Approximation of an arbitrary filter and its recursive implementation

Filtering is one of the important techniques in computer vision. It has been widely used in edge detection, image restoration, range image segmentation, etc. However, the efficient implementation of an arbitrary filter has been a challenging problem until now. In this paper, a novel method is proposed to implement an arbitrary filter. Firstly, an efficient recursive structure is proposed to implement any (polynomial) × (exponential)-type (PET) filter. The computational complexity and structure are independent of its filter mask size or its bandwidth. Secondly, a new method is proposed—Lagurre spectrum decomposition method—to obtain the PET approximation of any filters. As an example, the above method is applied to the approximation and implementation of Gaussian filters and experiments have shown that a perfect approximation can be obtained with only third-order Lagurre bases, and therefore only a fourth-order recursive filter is needed to implement Gaussian filters. Finally, the comparison of the present method with the known ones shows that (1) Lagurre polynomial bases are orthogonal with each other, so the filter approximation is simple, (2) the bases are complete and the completeness guarantees the approximation error can be reduced to zero, (3) the method can be used to design both Gaussian and any other filters.     

磁偶极子跟踪的渐进贝叶斯滤波方法

提出一种新的非线性滤波器应用于磁偶极子目标跟踪问题.建立了跟踪问题的状态空间模型, 在此基础上, 从高斯矩近似误差的角度分析了现有卡尔曼滤波更新在磁偶极子跟踪中的问题.对此, 将贝叶斯更新过程等效为求解连续时间上的渐进贝叶斯问题, 在线性高斯条件下推导了其解析解, 表明渐进贝叶斯更新可等效为定常系统的Kalman-Bucy滤波器; 进一步采用一阶Taylor展开得到非线性近似解表达式, 导出一种渐进贝叶斯滤波器, 从理论上分析了与现有方法的异同.仿真与实测磁目标跟踪试验结果表明, 渐进贝叶斯滤波器具有良好的精度和收敛性, 能够有效抑制磁目标跟踪中由于大初始误差导致的性能下降和滤波发散, 且计算效率与扩展卡尔曼滤波器相当, 适于实际应用.     

一种带多步随机延迟量测高斯滤波器的一般框架解

提出了一种适用于线性和非线性系统的带多步随机延迟量测高斯滤波器的一般框架解. 为了完成状态的递归更新估计, 噪声向量和先前时刻状态向量被扩展到当前时刻状态向量中. 然后基于贝叶斯方法推导了扩展后状态向量的一般框架解. 对于非线性系统, 通过利用不同的数值计算方法计算贝叶斯解中的高斯加权积分可以推导获得不同的高斯近似滤波器. 最后本文利用三阶球径容积准则来实施提出的方法, 并通过量测被随机延迟多步的目标跟踪模型对所提出的方法进行了仿真, 仿真结果验证了提出方法的有效性和优点.     

基于弱形式解的粒子流滤波器

针对粒子流滤波器中粒子速度场计算复杂,难以滤波求解的问题,提出一种基于弱形式解的粒子流滤波器.通过将粒子速度场等效为势函数的梯度,推导该速度场所满足的偏微分方程的弱形式;应用Galerkin有限元法和蒙特卡罗积分法,推导出一个易于计算的弱形式常数近似解. 仿真算例表明,在一定初始条件下,多峰型后验分布会使高斯假设滤波器局部收敛,而粒子流滤波器是有效的,且具有较高的跟踪精度和较好的鲁棒性.     

Analysis of single Gaussian approximation of Gaussian mixtures in Bayesian filtering applied to mixed multiple-model estimation

This paper examines the effect of the moment-matched single Gaussian approximation, which is made in various multiple-model filtering applications to approximate a Gaussian mixture, on the Bayesian filter performance. The estimation error caused by the approximation is analysed for both the prediction and the measurement updates of a Bayesian filter. An approximate formula is found for the covariance of the error caused by the approximation for a general Gaussian mixture with arbitrary components. The calculated error covariance is used for obtaining a mixed multiple-model estimation algorithm which has a performance near that of GPB2 with less computations.     

A skewed unscented Kalman filter

State estimation of nonlinear systems is a challenging task, especially when the Gaussian approximation fails. The unscented Kalman filter was proposed to deal with state estimation of nonlinear systems. We modify the traditional unscented Kalman filter to capture the third-order moment (skewness) of the state vector. Methods are also proposed to reduce the computation time of the suggested approach, and showing that the proposed algorithm is as fast as the unscented Kalman filter. Simulation results confirm that the method is better than, or at least as good as, the unscented Kalman filter.     

高斯渐进贝叶斯滤波器

渐进贝叶斯方法将先验分布到后验分布的演化描述为一阶动态系统,通过在伪时间上连续地引入观测信息实现后验状态估计.该方法的一般形式解,即动态系统的时间导数,是难以得到的.本文提出一种高斯型渐进贝叶斯滤波器.首先在线性高斯条件下推导了时间导数的解析解;然后证明了在该条件下,由该解析解确定的一阶动态系统与常量状态估计的Kalman-Bucy滤波器是一致的,且由此导出的高斯渐进贝叶斯滤波器与卡尔曼滤波器是一致的.最后利用一阶Taylor展开推导了滤波器在非线性高斯条件下的近似解表达式,并采用Monte Carlo方法给出了具体实现方法.通过若干仿真算例表明,新滤波器具有较高的精度,且在一定精度条件下的时间复杂度低于一般粒子滤波器.     

A convergent approximation of the continuous-time optimal parameterestimator

Continuous-time linear stochastic systems that are bilinear in the state and parameters are considered. A specific approximation to the optimal nonlinear filter used as a recursive parameter estimator is derived by retaining third-order moments and using a Gaussian approximation for higher order moments. With probability one, the specific approximation is proved to converge to a minimum of the likelihood function. The proof uses the ordinary differential equation technique and requires that the trajectories of the slow system be bounded on finite time intervals and that the fixed parameter fast system by asymptotically stable. The fixed parameter fast system is proved to be asymptotically stable if the parameter update gain is small enough. Essentially, the specific approximation is asympotically equivalent to the recursive prediction error method, thus inheriting its asymptotic rate of convergence. A numerical simulation for a simple example indicates that the specific approximation has better transient response than other commonly used convergent parameter estimators     

SIFER: Scale-Invariant Feature Detector with Error Resilience

We present a new method to extract scale-invariant features from an image by using a Cosine Modulated Gaussian (CM-Gaussian) filter. Its balanced scale-space atom with minimal spread in scale and space leads to an outstanding scale-invariant feature detection quality, albeit at reduced planar rotational invariance. Both sharp and distributed features like corners and blobs are reliably detected, irrespective of various image artifacts and camera parameter variations, except for planar rotation. The CM-Gaussian filters are approximated with the sum of exponentials as a single, fixed-length filter and equal approximation error over all scales, providing constant-time, low-cost image filtering implementations. The approximation error of the corresponding digital signal processing is below the noise threshold. It is scalable with the filter order, providing many quality-complexity trade-off working points. We validate the efficiency of the proposed feature detection algorithm on image registration applications over a wide range of testbench conditions.     

An adaptive robustizing approach to kalman filtering

The performance of a linear Kalman filter will degrade when the dynamic noise is not Gaussian. A robust Kalman filter based on the m-interval polynomial approximation (MIPA) method for unknown non-Gaussian noise is proposed. Two situations are considered: (a) the state is Gaussian and the observation noise is non-Gaussian; (b) the state is non-Gaussian and the observation noise is Gaussian. It is shown, as compared with other non-Gaussian filters, the MIPA Kalman filter is computationally feasible, unbiased, more efficient and robust. For the scalar model, Monte Carlo simulations are given to demonstrate the ideas involved.     

Near-optimal non-linear filtering using quasi-moment functions†

In previous publications (Fisher and Stear 1967 a, b), the authors dealt extensively with the general non-linear filtering problem. This paper utilizes the very desirable properties of the expansion of a probability density function in terms of its quasi-moment functions to obtain near-optimal ‘solutions’ to the general dynamical equation of the non-linear filtering problem. Specifically, stochastic differential equations are derived for the near-optimal non-linear filter using the theory of approximation of a probability density function by its lower-order quasi-moment functions. In addition, the result of approximating the conditional density function by a Gaussian density function is compared with the frequently used method of linearizing the non-linear equations about a nominal solution and applying Kalman linear filtering techniques.     

基于高斯似然近似的自适应球面径向积分滤波算法

针对量测噪声较小的环境下传统滤波算法容易出现偏差增大的实际问题, 基于高斯近似原理, 提出一种基于高斯似然近似的球面径向积分滤波(SRGLAF) 算法. 为进一步解决量测未知环境下的状态估计问题, 充分结合CKF 等确定性采样型滤波算法和SRGLAF 的优势, 设计一种基于高斯似然近似的自适应球面径向积分滤波(ASRGLAF) 算法. 仿真结果表明: SRGLAF 能够提高量测噪声较小环境下的估计精度, 而在量测噪声未知环境中, ASRGLAF 能够有效地进行状态估计, 具有明显的滤波优势.

     

Maximum correntropy unscented filter

The unscented transformation (UT) is an efficient method to solve the state estimation problem for a non-linear dynamic system, utilising a derivative-free higher-order approximation by approximating a Gaussian distribution rather than approximating a non-linear function. Applying the UT to a Kalman filter type estimator leads to the well-known unscented Kalman filter (UKF). Although the UKF works very well in Gaussian noises, its performance may deteriorate significantly when the noises are non-Gaussian, especially when the system is disturbed by some heavy-tailed impulsive noises. To improve the robustness of the UKF against impulsive noises, a new filter for non-linear systems is proposed in this work, namely the maximum correntropy unscented filter (MCUF). In MCUF, the UT is applied to obtain the prior estimates of the state and covariance matrix, and a robust statistical linearisation regression based on the maximum correntropy criterion is then used to obtain the posterior estimates of the state and covariance matrix. The satisfying performance of the new algorithm is confirmed by two illustrative examples.     

多目标跟踪的混合高斯PHD滤波

为解决目标数未知或随时间变化时的多目标跟踪问题,将多目标状态和观测信息表示为随机集的形式,建立了多目标跟踪的混合高斯概率假设密度（PHD）滤波方法。当目标初始的先验概率密度满足高斯分布的形式时,通过将状态噪声、观测噪声、目标的繁衍、新目标的产生、目标的存活概率和检测概率表示成混合高斯的形式,之后每个时刻的后验概率密度均能表示成混合高斯的形式。线性混合高斯PHD滤波方法将Kalman滤波引入到PHD滤波中,利用混合高斯成分预测和更新随机集的PHD,并估计出目标的状态。实验结果表明,在杂波环境下混合高斯PHD滤波方法可以有效地跟踪目标状态。     

Approximation problems with the divergence criterion for Gaussian variables and Gaussian processes

System identification for stationary Gaussian processes includes an approximation problem. Currently, the subspace algorithm for this problem enjoys much attention. This algorithm is based on a transformation of a finite time series to canonical variable form followed by a truncation. There is no proof that this algorithm is the optimal solution to an approximation problem with a specific criterion. In this paper it is shown that the optimal solution to an approximation problem for Gaussian random variables with the divergence criterion is identical to the main step of the subspace algorithm. An approximation problem for stationary Gaussian processes with the divergence criterion is formulated.     

一种自适应图像去噪混合滤波方法

结合自适应中值滤波技术和自适应压缩加权均值滤波技术,提出了一种新的图像混合噪声滤波算法。算法首先对受混合噪声污染的图像利用灰度极值检测出脉冲噪声,运用自适应中值滤波滤除脉冲噪声;其次对处理结果进行自适应压缩的加权均值滤波。实验结果说明算法不仅能有效地滤除脉冲与高斯混合噪声,而且可以较好地保护图像细节。     

BIAWGN信道中改进的LT码优化方法

在二进制输入加性高斯白噪声信道中传输LT码时,采用高斯近似方法预测置信传播译码算法的误比特率性能不够准确。为此,提出一种改进的高斯近似方法,其中,输入节点度分布采用泊松分布,相应的软信息为高斯混合物,在此基础上给出一种LT码度分布优化方法。仿真结果证明,该方法相比同类方法性能更优越。     

On the optimal shape parameter for Gaussian radial basis function finite difference approximation of the Poisson equation

We investigate the influence of the shape parameter in the meshless Gaussian radial basis function finite difference (RBF-FD) method with irregular centres on the quality of the approximation of the Dirichlet problem for the Poisson equation with smooth solution. Numerical experiments show that the optimal shape parameter strongly depends on the problem, but insignificantly on the density of the centres. Therefore, we suggest a multilevel algorithm that effectively finds a near-optimal shape parameter, which helps to significantly reduce the error. Comparison to the finite element method and to the generalised finite differences obtained in the flat limits of the Gaussian RBF is provided.     

基于组合导航系统的非线性高斯滤波算法

由于组合导航系统具有强非线性和模型不确定性的特点, 工程中扩展卡尔曼滤波无法满足组合导航系统实际应用的要求. 为此, 针对贝叶斯框架下高斯类非线性滤波算法的估计性能给出具体分析. 首先, 在估计点处对非线性函数进行泰勒展开获得泰勒近似, 通过一阶矩和二阶矩分析滤波算法的近似精度; 然后, 通过数值稳定性对非线性滤波算法进行分析; 最后, 分别采用低维和高维模型对各滤波算法进行对比分析, 为组合导航系统的实践提供借鉴.

     

Higher-Moments in Perturbation Solution of the Linear-Quadratic Exponential Gaussian Optimal Control Problem

The paper obtains two principal results. First, using a new definition ofhigher-order (>2) matrix derivatives, the paper derives a recursion forcomputing any Gaussian multivariate moment. Second, the paper uses this resultin a perturbation method to derive equations for computing the 4th-orderTaylor-series approximation of the objective function of the linear-quadraticexponential Gaussian (LQEG) optimal control problem. Previously, Karp (1985)formulated the 4th multivariate Gaussian moment in terms of MacRae'sdefinition of a matrix derivative. His approach extends with difficulty to anyhigher (>4) multivariate Gaussian moment. The present recursionstraightforwardly computes any multivariate Gaussian moment. Karp used hisformulation of the Gaussian 4th moment to compute a 2nd-order approximationof the finite-horizon LQEG objective function. Using the simpler formulation,the present paper applies the perturbation method to derive equations forcomputing a 4th-order approximation of the infinite-horizon LQEG objectivefunction. By illustrating a convenient definition of matrix derivatives in thenumerical solution of the LQEG problem with the perturbation method, the papercontributes to the computational economist's toolbox for solving stochasticnonlinear dynamic optimization problems.