Sze Fong Yau

A compact Cramer-Rao bound expression for parametric estimation of superimposed signals

The problem of parameter estimation of superimposed signals in white Gaussian noise is considered. Closed-form expressions of the Cramer-Rao bound for real or complex signals with vector parameters are derived, extending recent results by P. Stoica and A. Nehorai (1989). >

Maximum likelihood parameter estimation of superimposed signals by dynamic programming

The problem of fitting a model composed of a number of superimposed signals to noisy data using the maximum likelihood criterion is considered. It is shown, using the Cramer-Rao bound for the estimation accuracy, that in many instances, useful models for the composite signal can be restricted without loss of generality to component signals that directly interact only with one or two of their closest neighbors in parameter space. It is shown that for such models, the global extremum of the criterion can be found efficiently by dynamic programming. The computation requirements are linear in the number of signals, rather than exponential as in the case of exhaustive search. The technique applies for arbitrary sampling of the signals. The dynamic programming method is easily adapted to determining the number of signals as well, as is demonstrated using the minimum description length principle. Computer simulation results are given for several examples. >

Worst case Cramer-Rao bounds for parametric estimation of superimposed signals with applications

The problem of parameter estimation of superimposed signals in white Gaussian noise is considered. The effect of the correlation structure of the signals on the Cramer-Rao bounds is studied for both the single and multiple experiment cases. The best and worst conditions are found using various criteria. The results are applied to the example of parameter estimation of superimposed sinusoids, or plane-wave direction finding in white Gaussian noise, and best and worst conditions on the correlation structure and relative phase of the sinusoids are found. This provides useful information on the limits of the resolvability of sinusoid signals in time series analysis or of plane waves in array processing. The conditions are also useful for designing worst-case simulation studies of estimation algorithms, and for the design of minimax signal acquisition and estimation procedures, as demonstrated by an example. >

Parameter estimation of superimposed signals by dynamic programming

The problem of fitting a model composed of a number of superimposed signals to noisy data using the maximum-likelihood criterion is considered. A local interaction model is established through the study of Cramer-Rao bound. For such models, the global extremum of the criterion is found efficiently by dynamic programming. An approximate version of the algorithm is developed to further reduce the computation. Using the minimum description length principle, it is shown that the dynamic programming method can be easily adapted to determine the number of signals as well.<<ETX>>

On the robustness of parameter estimation of superimposed signals by dynamic programming

We analyze a recently proposed dynamic programming algorithm (REDP) for maximum likelihood (ML) parameter estimation of superimposed signals in noise. We show that it degrades gracefully with deviations from the key assumption of a limited interaction signal model (LISMO), providing exact estimates when the LISMO assumption holds exactly. In particular, we show that the deviations of the REDP estimates from the exact ML are continuous in the deviation of the signal model from the LISMO assumption. These deviations of the REDP estimates from the MLE are further quantified by a comparison to an ML algorithm with an exhaustive multidimensional search on a lattice in parameter space. We derive an explicit expression for the lattice spacing for which the two algorithms have equivalent optimization performance, which can be used to assess the robustness of REDP to deviations from the LISMO assumption. The values of this equivalent lattice spacing are found to be small for a classical example of superimposed complex exponentials in noise, confirming the robustness of REDP for this application.

Maximum likelihood parameter estimation and subspace fitting of superimposed signals by dynamic programming—an approximate method

Abstract The problem of fitting a model composed of a number of superimposed signals to noisy data using the criteria of maximum likelihood (ML) and subspace fitting is considered. Using a Local Interaction Signal Model, established through a study of the Cramer-Rao bound for the estimation accuracy, an approximate dynamic programming algorithm (ADP) for signal parameter estimation is proposed. It yields estimates close to the global maximum of the ML or subspace fitting criteria with a computational cost only linear in the number of signals, rather than exponential as in the case of exhaustive search. The ADP algorithm improves on our previously proposed Basic Dynamic Programming (BDP) and Robust Exact Dynamic Programming (REDP) algorithms by further reducing the computational requirements and by being applicable to the multiple snapshot scenario, at the cost of slightly reduced accuracy. For example, for m sinusoids the computation is reduced from O( mq 6 ) in BDP and REDP to O( mq 2 ), where q is the number of grid points per dimension for the parameter search, which determines the resolution of frequency. Like the BDP and REDP, the ADP algorithm is adapted to determine the number of signals, and is applicable to arbitrary signal sampling or sensor array geometry. The ADP estimates converge almost surely to their noiseless values with increasing number of snapshots. Computer simulation results of several examples are presented.

Image restoration by complexity regularization via dynamic programming

The restoration of an image modeled by piecewise-constant polygonal patches from its blurred (bandlimited) and noise corrupted version is considered. Under this model, the line-integral projections of the data image are piecewise linear signals, blurred and corrupted by noise. The break points and the associated amplitude parameters of each projection are estimated by minimizing the 1-D stochastic complexity of the projection using a recently proposed dynamic programming technique. The final image is reconstructed by convolution backprojection.<<ETX>>

A generalization of Bergstrom's inequality and some applications

Abstract Bergstroms inequality is generalized. Using the new inequality, several interesting results concerning Hadamard product of matrices are proved. More specifically, let P,Q ϵ C N×N such that P>0 and Q⩾0, Qii>0. Then we prove the following tight inequalities: (a) Qii[(P⊙Q)-1]ii⩽ (P-1ii; (b) Qii{[Re(P⊙Q)]-1}ii ⩽ max {[Re(φPφ∗)]-1}ii; (c) det(P⊙Q)⩾ (detP)πNi=1 Qii; (d) det Re(P⊙ Q) ⩾ min det[Re(φPφ∗)]πNi=1 Qii, where in (b) and (d) maximization and minimization are over all unitary diagonal matrices φϵ C N×N.

Performance analysis of parameter estimation of superimposed signals by dynamic programming

The problem of fitting a model composed of a number of superimposed signals to noisy data using the maximum likelihood (ML) criterion is considered. A dynamic programming (DP) algorithm which solves the problem efficiently is presented. An asymptotic property of the estimates is derived, and a bound on the bias of the estimates is given. The bound is then computed using perturbation analysis and compared with computer simulation results. The results show that the DP algorithm is a versatile and efficient algorithm for parameter estimation. In practical applications, the estimates can be refined by a local search (e.g., the Gauss-Newton method) of the exact ML criterion, initialized by the DP estimates.<<ETX>>

Image Restoration By Dynamic Programming

We consider the reconstruction of an image modeled by piecewise-constant polygonal patches from its blurred (bandlimited) and noise corrupted version. The problem is therefore 2dimensional deconvolution, which is well-known to be ill-posed. To regularize the problem without introducing undesirable smoothing, we adopt a parameter estimation approach to estimate the edge positions as well as the intensity level of each patch. Under the Gaussian noise assumption, the maximum likelihood estimation of the parameters can be solved in the 1-D case efficiently by dynamic progra,mming [l]. To apply this technique to a 2-D image, the line-integral projections of the data, image are first computed [2]. It is then clear that each projection can be modeled as a piecewise linear signal, blurred and corrupted by noise. The dynamic programming algorithm is then applied to each projection, and the original image obtained by convolution back projection. This approach is ideally suited for a parallel implementation. To improve the consistency of the individually estimated projections, the property that the 2-D Fourier transform of the projections has a finite “bowtie” shaped support [3] is enforced. Computer simulation results of several examples are shown to illustrate the method.