Nonlinear Filter

On the convergence of the wavelet-Galerkin method for nonlinear filteringThe aim of the paper is to examine the wavelet-Galerkin method for the solution of filtering equations. We use a wavelet biorthogonal basis with compact support for approximations of the solution. Then we compute the Zakai equation for our filtering problem and consider the implicit Euler scheme in time and the Galerkin scheme in space for the solution of the Zakai equation. We give theorems on convergence and its rate. The method is numerically much more efficient than the classical Galerkin method.

A nonlinear image technique for characterization of the optical nonlinearities is used to investigate the solid semiconductor ZnSe at 600 nm. The method based on a 4f nonlinear image technique with a phase object is used to obtain the diffraction pattern of the nonlinear filter in solid ZnSe located at the Fourier plane by a CCD camera. The nonlinear absorption coefficient and nonlinear refraction index were both obtained by fitting the nonlinear image. Good agreement between the experiment data and the simulated result are obtained indicating a sensitive and powerful method for nonlinear optical measurements.

Porting well known computer vision algorithms to low power, high performance computing devices such as SIMD linear processor arrays can be a challenging task. One especially useful such algorithm is the color-based particle filter, which has been applied successfully by many research groups to the problem of tracking non-rigid objects. In this paper, we propose an implementation of the color-based particle filter suitable for SIMD processors. The main focus of our work is on the parallel computation of the particle weights. This step is the major bottleneck of standard implementations of the color-based particle filter since it requires the knowledge of the histograms of the regions surrounding each hypothesized target position. We expect this approach to perform faster in an SIMD processor than an implementation in a standard desktop computer even running at much lower clock speeds.

The Teager-Kaiser operator is a discrete version of Teager's energy operator, advanced about 16 years ago. It is a filter of the moving window type and is commonly used as an estimator of the local energy contents of a signal; it is also used as a contrast enhancer of gray level images. We state some properties of a 2D version of the operator and its responses to common images. We characterize some of its root and preconstant images, and consider the case of separable images.

The celebrated work of Yau and Yau solved the nonlinear filtering problem in theory in the following manner. They reduced the problem of solving the Duncan-Mortensen-Zakai equation in real-time to the off-time solution of a Kolmogorov type equation. For the Yau filtering system, this Kolmogorov equation can be transformed as the Schrodinger equation. In this paper, we shall describe the fundamental solution of this Schrodinger equation with quartic potential.

This paper examines and contrasts the feasibility of joint state and parameter estimation of noise-driven chaotic systems using the extended Kalman filter (EKF), ensemble Kalman filter (EnKF), and particle filter (PF). In particular, we consider the chaotic vibration of a noisy Duffing oscillator perturbed by combined harmonic and random inputs ensuing a transition probability density function (pdf) of motion which displays strongly non-Gaussian features. This system offers computational simplicity while exhibiting a kaleidoscope of dynamical behavior with a slight change of input and system parameters. An extensive numerical study is undertaken to contrast the performance of various nonlinear filtering algorithms with respect to sparsity of observational data and strength of model and measurement noise. In general, the performance of EnKF is better than PF for smaller ensemble size, while for larger ensembles PF outperforms EnKF. For moderate measurement noise and frequent measurement data, EKF is able to correctly track the dynamics of the system. However, EKF performance is unsatisfactory in the presence of sparse observational data or strong measurement noise.

This paper proposes a novel multi-microphone nonlinear neural network based switched Griffiths-Jim beamformer structure for speech enhancement. The main objective of this algorithm is to reduce real-world interference signals such as radio, television or computer fan noise from an acquired speech signal. The proposed algorithm improves the current design of the switched Griffiths-Jim beamformer structure by introducing an adaptive nonlinear neural network filter for the noise reduction section. The network topology used here is a partially connected three-layer feedforward neural network structure. The error backpropagation algorithm is used here as the learning algorithm. A comparison analysis of the traditional three-microphone linear beamformer and the proposed three-microphone neural switched Griffiths-Jim beamformer structure is discussed here. They are both tested with different types of interference signal from the Noise-X database. All the experiments are conducted in real-world surroundings. The nonlinear approach introduced here shows remarkable improvement over the previous linear adaptive beamformer approach.

Introduction Perhaps the most challenging of all guid-ance and control problems is that of a mod-em tactical air-to-air missile in pursuit of a highly maneuverable aircraft. The problem consists of the estimation of target motion, the generation of guidance commands to op-timally ...

The paper presents an algorithm for syndromic surveillance of an epidemic outbreak formulated in the context of stochastic nonlinear filtering. The dynamics of the epidemic is modeled using a generalized compartmental epidemiological model with inhomogeneous mixing. The syndromic (typically non-medical) observations of the number of infected people (e.g. visits to pharmacies, sale of certain products, absenteeism from work/study etc.) are used for estimation. The state of the epidemic, including the number of infected people and the unknown parameters of the model, are estimated via a particle filter. The numerical results indicate that the proposed framework can provide useful early prediction of the epidemic peak if the uncertainty in prior knowledge of model parameters is not excessive.

Given a Markovian Brownian martingale $Z$, we build a process $X$ which is a martingale in its own filtration and satisfies $X_1 = Z_1$. We call $X$ a dynamic bridge, because its terminal value $Z_1$ is not known in advance. We compute explicitly its semimartingale decomposition under both its own filtration $\cF^X$ and the filtration $\cF^{X,Z}$ jointly generated by $X$ and $Z$. Our construction is heavily based on parabolic PDE's and filtering techniques. As an application, we explicitly solve an equilibrium model with insider trading, that can be viewed as a non-Gaussian generalization of Back and Pedersen's \cite{BP}, where insider's additional information evolves over time.