Filtering Equations Research Articles

A Novel Speckle Suppression Method with Quantitative Combination of Total Variation and Anisotropic Diffusion PDE Model

Speckle noise seriously affects synthetic aperture radar (SAR) image application. Speckle suppression aims to smooth the homogenous region while preserving edge and texture in the image. A novel speckle suppression method based on the combination of total variation and partial differential equation denoising models is proposed in this paper. Taking full account of the local statistics in the image, a quantization technique—which is different from the normal edge detection method—is supported by the variation coefficient of the image. Accordingly, a quantizer is designed to respond to both noise level and edge strength. This quantizer automatically determines the threshold of diffusion coefficient and controls the weight between total variation filter and anisotropic diffusion partial differential equation filter. A series of experiments are conducted to test the performance of the quantizer and proposed filter. Extensive experimental results have demonstrated the superiority of the proposed method with both synthetic images and natural SAR images.

Optimizing the Energy Efficiency of Unreliable Memories for Quantized Kalman Filtering

This paper presents a quantized Kalman filter implemented using unreliable memories. We consider that both the quantization and the unreliable memories introduce errors in the computations, and we develop an error propagation model that takes into account these two sources of errors. In addition to providing updated Kalman filter equations, the proposed error model accurately predicts the covariance of the estimation error and gives a relation between the performance of the filter and its energy consumption, depending on the noise level in the memories. Then, since memories are responsible for a large part of the energy consumption of embedded systems, optimization methods are introduced to minimize the memory energy consumption under the desired estimation performance of the filter. The first method computes the optimal energy levels allocated to each memory bank individually, and the second one optimizes the energy allocation per groups of memory banks. Simulations show a close match between the theoretical analysis and experimental results. Furthermore, they demonstrate an important reduction in energy consumption of more than 50%.

Excitation–response relationships for linear structural systems with singular parameter matrices: A periodized harmonic wavelet perspective

Novel wavelet-based input–output (excitation–response) relationships are developed referring to stochastically excited linear structural systems with singular parameter matrices. This is done by relying on the family of periodized generalized harmonic wavelets for expanding the excitation and response processes, and by resorting to the concept of Moore–Penrose matrix inverse for solving the resulting overdetermined linear system of algebraic equations to calculate the response wavelet coefficients. In this regard, system response statistics in the joint time–frequency domain, such as the response evolutionary power spectrum matrix, can be determined in a straightforward manner based on the herein derived input–output relationships. The developed technique can be construed as a generalization of earlier efforts in the literature to account for singular parameter matrices in the governing equations of motion. The reliability of the technique is demonstrated by comparing the analytical results with pertinent Monte Carlo simulation data. This is done in conjunction with various diverse numerical examples pertaining to energy harvesters with coupled electromechanical equations, oscillators subject to non-white excitations modeled via auxiliary filter equations and structural systems modeled by a set of dependent coordinates.

Optimized Design of 3D Spatial Images Based on Kalman Filter Equation

This paper takes the advantageous ability of Kalman filter equation as a means to jointly realize the accurate and reliable extraction of 3D spatial information and carries out the research work from the extraction of 3D spatial position information from multisource remote sensing optical stereo image pairs, recovery of 3D spatial structure information, and joint extraction of 3D spatial information with optimal topological structure constraints, respectively. Taking advantage of the stronger effect capability of Wiener recovery and shorter computation time of Kalman filter recovery, Wiener recovery is combined with Kalman filter recovery (referred to as Wiener-Kalman filter recovery method), and the mean square error and peak signal-to-noise ratio of the recovered image of this method are comparable to those of Wiener recovery, but the subjective evaluation concludes that the recovered image obtained by the Wiener-Kalman filter recovery method is clearer. To address the problem that the Kalman filter recovery method has the advantage of short computation time but the recovery effect is not as good as the Wiener recovery method, an improved Kalman filter recovery algorithm is proposed, which overcomes the fact that the Kalman filter recovery only targets the rows and columns of the image matrix for noise reduction and cannot utilize the pixel point information among the neighboring rows and columns. The algorithm takes the first row of the matrix image as the initial parameter of the Kalman filter prediction equation and then takes the first row of the recovered image as the initial parameter of the second Kalman filter prediction equation. The algorithm does not need to estimate the degradation function of the degradation system based on the degraded image, and the recovered image presents the image edge detail information more clearly, while the recovery effect is comparable to that of the Wiener recovery and Wiener-Kalman filter recovery method, and the improved Kalman filter recovery method has stronger noise reduction ability compared with the Kalman filter recovery method. The problem that the remote sensing optical images are seriously affected by shadows and complex environment detail information when 3D spatial structure information is extracted and the data extraction feature edge is not precise enough and the structure information extraction is not stable enough is addressed. A global optimal planar segmentation method with graded energy minimization is proposed, which can realize the accurate and stable extraction of the topological structure of the top surface by combining the edge information of remote sensing optical images and ensure the accuracy and stability of the final extracted 3D spatial information.

Embedded Image and Video Coding Algorithm Based on Adaptive Filtering Equation

Based on the improved adaptive filtering method, this paper conducts in-depth discussion and research on embedded graphics and video coding and chooses to improve the adaptive filtering algorithm from three aspects: starting point prediction, search template, and window partitioning. The algorithm is imported into the encoder for video capture and encoding. By capturing videos of different formats, resolutions, and times, the memory size of the video files collected before and after the algorithm optimization is compared, and the optimized algorithm occupies the memory space of the video file in the actual system. The conclusion of less and higher coding rates. The collected video information is stored on a personal computer equipped with a freeness, and external electronic devices only need to download and install the browser, and the collected video information can be accessed in the local area network through the protocol. The improved coding algorithm has higher coding efficiency and can reduce the storage space occupied by the video.

Optimization of Correlation Filters Using Extended Particle Swarm Optimization Technique.

In the past few decades, the field of image processing has seen a rapid advancement in the correlation filters, which serves as a very promising tool for object detection and recognition. Mostly, complex filter equations are used for deriving the correlation filters, leading to a filter solution in a closed loop. Selection of optimal tradeoff (OT) parameters is crucial for the effectiveness of correlation filters. This paper proposes extended particle swarm optimization (EPSO) technique for the optimal selection of OT parameters. The optimal solution is proposed based on two cost functions. The best result for each target is obtained by applying the optimization technique separately. The obtained results are compared with the conventional particle swarm optimization method for various test images belonging from different state-of-the-art datasets. The obtained results depict the performance of filters improved significantly using the proposed optimization method.

Embedding non-linear filtering in autonomous underwater vehicle dynamics via the Kolmogorov backward equation

This paper revisits the state vector of an autonomous underwater vehicle (AUV) dynamics coupled with the underwater Markovian stochasticity in the ‘non-linear filtering’ context. The underwater stochasticity is attributed to atmospheric turbulence, planetary interactions, sea surface conditions and astronomical phenomena. In this paper, we adopt the Itô process, a homogeneous Markov process, to describe the AUV state vector evolution equation. This paper accounts for the process noise as well as observation noise correction terms by considering the underwater filtering model. The non-linear filtering of the paper is achieved using the Kolmogorov backward equation and the evolution of the conditional characteristic function. The non-linear filtering equation is the cornerstone formalism of stochastic optimal control systems. Most notably, this paper introduces the non-linear filtering theory into an underwater vehicle stochastic system by constructing a lemma and a theorem for the underwater vehicle stochastic differential equation that were not available in the literature.

Modified Filter Equation with Improved Fuzzy Logic System Based Directional Median Filter for Mixed Noise

DWM (Directional weighted median) filter is very popular in filtering digital image and remove mixed noise. Fuzzy logic is implemented with median filters to improve its performance. In the previous work, fuzzy logic system is implemented with switching median filter and gives better performance than directional median filter as well as switching median filter. Experimenting directional median filter with same fuzzy logic system didn’t yield to better results therefore fuzzy logic parameters has been changes as per strong points of directional weighted median filter and a constant has been included in the filtering equation to improve the results. So in this proposed work, we have successfully implemented directional weighted median filter with fuzzy logic system which is proving better results than DWM and FSMF (Fuzzy Switching Median Filter). PSNR (Peak Signal to Noise Ratio)is used for qualitative analysis of results.

Linear-quadratic non-zero sum differential game for mean-field stochastic systems with asymmetric information

In this study, we consider a class of asymmetric information linear-quadratic non-zero sum stochastic differential game problems. The system dynamics are governed by a forward linear mean-field stochastic differential equation and the cost functional is quadratic. In order to facilitate some practical applications, the mean-field terms for the state process and control process are considered in both the system dynamics and cost functional. Using the classical calculus of variation and dual methods, the open-loop Nash equilibrium point can be expressed by introducing an auxiliary mean-field forward backward stochastic differential equation, which comprises one forward and two backward components. Due to some Riccati equations and ordinary differential equations that possess unique solutions, the Nash equilibrium point can also be represented in feedback form for several special cases under asymmetric information. The corresponding filtering equations are derived, and the existence and uniqueness of the solutions are proved. An investment problem in finance is discussed to demonstrate the good performance of the theoretical results.

Efficient Attitude Estimators: A Tutorial and Survey

Inertial sensors based on micro-electro-mechanical systems (MEMS) technology, such as accelerometers and angular rate sensors, are cost-effective solutions used in inertial navigation systems in a broad spectrum of applications that estimate position, velocity and orientation of a system with respect to an inertial reference frame. Although they present several advantages in terms of cost and form factor, they are prone to various disturbances such as noise, biases, and random walk that degrade their orientation estimation. The task of an orientation filter is to compute an optimal solution for the attitude state, consisting of roll, pitch and yaw, through the fusion of angular rate, accelerometer, and magnetometer measurements, regardless of the underlying environmental constraints. The aim of this paper is threefold: first, it serves researchers and practitioners in the signal processing community seeking the most appropriate attitude estimators that fulfills their needs, shedding light on the drawbacks and the advantages of a wide variety of designs. Second, it serves as a survey and tutorial for existing estimator designs in the literature, assessing their design aspects and components, and dissecting their hidden details for the benefit of researchers. Third, a comprehensive list of algorithms is discussed for a fully functional inertial navigation system, starting from the navigation equations and ending with the filter equations, keeping in mind their suitability for power-limited embedded processors. The source code of all algorithms is published, with the aim of it being an out-of-box solution for researchers in the field. The reader will take away the following concepts from this article: understand the key concepts of an inertial navigation system; be able to implement and test a complete stand alone solution; be able to evaluate and understand different algorithms; understand the trade-offs between different filter architectures and techniques; and understand efficient embedded processing techniques, trends and opportunities.

Discrete-time estimation of switched stochastic polynomial systems and its application to reconstruction of induction motor variables

This paper presents a solution to the filtering problem for a class of stochastic discrete-time nonlinear polynomial systems with switching in the state equation over linear observations and its application to a mechatronic system. The switching in the state equation is performed between two different nonlinear functions according to a sequence of independent Bernoulli random variables that take the quantities of zero and one. The mean-square filtering solution is obtained for a general nonlinear discrete-time polynomial system and a finite-dimensional system of filtering equations is then obtained for a second degree polynomial system as a particular case. The mean-square estimates of polynomial state terms are expressed as functions of the estimate and covariance matrix. Finally, some numerical simulations are carried out to reconstruct the variable states given a vector output measurement for a linear system, a second degree polynomial system, and an induction motor model to show effectiveness of the proposed algorithm. The proposed method is compared with an extended Kalman filter-based algorithm for discrete-time switched nonlinear systems.

A Noise Reduction Method Based on Modified LMS Algorithm of Real time Speech Signals

In real time speech de-noising, adaptive filtering technique with variable length filters are used which is used to track the noise characteristics and through those characteristics the filter equations are selected The main features that attracted the use of the LMS algorithm are low computational complexity, proof of convergence in stationary environment. In this paper, modified LMS algorithm is proposed which is used to denoise real time speech signal. The proposed algorithm is made by combining general LMS algorithm with Diffusion least mean-square algorithm which increase the capabilities of adaptive filtering. The performance parameter calculation shows that the proposed algorithm is effective to de-noise speech signal. A full programming routine written in MATLAB software is provided for replications and further research applications.

Innovation Feedback Kalman Filter

The conventional Kalman filter demands complete knowledge of the actual filter model, which is usually inaccessible in practical problems. When dealing with the filtering problem with uncertainty, any improper prediction of the uncertainty may degrade the filtering performance and even lead to divergence phenomenon. In this paper, the innovation feedback Kalman filter, which introduces the innovation feedback controller to the Kalman filter equations, is newly proposed on the basis of automatic control theory to address the filtering problem with uncertainty and is different from other filtering methods. By studying the estimate error equations, the estimate bias is extracted and its propagation mechanism is formulated. The estimate bias propagation equations reveal that eliminating the estimate bias essentially equivalents to an output regulation problem with uncertain exosystem. And a concise yet effective innovation feedback controller is subsequently given as an example. The proposed method is applied to one-dimension target tracking scenario and its high estimation accuracy performance and good stability are simulated through a comparative analysis with the ideal filtering results.

Noise covariance matrix estimation with subspace model identification for Kalman filtering

SummaryA problem frequently encountered in Kalman filtering is the tuning of the noise covariance matrices. Indeed, misspecifying their values can drastically reduce the performance of the Kalman filter. Unfortunately, in most practical cases, noise statistics are not known a priori. This paper focuses on a method relying on subspace model identification theory to determine them accurately. This solution is developed for linear time invariant systems with stationary random disturbances having constant covariance matrices. Practically, these noise covariance matrices are determined from the comparison between an estimated state space representation and the discrete time state space representation involved in the Kalman filter equations. The method developed in this paper departs from most of the solutions available in the literature by the fact that it does not need any tuning parameter to be chosen by the user. After discussing theoretical results, several numerical examples are given to demonstrate the efficiency of the approach.

발사체 분류를 위한 탄도계수 추정 방법 비교

The ballistic coefficient is a measure of a projectile’s ability to overcome air resistance. Because the motion equations of the projectile are functions of the ballistic coefficient, several studies estimating the projectile’s ballistic coefficient have been conducted to track its trajectory. In this paper, two methods are proposed to estimate the ballistic coefficient for classification of the projectile. The primary techniques used are the extended Kalman filter and ordinary differential equation parameter estimation. The true projectile trajectory is created via simulation, and the proposed method is used to estimate the ballistic coefficient. Furthermore their performances are compared and analyzed.

Two-Dimensional Separable Gridless Direction-of-Arrival Estimation Based on Finite Rate of Innovation

In order to solve the problem that the gridless DOA estimation algorithms based on generalized finite rate of innovation (FRI) signal reconstruction model are not suitable for two-dimensional DOA estimation using planar array, a separable gridless DOA estimation algorithm exploiting bi-orthogonal sparse linear array (BSLA) structure is proposed in this article, which is called 2D-SGFRI. The 2D-SGFRI algorithm firstly recovers the covariance data of the virtual array formed by BSLA through the matrix completion method, so as to obtain the complete covariance data vectors about two independent parameters respectively. Next, since the covariance data vector satisfies the constraints of annihilation filter equations, the generalized FRI signal reconstruction model can be utilized to retrieve DOA from the covariance data vector. Compared with the existing DOA estimation algorithms based on generalized FRI signal reconstruction model, the 2D-SGFRI algorithm can be can be effectively applied to two-dimensional DOA estimation, and can obtain stable estimation results. At the same time, due to the reduction of the dimension of positive semidefinite matrix, the 2D-SGFRI algorithm can significantly reduce the computational complexity compared with the two-dimensional DOA estimation algorithms based on atomic norm minimization (ANM). A series of simulation experiments are shown to verify the effectiveness and superiority of 2D-SGFRI algorithm.

Structural health monitoring with non-linear sensor measurements robust to unknown non-stationary input forcing

Bayesian filtering based structural health monitoring algorithms typically assume stationary white Gaussian noise models to represent an unknown input forcing. However, typical structural damages occur mostly under the action of extreme loading conditions, like earthquake or high wind/waves, which are characteristically non-stationary and non-Gaussian. Clearly, this invalidates this basic assumption, causing these algorithms to perform poorly under non-stationary noise conditions. This paper extends an existing interacting filtering algorithm to efficiently estimate structural damages while being robust to unknown non-stationary non-Gaussian input forcing. Furthermore, this approach is generalized beyond linear measurements to encompass the case of non-linear measurements such as strains. The joint estimation of state and parameters is performed by combining Ensemble Kalman filtering, for non-linear system state estimation, and Particle filtering to estimate changes in the structural parameters. The robustness against input forcing is achieved through an output injection approach embedded in the state filter equation. Numerical simulations for two kinds of response measurements (acceleration and strain) are performed on a 3D frame structure under different damage location and severity scenarios. The sensitivity with respect to noise and the impact of different sensor combinations have also been investigated.

On Entropy of Probability Integral Transformed Time Series.

The goal of this paper is to investigate the changes of entropy estimates when the amplitude distribution of the time series is equalized using the probability integral transformation. The data we analyzed were with known properties—pseudo-random signals with known distributions, mutually coupled using statistical or deterministic methods that include generators of statistically dependent distributions, linear and non-linear transforms, and deterministic chaos. The signal pairs were coupled using a correlation coefficient ranging from zero to one. The dependence of the signal samples is achieved by moving average filter and non-linear equations. The applied coupling methods are checked using statistical tests for correlation. The changes in signal regularity are checked by a multifractal spectrum. The probability integral transformation is then applied to cardiovascular time series—systolic blood pressure and pulse interval—acquired from the laboratory animals and represented the results of entropy estimations. We derived an expression for the reference value of entropy in the probability integral transformed signals. We also experimentally evaluated the reliability of entropy estimates concerning the matching probabilities.

On the continuous time limit of the ensemble Kalman filter

We present recent results on the existence of a continuous time limit for Ensemble Kalman Filter algorithms. In the setting of continuous signal and observation processes, we apply the original Ensemble Kalman Filter algorithm proposed by Burgers, van Leeuwen, and Evensen [Monthly Weather Review 126 (1998), pp. 1719–1724] as well as a recent variant of de Wiljes, Reich, and Stannat [SIAM J. Appl. Dyn. Syst. 17 (2018), no. 2, pp. 1152–1181] to the respective discretizations and show that in the limit of decreasing stepsize the filter equations converge to an ensemble of interacting (stochastic) differential equations in the ensemble-mean-square sense. Our analysis also allows for the derivation of convergence rates with respect to the stepsize. An application of our analysis is the rigorous derivation of continuous-time ensemble filtering algorithms consistent with discrete-time approximation schemes. Conversely, the continuous time limit allows for a better qualitative and quantitative analysis of the discrete-time counterparts using the rich theory of dynamical systems in continuous time.

Kalman filter tracking of sequence spot centroid ablated by femtosecond laser

AbstractIn the process of processing monocrystalline silicon by femtosecond laser, there is a phenomenon of derived plasma luminescence, which formed the time series spot. The laser spot image contains some effective information related to process parameters, such as ablation power of femtosecond laser, processing speed and so on. Through the comprehensive analysis of the geometric features of the sequential spot image, it can be used in the closed‐loop feedback system of micromachining based on the features of the plasma spot image. Because the spot centroid can be used to locate the spot position accurately, the sequential spot centroid obtained by gray centroid method was used as the sampling data, which was tracked by Kalman filter, and then the movement law of the spot was observed and analyzed. In this paper, the discrete linear Kalman filter algorithm is used to establish motion model of the spot centroid, and the next movement state of spot is predicted, and then the trajectory of the spot is obtained. Generally, the traditional Kalman filter has some shortcomings, such as inaccurate model, too much noise, and it is difficult to achieve the optimal estimation, therefore, the application of Kalman filter in spot centroid tracking is optimized. Firstly, the process noise is updated in real time to correct the direction of acceleration accurately and reduce the uncontrollability of process noise by introducing the factor l; Secondly, combined with Wiener filter, the observation value is optimized to minimize the influence of observation noise; Thirdly, starting from the Kalman filter equation, the actual value of the previous moment is connected with the optimal estimated value by weighted addition based on the “Identity of A‐OE” operator, so as to improve the prediction accuracy of the current time. Finally, the EF‐BP neural network is introduced on the basis of the “Identity of A‐OE” operator, and a stable training system is used to optimize the weight repeatedly, so that the error can reach the expected value. The results show that the error of the observation value is reduced by 8.2503%, the error of the optimal filter estimation value is reduced to 0.3490, and the error reduction range is 6.7662. The centroid tracking accuracy of the ablated spot image is improved obviously.