Optimal nonlinear filtering for independent increment processes--Part I

A new formulation of the multidimensional optimal nonlinear filtering problem is presented in this two-part paper. This formulation permits generalization and unification of some well-known recent results on optimal nonlinear filtering theory. [1]-[7] Specifically, the problem investigated is that of determining the conditional probability density function ofx(t)given{y(tau); t_{0} leq tau leq t}, wherex(t)is then-dimensional state vector of a non-linear system perturbed by an independent increment noise process, andy(t)is anm-dimensional measurement vector which is a nonlinear function ofxand contains an additive independent increment noise process. The results are obtained through use of characteristic functions and the theory of independent increment processes. The foundation for the treatment of general independent increment noise processes is given in Part I, but the final results in Part I are restricted to Gaussian independent increment noise processes. The extension to general independent increment noise processes is considered in Part II. It is shown in Part I that the results for the linear-Gaussian case can be obtained in two different ways, one of which cannot be used for the general case. Some important properties of general independent increment processes and a special property of Gaussian independent increment processes are discussed.     

Optimal nonlinear filtering with constraints—Part II

The impulse response of nonlinear systems with bounded memory and energy have been considered. On the base of the minimum mean square error criterion a generalization of the Wiener-Hopf equation is developed.     

A note on filtering for independent increment processes (Corresp.)

Using the equations of Fisher and Stear, a suboptimal filter for estimating the state of a linear system subject to non-Gaussian noise is developed. Some qualitative properties of this filter are derived.     

Optimal nonlinear filtering with constraints

The impulse response of nonlinear systems with limitation of band has been considered. On the base of the minimum mean square error criterion the generalisation of the Wiener-Hopf's equation has been calculated.     

Optimal filtering for multirate systems

For a multirate system where the output sampling is slower than the input updating, this brief aims at designing filters for fast state estimation in the H/sub 2/ and H/sub /spl infin// settings. Because of the multirate nature, linear matrix inequality solutions to the design problems involve a nonconvex constraint, which is numerically tackled by the product reduction algorithm. Finally, a design example is given and the effectiveness of the approach is illustrated.     

Optimum nonlinear filtering

This paper is composed of two parts. The first part surveys the literature regarding optimum nonlinear filtering from the (continuous-time) stochastic analysis point of view, and the other part explores the impact of recent applications of neural networks (in a discrete-time context) to nonlinear filtering. In particular, the results obtained by using a regularized form of radial basis function (RBF) networks are presented in fair detail     

Optimal digital filtering for tremor suppression

Remote manually operated tasks such as those found in teleoperation, virtual reality, or joystick-based computer access, require the generation of an intermediate electrical signal which is transmitted to the controlled subsystem (robot arm, virtual environment, or a cursor in a computer screen). When human movements are distorted, for instance, by tremor, performance can be improved by digitally filtering the intermediate signal before it reaches the controlled device. This paper introduces a novel tremor filtering framework in which digital equalizers are optimally designed through pursuit tracking task experiments. Due to inherent properties of the man-machine system, the design of tremor suppression equalizers presents two serious problems: 1) performance criteria leading to optimizations that minimize mean-squared error are not efficient for tremor elimination and 2) movement signals show ill-conditioned autocorrelation matrices, which often result in useless or unstable solutions. To address these problems, a new performance indicator in the context of tremor is introduced, and the optimal equalizer according to this new criterion is developed. Ill-conditioning of the autocorrelation matrix is overcome using a novel method which we call pulled-optimization. Experiments performed with artificially induced vibrations and a subject with Parkinson's disease show significant improvement in performance. Additional results, along with MATLAB source code of the algorithms, and a customizable demo for PC joysticks, are available on the Internet at http:?tremor-suppression.com.     

Linear and nonlinear filtering

This paper provides a review of the theory of filtering for stochastic processes. In particular, the sequential theory of linear filtering is reviewed as well as the theory of nonlinear filtering.     

Posterior Cramer-Rao bounds for discrete-time nonlinear filtering

A mean-square error lower bound for the discrete-time nonlinear filtering problem is derived based on the van Trees (1968) (posterior) version of the Cramer-Rao inequality. This lower bound is applicable to multidimensional nonlinear, possibly non-Gaussian, dynamical systems and is more general than the previous bounds in the literature. The case of singular conditional distribution of the one-step-ahead state vector given the present state is considered. The bound is evaluated for three important examples: the recursive estimation of slowly varying parameters of an autoregressive process, tracking a slowly varying frequency of a single cisoid in noise, and tracking parameters of a sinusoidal frequency with sinusoidal phase modulation     

Efficient and reliable schemes for nonlinear diffusion filtering

Nonlinear diffusion filtering in image processing is usually performed with explicit schemes. They are only stable for very small time steps, which leads to poor efficiency and limits their practical use. Based on a discrete nonlinear diffusion scale-space framework we present semi-implicit schemes which are stable for all time steps. These novel schemes use an additive operator splitting (AOS), which guarantees equal treatment of all coordinate axes. They can be implemented easily in arbitrary dimensions, have good rotational invariance and reveal a computational complexity and memory requirement which is linear in the number of pixels. Examples demonstrate that, under typical accuracy requirements, AOS schemes are at least ten times more efficient than the widely used explicit schemes.     

An adaptive nonlinear diffusion algorithm for filtering medicalimages

The nonlinear anisotropic diffusive process has shown the good property of eliminating noise while preserving the accuracy of edges and has been widely used in image processing. However, filtering depends on the threshold of the diffusion process, i.e., the cut-off contrast of edges. The threshold varies from image to image and even from region to region within an image. The problem compounds with intensity distortion and contrast variation. We have developed an adaptive diffusion scheme by applying the central limit theorem to selecting the threshold. Gaussian distribution and Rayleigh distribution are used to estimate the distributions of visual objects in images. Regression under such distributions separates the distribution of the major object from other visual objects in a single-peak histogram. The separation helps to automatically determine the threshold. A fast algorithm is derived for the regression process. The method has been successfully used in filtering various medical images     

A nonlinear Bayesian filtering framework for ECG denoising

In this paper, a nonlinear Bayesian filtering framework is proposed for the filtering of single channel noisy electrocardiogram (ECG) recordings. The necessary dynamic models of the ECG are based on a modified nonlinear dynamic model, previously suggested for the generation of a highly realistic synthetic ECG. A modified version of this model is used in several Bayesian filters, including the Extended Kalman Filter, Extended Kalman Smoother, and Unscented Kalman Filter. An automatic parameter selection method is also introduced, to facilitate the adaptation of the model parameters to a vast variety of ECGs. This approach is evaluated on several normal ECGs, by artificially adding white and colored Gaussian noises to visually inspected clean ECG recordings, and studying the SNR and morphology of the filter outputs. The results of the study demonstrate superior results compared with conventional ECG denoising approaches such as bandpass filtering, adaptive filtering, and wavelet denoising, over a wide range of ECG SNRs. The method is also successfully evaluated on real nonstationary muscle artifact. This method may therefore serve as an effective framework for the model-based filtering of noisy ECG recordings.     

Artificial neural networks for nonlinear time-domain filtering ofspeech

A multilayer perceptron (MLP) is applied as a time domain nonlinear filter to two classes of degraded speech, namely Gaussian white noise and nonlinear system degradation introduced by a low bit-rate CELP coder. The goal of the study is to examine the influence of the inherent nonlinearity within the MLP, and this is achieved by varying the levels of nonlinearity within the structure. Direct comparisons of MLPs and linear filters show that with CELP degradation the SNR improvements achieved by the MLP is measurably better than with an equivalent linear structure (3 dB cf 1.5 dB) but when the degradation is additive noise the two structures perform equally well. The study highlights the importance of scaling to achieve optimum performance, and of matching the enhancer to the degradation     

Robust filtering for discrete nonlinear fractional transformation systems

In this brief, we consider robust filtering problems for uncertain discrete-time systems. The uncertain plants under consideration possess nonlinear fractional transformation (NFT) representations which are a generalization of the classical linear fractional transformation (LFT) representations. The proposed NFT is more practical than the LFT, and moreover, it leads to substantial performance gains as well as computational savings. For this class of systems, we derive linear-matrix inequality characterizations for H/sub 2/, & H/sub /spl infin//, and mixed filtering problems. Our approach is finally validated through a number of examples.     

Unscented filtering and nonlinear estimation

The extended Kalman filter (EKF) is probably the most widely used estimation algorithm for nonlinear systems. However, more than 35 years of experience in the estimation community has shown that is difficult to implement, difficult to tune, and only reliable for systems that are almost linear on the time scale of the updates. Many of these difficulties arise from its use of linearization. To overcome this limitation, the unscented transformation (UT) was developed as a method to propagate mean and covariance information through nonlinear transformations. It is more accurate, easier to implement, and uses the same order of calculations as linearization. This paper reviews the motivation, development, use, and implications of the UT.     

Optimal reduced-rank estimation and filtering

This paper provides a unified view of, and a further insight into, a class of optimal reduced-rank estimators and filters. An alternating power (AP) method for computing the optimal reduced-rank estimators and filters is derived and analyzed. The AP method is a generalization of the conventional power method for subspace computation, which is shown to be globally and exponentially convergent under weak conditions. When the rank reduction is relatively large, the AP method is computationally more efficient than the conventional methods. The AP method is useful for adaptive computation of the canonical components of a desired reduced-rank estimate, which in turn facilitates the detection of a time-varying rank. The study shown in this paper is particularly useful for applications that involve a large number of sources and a large number of receivers, where rank reduction is either inherent in the multivariate system or required to reduce the model complexity and/or the computational load     

Remarks on linear and nonlinear filtering

This communication tries to give some insight into relationships existing between Viterbi and the forward-backward algorithm (used in the context of hidden Markov models) on the one hand and Kalman filtering and Rauch-Tung Striebel smoothing on the other. We give a unifying view which shows how those algorithms are related and give an example of a nonlinear hybrid system that can be filtered through a mixed algorithm     

On a nonlinear feedback filtering system

A nonlinear filtering scheme with noiseless feedback is presented, based on a consideration of the minimum-mean-squared error filtering of independent signal samples corrupted by additive noise. The explicit solution for the general case is very complex. However, if the signal-to-noise ratio is assumed to be large and the nonlinear estimating filter has zero memory, the problem may be simplified by reducing it to the zero-memory prefiltering problem combined with predictive feedback. The improvement over the linear case without feedback is shown to be the product of the improvements due to the zero-memory non-linearities and the feedback. An example is considered to illustrate the improvements in the error,     

非线性滤波算法性能对比

随着目标运动的多样性和复杂化,雷达非线性目标跟踪算法越来越受到重视。本文对目前非线性滤波的主要算法即扩展卡尔曼滤波、不敏卡尔曼滤波、粒子滤波的滤波模型、适用条件、性能进行了分析比较,通过一个非线性非高斯模型进行了仿真,验证了这些算法的性能,仿真结果表明非线性条件下粒子滤波算法要明显优于其它两种滤波算法。