In this paper, we propose a recursive filtering algorithm to restore monochromatic images which are corrupted by general dependent additive noise. It is assumed that the equation which describes the image field is not available and a filtering algorithm is obtained using the information provided by the covariance functions of the signal, noise that affects the measurement equation, and the fourth-order moments of the signal. The proposed algorithm is obtained by an innovation approach which provides a simple derivation of the least mean-squared error linear estimators. The estimation of the grey level in each spatial coordinate is made taking into account the information provided by the grey levels located on the row of the pixel to be estimated. The proposed filtering algorithm is applied to restore images which are affected by general signal-dependent additive noise.

Least-squares second-order polynomial filter and fixed-point smoother are derived in systems with uncertain observations, when the variables describing the uncertainty are non-independent. The proposed estimators do not require the knowledge of the state-space model of the signal. The available information is only the moments, up to the fourth one, of the involved processes, the probability that the signal exists in the observations and the (2,2) element of the conditional probability matrices of the sequence describing the uncertainty.

This paper presents recursive algorithms for the least mean-squared error linear filtering and fixed-interval smoothing estimators, from uncertain observations for the case of white and white plus coloured observation noises. The estimators are obtained by an innovation approach and do not use the state-space model, but only covariance information about the signal and the observation noises, as well as the probability that the signal exists in the observed values. Therefore the algorithms are applicable not only to signal processes that can be estimated by the conventional formulation using the state-space model but also to those for which a realization of the state-space model is not available. It is assumed that both the signal and the coloured noise autocovariance functions are expressed in a semi-degenerate kernel form. Since the semi-degenerate kernel is suitable for expressing autocovariance functions of non-stationary or stationary signal processes, the proposed estimators provide estimates of general signal processes.

This paper considers the least-squares linear estimation problem of signals from randomly delayed observations when the additive white noise is correlated with the signal. The delay values are treated as unknown variables, modelled by a binary white noise with values zero or one; these values indicate that the measurements arrive in time or they are delayed by one sampling time. A recursive one-stage prediction and filtering algorithm is obtained by an innovation approach and do not use the state-space model of the signal. It is assumed that both, the autocovariance functions of the signal and the crosscovariance function between the signal and the observation noise are expressed in a semi-degenerate kernel form; using this information and the delay probabilities, the estimators are recursively obtained.

This paper treats the least-squares linear filtering and smoothing problems of discrete-time signals from uncertain observations when the random interruptions in the observation process are modelled by a sequence of independent Bernoulli random variables. Using an innovation approach we obtain the filtering algorithm and a general expression for the smoother which leads to fixed-point, fixed-interval and fixed-lag smoothing recursive algorithms. The proposed algorithms do not require the knowledge of the state-space model generating the signal, but only the covariance information of the signal and the observation noise, as well as the probability that the signal exists in the observed values.

The least-squares linear filtering and fixed-point smoothing problems of uncertainly observed signals are considered when the signal and the observation additive noise are correlated at any sampling time. Recursive algorithms, based on an innovation approach, are proposed without requiring the knowledge of the state-space model generating the signal, but only the autocovariance and crosscovariance functions of the signal and the observation white noise, as well as the probability that the signal exists in the observations.

This paper discusses the least-squares linear filtering and smoothing (fixed-point and fixed-interval) problems of discrete-time signals from observations, perturbed by additive white noise, which can be randomly delayed by one sampling time. It is assumed that the Bernoulli random variables characterizing delay measurements are correlated in consecutive time instants. The marginal distribution of each of these variables, specified by the probability of a delay in the measurement, as well as their correlation function, are known. Using an innovation approach, the filtering, fixed-point and fixed-interval smoothing recursive algorithms are obtained without requiring the state-space model generating the signal; they use only the covariance functions of the signal and the noise, the delay probabilities and the correlation function of the Bernoulli variables. The algorithms are applied to a particular transmission model with stand-by sensors for the immediate replacement of a failed unit.

This paper discusses the least-squares linear filtering and fixed-lag smoothing problems of discrete-time signals from uncertain observations when the random interruptions in the observation process are modelled by a sequence of not necessarily independent Bernoulli variables. It is assumed that the observations are perturbed by white noise and the autocovariance function of the signal is factorizable. Using an innovation approach we obtain the filtering and fixed-lag smoothing recursive algorithms, which do not require the knowledge of the state-space model generating the signal. Besides the observed values, they use only the matrix functions defining the factorizable autocovariance function of the signal, the noise autocovariance function, the marginal probabilities and the (2,2)-element of the conditional probability matrices of the Bernoulli variables. The algorithms are applied to estimate a scalar signal which may be transmitted through one of two channels.