The generalized Gabor scheme of image representation in biologicaland machine vision

A scheme suitable for visual information representation in a combined frequency-position space is investigated through image decomposition into a finite set of two-dimensional Gabor elementary functions (GEF). The scheme is generalized to account for the position-dependent Gabor-sampling rate, oversampling, logarithmic frequency scaling and phase-quantization characteristic of the visual system. Comparison of reconstructed signal highlights the advantages of the generalized Gabor scheme in coding typical bandlimited images. It is shown that there exists a tradeoff between the number of frequency components used per position and the number of such clusters (sampling rate) utilized along the spatial coordinate     

Optimal ARMA parameter estimation based on the sample covariancesfor data with missing observations

The problem of spectral estimation through the autoregressive moving-average (ARMA) modeling of stationary processes with missing observations is considered. A class of estimators based on the sample covariances is presented, and an asymptotically optimal estimator in this class is proposed. The proposed algorithm is based on a nonlinear-least-squares fit of the sample covariances computed from the data to the true covariances of the assumed ARMA model. The statistical properties of the algorithm are explored and used to show that it is asymptotically optimal, in the sense of achieving the smallest possible asymptotic variance. The performance of the algorithm is illustrated by some numerical examples     

Direction finding algorithms based on high-order statistics

Two direction finding algorithms are presented for nonGaussian signals, which are based on the fourth-order cumulants of the data received by the array. The first algorithm is similar to MUSIC, while the second is asymptotically minimum variance in a certain sense. The first algorithm requires singular value decomposition of the cumulant matrix, while the second is based on nonlinear minimization of a certain cost function. The performance of the minimum variance algorithm can be assessed by analytical means, at least for the case of discrete probability distributions of the source signals and spatially uncorrelated Gaussian noise. The numerical experiments performed seem to confirm the insensitivity of these algorithms to the (Gaussian) noise parameters     

Efficient construction of canonical ladder forms for vector autoregressive processes

The paper treats the problem of constructing canonical ladder realizations for vector autoregressive (AR) processes specified by their characteristic matrix polynomials. The difficulty of this problem is rooted in the fact that the backward matrix polynomial corresponding to a given vector AR process is a nontrivial function of the forward matrix polynomial. The construction calls for solving a discrete Lyapunov equation in block-controller form. Two efficient procedures for solving this equation are presented, both requiring a number of operations that is proportional to at most the square of the model order. Applications of the new procedures to stability, tests, simulation of AR processes, and model reduction are described.     

Parameter estimation of continuous-time stationary Gaussian processes with rational spectra

This paper considers the problem of estimating the parameters of continuous-time stationary Gaussian processes with rational spectra, from uniformly sampled measurements. The sampled process is shown to be an autoregressive moving-average process, and explicit relationships between the parameters of the continuous-time and the sampled processes are derived. These relationships are then used to derive a lower bound on the variances of biased estimates of the continuous-time parameters, and on the generalized variance of such estimates. It is shown by some examples that the bound on the generalized variance depends on the sampling interval in a nonmonotonic manner. In particular, for each specific set of parameters there exists a sampling interval for which the lower bound is minimized.This work was supported by the Army Research Office under Contract Number DAAG29-83-C-0027.     

Blind identification of FIR systems excited by discrete-alphabetinputs

An algorithm for the identification of finite-impulse-response (FIR) system parameters from output measurements, for systems excited by discrete-alphabet inputs, is described. The approach taken is algebraic. It does not rely directly on the statistical properties of the measurements, but rather it essentially solves the nonlinear equations appearing in the problem by converting them to equivalent linear equations, using the discrete-alphabet property of the input signal. The proposed algorithm was tested by computer simulations and some of these simulations are illustrated     

Square root covariance ladder algorithms

Square root normalized ladder algorithms provide an efficient recursive solution to the problem of multichannel autoregressive model fitting. A simplified derivation of the general update formulas for such ladder forms is presented, and is used to develop the growing memory and sliding memory covariance ladder algorithms. New ladder form realizations for the identified models are presented, leading to convenient methods for computing the model parameters from estimated reflection coefficients. A complete solution to the problem of possible singularity in the ladder update equations is also presented.     

The second-order moments of the sample covariances for time serieswith missing observations

The estimation of the covariances of a stationary time series with missing observations is considered. General formulas for the asymptotic second-order moments of the sample covariances of such a time series are given for either random or deterministic patterns of misses. Closed-form expressions are derived for the random Bernoulli pattern and for the deterministic periodic pattern of missing observations and are explicitly evaluated for autoregressive moving-average time series. These results are useful for constructing and analyzing parameter or spectrum estimation algorithms based on the sample covariances for a stationary time series with missing observations     

Bounds on the accuracy of Gaussian ARMA parameter estimation methods based on sample covariances

The asymptotic accuracy of Gaussian ARMA parameter estimation methods based on a fixed number of sample covariances is considered. Several key results are briefly reviewed, including: i) a general asymptotic expression for the error covariance of the ARMA parameter estimates; ii) the fact that this error covariance is always greater than a certain lower bound; iii) the fact that this lower bound is strictly greater than the Cramer-Rao bound; iv) an explicit ARMA estimation technique that asymptotically achieves the bound. The key result of this note is a proof that this lower bound approaches the Cramer-Rao bound as the number of sample covariances tends to infinity.     

Computer image generation for flight simulators: the Gabor approach

A formalism for image representation in the combined frequency-position space is presented using the generalized Gabor approach. This approach uses elementary functions to which the human visual system is particularly sensitive and which are efficient for the analysis and synthesis of visual imagery. The formalism is also compatible with the implementation of a variable resolution system wherein image information is nonuniformly distributed across the visual field in accordance with the human visual system's ability to process it. When used with a gaze-slaved visual display system, imagery generated using the techniques described here affords a combination of high resolution and wide field-of-view. This combination is particularly important in high-fidelity, computergenerated, visual environments as required, for instance, in flight simulators.