Approximate Filter Research Articles

Uncertainty quantification of nonlinear Lagrangian data assimilation using linear stochastic forecast models

Lagrangian data assimilation exploits the trajectories of moving tracers as observations to recover the underlying flow field. One major challenge in Lagrangian data assimilation is the intrinsic nonlinearity that impedes using exact Bayesian formulae for the state estimation of high-dimensional systems. In this paper, an analytically tractable mathematical framework for continuous-in-time Lagrangian data assimilation is developed. It preserves the nonlinearity in the observational processes while approximating the forecast model of the underlying flow field using linear stochastic models (LSMs). A critical feature of the framework is that closed analytic formulae are available for solving the posterior distribution, which facilitates mathematical analysis and numerical simulations. First, an efficient iterative algorithm is developed in light of the analytically tractable statistics. It accurately estimates the parameters in the LSMs using only a small number of the observed tracer trajectories. Next, the framework facilitates the development of several computationally efficient approximate filters and the quantification of the associated uncertainties. A cheap approximate filter with a diagonal posterior covariance derived from the asymptotic analysis of the posterior estimate is shown to be skillful in recovering incompressible flows. It is also demonstrated that randomly selecting a small number of tracers at each time step as observations can reduce the computational cost while retaining the data assimilation accuracy. Finally, based on a prototype model in geophysics, the framework with LSMs is shown to be skillful in filtering nonlinear turbulent flow fields with strong non-Gaussian features.

A PRticle filter algorithm for nonparametric estimation of multivariate mixing distributions

Predictive recursion (PR) is a fast, recursive algorithm that gives a smooth estimate of the mixing distribution under the general mixture model. However, the PR algorithm requires evaluation of a normalizing constant at each iteration. When the support of the mixing distribution is of relatively low dimension, this is not a problem since quadrature methods can be used and are very efficient. But when the support is of higher dimension, quadrature methods are inefficient and there is no obvious Monte Carlo-based alternative. In this paper, we propose a new strategy, which we refer to as PRticle filter, wherein we augment the basic PR algorithm with a filtering mechanism that adaptively reweights an initial set of particles along the updating sequence which are used to obtain Monte Carlo approximations of the normalizing constants. Convergence properties of the PRticle filter approximation are established and its empirical accuracy is demonstrated with simulation studies and a marked spatial point process data analysis.

Attacking the Privacy of Approximate Membership Check Filters by Positive Concentration

Approximate membership check filters are increasingly used to speed up data processing in many applications. Also, privacy is becoming a key design objective for many systems and thus, the privacy of filters needs to be carefully considered. Previous works have shown that an attacker that knows the implementation details of the filter and has access to its content, may be able to extract some information about the elements stored in the filter. This attack is, however, specific to Bloom filters and requires that the universe of elements must be small. In this article, we show that in many practical settings, an attacker that has only a black-box access to the filter, can extract information about the elements stored in the filter regardless of the specific filter type and the universe size. This is possible based on the key observation that in many applications, the elements stored in the filter are not randomly chosen, but they are concentrated in one or more parts of the universe of elements. To identify these parts, the positive probability can be measured on different parts of the universe; the parts having significantly larger values than the average positive probability for the filter are the ones on which the filter elements are concentrated. This approach is formalized and applied to several case studies showing the process by which the attacker can get additional information about the elements stored for the filters in a wide range of scenarios.

IPDT Model-Based Ziegler-Nichols Tuning Generalized to Controllers with Higher-Order Derivatives.

The paper extends the earlier work entitled "Making the PI and PID Controller Tuning Inspired by Ziegler and Nichols Precise and Reliable", to higher-order controllers and a broader range of experiments. The original series PI and PID controllers, based on automatic reset calculated by filtered controller outputs, are now augmented by higher-order output derivatives. This increases the number of degrees of freedom that can be used to modify the resulting dynamics, accelerates transient responses, and increases robustness to unmodeled dynamics and uncertainties. The fourth order noise attenuation filter used in the original work allows for the addition of an acceleration feedback signal, thus resulting in a series PIDA controller or even a jerk feedback that leads to a PIDAJ series controller. Such a design can further use the original process and filter approximation of the step responses through the integral-plus-dead-time (IPDT) model, while allowing experimentation with disturbance and setpoint step responses of the series PI, PID, PIDA and PIDAJ controllers, and thus, evaluating the role of output derivatives and noise attenuation from a broader perspective. All controllers considered are tuned using the Multiple Real Dominant Pole (MRDP) method, which is complemented by a factorization of the controller transfer functions to achieve the smallest possible time constant for automatic reset. The smallest time constant is chosen to improve the constrained transient response of the considered controller types. The obtained excellent performance and robustness allow the proposed controllers to be applied to a wider range of systems with dominant first-order dynamics. The proposed design is illustrated on a real-time speed control of a stable direct-current (DC) motor, which is approximated (together with a noise attenuation filter) by an IPDT model. The transient responses obtained are nearly time-optimal, with control signal limitations active for most setpoint step responses. Four controllers with different degrees of derivative with generalized automatic reset were used for comparison. It was found that controllers with higher-order derivatives may significantly improve the disturbance performance and virtually eliminate overshoots in the setpoint step responses in constrained velocity control.

Low-energy Pipelined Hardware Design for Approximate Medium Filter

Image and video processing algorithms are currently crucial for many applications. Hardware implementation of these algorithms provides higher speed for large computation applications. Removing noise is often a typical pre-processing step to enhance the results of later analysis and processing. Median filter is a typical nonlinear filter that is very commonly used for impulse noise elimination in digital image processing. This article suggests a low-energy median filter hardware design for battery-based hardware applications. An approximate solution with high accuracy is investigated to speed up the filtering operation, reduce the area, and consume less power/energy. Pipelining and parallelism are used to optimize the speed and power of this technique. Non-pipelined, two different pipelined structures, and two parallel architectures versions are designed. The design versions are implemented first with a Virtex-5 LX110T FPGA and then using the UMC 130nm standard cell ASIC technology. The selection and the even-odd sorting-based median filters are also implemented for an equitable comparison with the standard median filtering techniques. The suggested non-pipelined median filter design enhances the throughput 35% more than the highest investigated state of the art. The pipelining enhances the throughput to more than twice its value. Additionally, the parallel architecture decreases the area and the consumed power by around 40%. The simulation results reveal that one of the suggested designs significantly decreases the area, with the same speed as the fastest design in the literature, without noticeably degrading the accuracy, and a significant decrease in energy consumption by about 60%.

A novel robust IMM filter for jump Markov systems with heavy-tailed process and measurement noises

In some tracking application scenarios, the performances of the traditional interacting multiple model (IMM) approach may degrade dramatically since the presence of outliers in process and measurement noises. Aiming at solving the filtering problem for jump Markov systems containing heavy-tailed process and measurement noises, a new robust student's t-based Gaussian approximate filter utilizing the IMM filtering framework is presented in this paper. Firstly, the heavy-tailed process and measurement noises are assumed to obey student's t-distributions. Secondly, the degree of freedom parameters and the scale matrices are modeled as Gamma distributions and inverse-Wishart distributions respectively. Finally, the system state, degree of freedom parameters, scale matrices and the mode probabilities are inferred simultaneously by introducing the variational Bayesian (VB) theory. Maneuvering target tracking simulation verifies that the designed algorithm achieves superior estimation performances among the state-of-the-art filters.

Design of robust Gaussian approximate filter and smoother with latency probability identification

The Kalman filter is a linear optimal estimator when it can accurately receive measurements and the noise obeys a Gaussian distribution. However, the measurement noise may have heavy-tailed characteristics because of outlier interference. Actually, real-world measurement information may have one-step randomly delayed measurements (ORDMs) with unknown latency probability (LP) due to exogenous disturbances. Moreover, systems are usually not linear. To overcome the influence of the state estimator of nonlinear systems from heavy-tailed measurement noise (HMN) and the unknown LP, this paper focuses on the robust Gaussian approximate (GA) filter and smoother for the above issues. First, the one-step predicted probability density function (PDF) is modeled as a Gaussian distribution, and the likelihood PDF is modeled as a Normal–Gamma–Beta (NGBM) distribution. Furthermore, novel robust Gaussian approximate filter and smoother are proposed based on the variational Bayesian technique. They provide general estimation frameworks for estimating state of nonlinear systems with HMN and ORDMs, and different nonlinear estimators can be implemented using different numerical techniques. In addition, the computational complexity of the proposed algorithms is calculated. Finally, taking a univariate nonstationary growth model, a passive ranging problem, and target tracking as examples, the effectiveness and superiority of the proposed robust algorithms contrasted with the existing algorithms are shown. The outcomes show that the proposed methods have higher estimation precision; however, the computational burden is marginally increased.

Design of Approximate Bilateral Filters for Image Denoising on FPGAs

This paper presents the hardware design of fast and low-cost denoising filters suitable to be exploited in the enabling technologies for Industry 5.0. A novel approximate computing strategy is introduced to reduce the computational complexity of the image denoising operation and to comply with real-time requirements. Firstly, it is demonstrated that the novel approximate approach can be helpfully exploited in the design of reconfigurable denoising filters able to reach image qualities as close as possible to the precise software counterparts. The reconfigurability leads to hardware architectures run-time adaptable to different levels of noise, whereas the adopted approximation strategy limits hardware resources and energy requirements. Quality tests, performed at various image and kernel sizes, and noise standard deviations, demonstrate that the approximate denoising approach presented here reaches PSNR and SSIM comparable with the precise denoise filtering. In comparison with state-of-the-art FPGA-based competitors, the novel filters reduce the resources requirements by up to 70%, achieve frame rates up to 35 times higher, and dissipate more than 45% lower power. When implemented within the XC7Z7020 FPGA device, a <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$5\times 5$ </tex-math></inline-formula> filter designed as proposed here denoises <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$512\times 512$ </tex-math></inline-formula> grayscale images using only 1689 LUTs, 2635 Flip-Flops and 32 DSPs. Moreover, it processes up to 926.8 frames per second, consumes just 63mW @ 244MHz and, with a noise standard deviation equal to 10, it achieves an average PSNR of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\sim 33$ </tex-math></inline-formula> dB with an average SSIM of ~0.86.

Exact Solution and Projection Filters for Open Quantum Systems Subject to Imperfect Measurements

In this letter, we consider an open quantum system undergoing imperfect and indirect measurement. For quantum non-demolition (QND) measurement, we show that the system evolves on an appropriately chosen manifold and we express the exact solution of the quantum filter equation in terms of the solution of a lower dimensional stochastic differential equation. In order to further reduce the dimension of the system under study, we consider the projection on the lower dimensional manifold originally introduced in Gao et al. (2019) for the case of perfect measurements. An error analysis is performed to evaluate the precision of this approximate quantum filter, focusing on the case of QND measurement. Simulations suggest the efficiency of the proposed quantum projection filter, even in presence of a stabilizing feedback control which depends on the projection filter.

A Data-Driven Hybrid Methodology Using Randomized Low-Rank DMD Approximation and Flat-Top FIR Filter for Voltage Fluctuations Monitoring in Grid-Connected Distributed Generation Systems

The voltage fluctuations are caused by variations in renewable energy source outputs, increased usage of nonlinear loads, high reactive power consumption of loads, etc. This results in damage to electric components and enormous economic losses. It is necessary to measure the parameters of voltage fluctuations and flicker levels to achieve a secure supply of power. However, higher-order harmonics and noises in voltage waveform make such assessment a burdensome task. The traditional techniques fail to provide exact monitoring due to the uncharacteristic variations in the input voltage signal. Nowadays, data-driven methods have become a prominent choice for non-stationary, nonlinear signal analysis as it efficiently tackles atypical changes by its capability to identify spatio-temporal information from given data. The present paper proposes a data-driven hybrid methodology for monitoring voltage fluctuations using the low-rank approximation of dynamic mode decomposition (DMD) and flat-top finite impulse response (FIR) filter. The low-rank approximation of DMD enables to identify the spatio-temporal coherent structures from the data by mapping the data into a lower-dimensional space. Adaptive flat-top FIR filter tracks the fundamental dynamic phasor by identifying the instantaneous frequency, magnitude and phase variations. The performance of the proposed hybrid methodology is assessed using different test cases, and is compared with various existing approaches. The promising results suggest that the proposed methodology can be used for accurate parameter identification and monitoring voltage fluctuations in smart grid scenarios. It can also be used as an accurate tool in distribution grids to detect higher-order harmonics.

A Universal Filter Approximation of Edge Diffraction for Geometrical Acoustics

Sound propagation in urban and indoor environments often involves diffraction at corners, finite objects and openings, resulting in perceptually relevant frequency-dependent attenuation. Geometrical acoustics (GA) have become a de-facto standard for the prediction and simulation of sound propagation and real-time virtual acoustics, including effects of edge diffraction. However, methods to account for edge diffraction often assume infinite edges, such as the uniform theory of diffraction, or are computationally involved, such as the Biot-Tolstoy-Medwin-Svensson (BTMS) method. Particularly for interactive auralization, an efficient approximation of edge diffraction is desirable. Here, an extension of GA to account for diffraction with simple-to-derive parameters and a time-domain recursive filter implementation is suggested. This universal diffraction filter approximation (UDFA) describes diffraction from infinite and finite wedges using a combination of first-order and (fractional) half-order low-pass filters to account for the frequency-dependent attenuation of the diffracted incident and reflected sound field. The physically-based UDFA exactly matches asymptotic solutions for infinite wedges and provides approximations for finite wedges. It is demonstrated that first-order diffraction from flat finite objects like a plate or an aperture can be described by combining the filters for each edge. To account for effects of higher-order diffraction, an additional heuristic filter extension is suggested.

Breaking down Polyblur: Fast Blind Correction of Small Anisotropic Blurs

Polyblur is a two stage blind deblurring technique for removing small-sized blurs, like small camera shake or the lens point-spread function, proposed in 2021 by Delbracio et al. First, the blur is modeled with a zero-mean anisotropic Gaussian kernel whose parameters are rapidly estimated from the oriented blurry image gradients. Second, a sharp estimate is obtained by applying an approximate deconvolution filter, which is designed as a polynomial function of the estimated blurring kernel. Since in practice true blurs are not exactly Gaussian filters, the residual blur is gradually removed by repeating this two-stage procedure. Because it relies only on simple image manipulations, Polyblur is a quick blind deblurring technique, running in a fraction of a second on a smartphone. In this presentation, we analyze its key ingredients, showcase several use cases on real images, and provide Numpy and Pytorch implementations.

Polynomial chaos Kalman filter for target tracking applications

In this paper, an approximate Gaussian state estimator is developed based on generalised polynomial chaos expansion for target tracking applications. Motivated by the fact that calculating conditional moments in an approximate Gaussian filter involves computing integrals with respect to Gaussian density, the authors approximate the non-linear dynamics using polynomial chaos expansion. Second-order as well as third-order polynomial chaos expansions were used for approximate filtering, to derive the necessary recursive algorithm and also provide certain algebraic simplifications which reduce the computational burden without significantly affecting the filtering performance. Two comprehensive numerical experiments for multivariate systems, including one for a multi-model system, demonstrate the potential of the new algorithms.

Switching Gaussian-heavy-tailed distribution based robust Gaussian approximate filter for INS/GNSS integration

In inertial navigation system and global navigation satellite system (INS/GNSS) integration, the practical stochastic measurement noise may be non-stationary heavy-tailed distribution due to outlier measurements induced by multipath and/or non-line-of-sight receptions of the original GNSS signals. To address the problem, a new switching Gaussian-heavy-tailed (SGHT) distribution is presented, which models the measurement noise with the help of switching between the Gaussian and the an existing heavy-tailed distribution. Then, utilizing two auxiliary parameters satisfying categorical and Bernoulli distributions respectively, we construct the SGHT distribution as a hierarchical Gaussian presentation. Furthermore, applying variational Bayesian inference, a novel SGHT distribution based robust Gaussian approximate filter is derived. Meanwhile, to reduce the computational complexity of the filtering process, an improved fixed-point iteration method is designed. Finally, the simulation of integrated navigation for an aircraft illustrates effectiveness and superiority of the proposed filter as compared the existing robust filters.

Approximate Length Scale Filter in Topology Optimization using Fourier Enhanced Neural Networks

Length scale control is imposed in topology optimization (TO) to make designs amenable to manufacturing and other functional requirements. Broadly, there are two types of length-scale control in TO: exact and approximate . While exact control is desirable, its implementation requires imposing additional constraints during optimization. Here we introduce an approximate length scale filter that does not involve additional constraints. In this paper, we propose an approximate length scale filter strategy for TO, by extending a recently proposed density-based TO formulation using neural networks (TOuNN). Specifically, we enhance TOuNN with a Fourier space projection, to approximately control the minimum and/or maximum length scales. The proposed method does not involve additional constraints, and the sensitivity computations are automated by expressing the computations in an end-end differentiable fashion using the neural net’s library. The proposed method is illustrated through several numerical experiments for single and multi-material designs. • A neural network-based topology optimization method with length scale control is presented. • The weights and bias associated with the NN serve as the design variables. • The length scale is controlled through Fourier projection. • The sensitivity computations are automated by expressing the computations in an end-end differentiable fashion using the NN’s library.

Data-Driven Fuzzy Transform

The Fuzzy transform is applied mainly to 1-D signals and 2-D data organized as a regular grid (e.g., 2-D images), thus, limiting its potential application to arbitrary data in terms of dimensionality and structure. This article defines and analyzes the properties of the data-driven F-transform, with a focus on the construction of the class of data-driven membership functions, which are multiscale, local, linearly independent, intrinsic, and robust to data discretization. Data-driven membership functions are defined by applying a 1-D filter to the Laplace–Beltrami operator, which encodes the geometric and topological properties of the input data. Then, we address the efficient computation of the data-driven F-transform through a polynomial or a rational polynomial approximation of the input filter. In this way, the computation of the data-driven F-transform is independent of the evaluation of the membership functions at any point of the input domain and reduces to the solution of a small set of sparse and symmetric linear systems. Finally, the data-driven F-transform is efficiently evaluated on large and arbitrary data, in terms of dimensionality, structure, and size.

Active learning strategy: Computer aided numerical class project on pole-zero plot and transfer function of five low pass filter approximation functions

Active learning strategy: Computer aided numerical class project on pole-zero plot and transfer function of five low pass filter approximation functions

Active learning strategy: Computer aided numerical class project on pole-zero plot and transfer function of five low pass filter approximation functions

The conversion of the concept and the teaching of approximation function for analog low pass filter to computer aided numerical class project as an active learning strategy is presented. Governing equations for five approximation functions are presented and discussed. It is recommended that students are tasked to develop computer programs in any convenient computer programming language to make the pole-zero plot, obtain the transfer function, and plot the frequency response of low pass filters. Steps for such programs are suggested. Typical programs were developed and tested. The pole-zero plots and frequency response graphs generated are good illustrative and innovative teaching aids. Linear-phase response over the passband is a characteristic of Bessel approximation function but suffers less amplitude discrimination in the stopband. Chebyshev and Inverse Chebyshev approximation functions have the same order requirement which is usually greater than that of Elliptic approximation function but less than that of Butterworth approximation function. Elliptic approximation function is the best choice although it is more complex than the others. Through this computer aided numerical class project, the authors themselves gained more insight in the subject matter; certainly, students will learn actively.

Approximation and realization of power-law all-pass filters

Non-integer power-law all-pass transfer functions, are approximated by suitable integer-order transfer functions in this work. The derivation of the integer-order transfer functions is based on the analytic expansion of the (non-integer) power-law transfer functions through the utilization of the binomial theorem. The offered benefit is the derivation of stable integer-order transfer functions. This study is supported by experimental results, obtained using a Field Programmable Analog Array device, after the employment of curve fitting based approximation techniques.

Adaptive fuzzy neural network-aided progressive Gaussian approximate filter for GPS/INS integration navigation

To address the difficulty of high uncertainty priori information in nonlinear filtering that occurs in tightly coupled navigation and positioning systems, a novel adaptive fuzzy neural network-aided progressive Gaussian filter is proposed in this paper to further improve the robustness and accuracy of the filter. A joint estimation of the step size and measurement noise covariance matrix is performed by a variational Bayesian approach. The estimation accuracy of the progressive Gaussian approximation filter is thereby enhanced. In particular, the measurement noise covariance matrix is adjusted using an adaptive fuzzy algorithm, and the state estimation error covariance matrix is optimized exploiting a neural network model. The proposed filter is experimentally verified to have higher accuracy than the existing state-of-the-art filters.