constraint-based methods. Another is how to construct the true causal graph from a set of Markov equivalence classes returned by these methods.For the first issue, we design a recursive decomposition approach where the original data (a set of variables) is first decomposed into three small subsets, each of which is then recursively decomposed into three smaller subsets until none of subsets can be decomposed further. Consequently, redundant CI tests can be reduced by inferring causality from these subsets. Advantage of this decomposition scheme lies in two aspects: 1) it requires only low-order CI tests, and 2) it does not violate d-separation. Thus, the complete causality can be reconstructed by merging all the partial results of the subsets.For the second issue, we employ regression-based conditional independence test to check CIs in linear non-Gaussian additive noise cases, which can identify more causal directions by x−E(x|Z)⊥z (or y−E(y|Z)⊥z). Therefore, causal direction learning is no longer limited by the number of returned Vstructures and the consistent propagation.Extensive experiments show that the proposed method can not only substantially reduce redundant CI tests but also effectively distinguish the equivalence classes, thus is superior to the state of the art constraint-based methods in causality inference.

DoPAMINE: Double-Sided Masked CNN for Pixel Adaptive Multiplicative Noise Despeckling

We propose DoPAMINE, a new neural network based multiplicative noise despeckling algorithm. Our algorithm is inspired by Neural AIDE (N-AIDE), which is a recently proposed neural adaptive image denoiser. While the original NAIDE was designed for the additive noise case, we show that the same framework, i.e., adaptively learning a network for pixel-wise affine denoisers by minimizing an unbiased estimate of MSE, can be applied to the multiplicative noise case as well. Moreover, we derive a double-sided masked CNN architecture which can control the variance of the activation values in each layer and converge fast to high denoising performance during supervised training. In the experimental results, we show our DoPAMINE possesses high adaptivity via fine-tuning the network parameters based on the given noisy image and achieves significantly better despeckling results compared to SAR-DRN, a state-of-the-art CNN-based algorithm.

Signal-to-noise ratio constrained feedback control: Robust stability analysis

The explicit consideration of a communication channel model in a feedback control loop is known to be constrained by a fundamental limitation for stabilizability on the channel signal-to-noise ratio (SNR) when the linear time invariant (LTI) plant model is unstable. The LTI modelling approach for real, usually nonlinear, processes compromises accuracy versus complexity of the resulting model. This in turn introduces a gap between the proposed model and the real process, which is known as the model uncertainty. In this paper we then study SNR limitations by considering the continuous-time scenario and the case of an additive coloured Gaussian noise (ACGN) channel with bandwidth limitation, for which we then quantify the infimal SNR subject to the simultaneous presence of plant, channel and noise model uncertainties. We observe, for the special case of memoryless additive white Gaussian noise (AWGN) channels, that the obtained SNR limitation subject to plant model uncertainty can be redefined as a channel capacity limitation for stabilization.

Noise-resistant signal processing in a ballistic laser gravimeter with a symmetrical way of measuring the gravitational acceleration

The paper considers methods for noise-resistant processing of signals in a ballistic laser gravimeter (BLG) with a symmetrical way of measuring the gravitational acceleration (GA).Seismic oscillations of the ground surface are the main source of errors when measuring the GA using a BLG. Gravimeters with the symmetrical way of measuring have a relatively short bump of the catapult, which causes basis oscillations, called auto-seismic. Systematic error of measuring the GA appears due to such the oscillations. In addition, it is necessary to take into account the influence of the discretization noises that appear when measuring trajectory parameters of the test body.The least squares (LS) method, traditionally used in ballistic gravimeters, is optimal only for the case of uncorrelated additive noise and under the condition of homoscedasticity. The mentioned conditions are not met for ballistic gravimetry. Thus, the problem of developing new methods of signal processing in a BLG is relevant.Our main attention is focused on synthesizing gravimeter weight functions aimed at minimizing the influence of the following types of additive noise: 1) external seismic noise; 2) auto-seismic noise; 3) discretization noise.A combination of quality factors was used to comprehensively take into account various noises. To determine weights of observation, the weighted least squares (WLS) method was used in this paper. The WLS method allows to use window functions, in particular the Hann window. In contrast to known methods of regression analysis, the form of such the window functions is determined not by variation of the noise variance, but rather to be selected to maximize suppression of the noise impact on the result of measuring the GA.Properties of gravimeter weight functions are given. Analysis of the impact of the form of the Hann window functions on the factors of noise resistance of the BLG is performed. It was shown that increase in the speed of window function drop from the center to the tails leads to significant decrease in the auto-seismic component of the error of measuring the GA at the expense of some negligible increase in other noise-resistance factors. The impact of omission of observations around the highest point of the test body trajectory on results of GA measurements is investigated. It is shown that fulfillment of the Nyquist condition for the weight function of the gravimeter allows to decrease the auto-seismic component of the GA measurement error.

Fundamental limits of symmetric low-rank matrix estimation

We consider the high-dimensional inference problem where the signal is a low-rank symmetric matrix which is corrupted by an additive Gaussian noise. Given a probabilistic model for the low-rank matrix, we compute the limit in the large dimension setting for the mutual information between the signal and the observations, as well as the matrix minimum mean squared error, while the rank of the signal remains constant. We also show that our model extends beyond the particular case of additive Gaussian noise and we prove an universality result connecting the community detection problem to our Gaussian framework. We unify and generalize a number of recent works on PCA, sparse PCA, submatrix localization or community detection by computing the information-theoretic limits for these problems in the high noise regime. In addition, we show that the posterior distribution of the signal given the observations is characterized by a parameter of the same dimension as the square of the rank of the signal (i.e. scalar in the case of rank one). This allows to locate precisely the information-theoretic thresholds for the above mentioned problems. Finally, we connect our work with the hard but detectable conjecture in statistical physics.

Likelihood for transcriptions in a genetic regulatory system under asymmetric stable Lévy noise.

This work is devoted to investigating the evolution of concentration in a genetic regulation system, when the synthesis reaction rate is under additive and multiplicative asymmetric stable Lévy fluctuations. By focusing on the impact of skewness (i.e., non-symmetry) in the probability distributions of noise, we find that via examining the mean first exit time (MFET) and the first escape probability (FEP), the asymmetric fluctuations, interacting with nonlinearity in the system, lead to peculiar likelihood for transcription. This includes, in the additive noise case, realizing higher likelihood of transcription for larger positive skewness (i.e., asymmetry) index β, causing a stochastic bifurcation at the non-Gaussianity index value α = 1 (i.e., it is a separating point or line for the likelihood for transcription), and achieving a turning point at the threshold value β≈-0.5 (i.e., beyond which the likelihood for transcription suddenly reversed for α values). The stochastic bifurcation and turning point phenomena do not occur in the symmetric noise case (β = 0). While in the multiplicative noise case, non-Gaussianity index value α = 1 is a separating point or line for both the MFET and the FEP. We also investigate the noise enhanced stability phenomenon. Additionally, we are able to specify the regions in the whole parameter space for the asymmetric noise, in which we attain desired likelihood for transcription. We have conducted a series of numerical experiments in "regulating" the likelihood of gene transcription by tuning asymmetric stable Lévy noise indexes. This work offers insights for possible ways of achieving gene regulation in experimental research.