Arie Feuer

Restoration of a single superresolution image from several blurred, noisy, and undersampled measured images

The three main tools in the single image restoration theory are the maximum likelihood (ML) estimator, the maximum a posteriori probability (MAP) estimator, and the set theoretic approach using projection onto convex sets (POCS). This paper utilizes the above known tools to propose a unified methodology toward the more complicated problem of superresolution restoration. In the superresolution restoration problem, an improved resolution image is restored from several geometrically warped, blurred, noisy and downsampled measured images. The superresolution restoration problem is modeled and analyzed from the ML, the MAP, and POCS points of view, yielding a generalization of the known superresolution restoration methods. The proposed restoration approach is general but assumes explicit knowledge of the linear space- and time-variant blur, the (additive Gaussian) noise, the different measured resolutions, and the (smooth) motion characteristics. A hybrid method combining the simplicity of the ML and the incorporation of nonellipsoid constraints is presented, giving improved restoration performance, compared with the ML and the POCS approaches. The hybrid method is shown to converge to the unique optimal solution of a new definition of the optimization problem. Superresolution restoration from motionless measurements is also discussed. Simulations demonstrate the power of the proposed methodology.

Convergence analysis of LMS filters with uncorrelated Gaussian data

Statistical analysis of the least mean-squares (LMS) adaptive algorithm with uncorrelated Gaussian data is presented. Exact analytical expressions for the steady-state mean-square error (mse) and the performance degradation due to weight vector misadjustment are derived. Necessary and sufficient conditions for the convergence of the algorithm to the optimal (Wiener) solution within a finite variance are derived. It is found that the adaptive coefficient μ, which controls the rate of convergence of the algorithm, must be restricted to an interval significantly smaller than the domain commonly stated in the literature. The outcome of this paper, therefore, places fundamental limitations on the mse performance and rate of convergence of the LMS adaptive scheme.

Super-resolution reconstruction of image sequences

In an earlier work (1999), we introduced the problem of reconstructing a super-resolution image sequence from a given low resolution sequence. We proposed two iterative algorithms, the R-SD and the R-LMS, to generate the desired image sequence. These algorithms assume the knowledge of the blur, the down-sampling, the sequences motion, and the measurements noise characteristics, and apply a sequential reconstruction process. It has been shown that the computational complexity of these two algorithms makes both of them practically applicable. In this paper, we rederive these algorithms as approximations of the Kalman filter and then carry out a thorough analysis of their performance. For each algorithm, we calculate a bound on its deviation from the Kalman filter performance. We also show that the propagated information matrix within the R-SD algorithm remains sparse in time, thus ensuring the applicability of this algorithm. To support these analytical results we present some computer simulations on synthetic sequences, which also show the computational feasibility of these algorithms.

Superresolution restoration of an image sequence: adaptive filtering approach

This paper presents a new method based on adaptive filtering theory for superresolution restoration of continuous image sequences. The proposed methodology suggests least squares (LS) estimators which adapt in time, based on adaptive filters, least mean squares (LMS) or recursive least squares (RLS). The adaptation enables the treatment of linear space and time-variant blurring and arbitrary motion, both of them assumed known. The proposed new approach is shown to be of relatively low computational requirements. Simulations demonstrating the superresolution restoration algorithms are presented.

Robust optimal experiment design for system identification

This paper develops the idea of min-max robust experiment design for dynamic system identification. The idea of min-max experiment design has been explored in the statistics literature. However, the technique is virtually unknown by the engineering community and, accordingly, there has been little prior work on examining its properties when applied to dynamic system identification. This paper initiates an exploration of these ideas. The paper considers linear systems with energy (or power) bounded inputs. We assume that the parameters lie in a given compact set and optimise the worst case over this set. We also provide a detailed analysis of the solution for an illustrative one parameter example and propose a convex optimisation algorithm that can be applied more generally to a discretised approximation to the design problem. We also examine the role played by different design criteria and present a simulation example illustrating the merits of the proposed approach.

Convergence and performance analysis of the normalized LMS algorithm with uncorrelated Gaussian data

It is demonstrated that the normalized least mean square (NLMS) algorithm can be viewed as a modification of the widely used LMS algorithm. The NLMS is shown to have an important advantage over the LMS, which is that its convergence is independent of environmental changes. In addition, the authors present a comprehensive study of the first and second-order behavior in the NLMS algorithm. They show that the NLMS algorithm exhibits significant improvement over the LMS algorithm in convergence rate, while its steady-state performance is considerably worse. >

On sparse representation in pairs of bases

In previous work, Elad and Bruckstein (EB) have provided a sufficient condition for replacing an l/sub 0/ optimization by linear programming minimization when searching for the unique sparse representation. We establish here that the EB condition is both sufficient and necessary.

Potential benefits of hybrid control for linear time invariant plants

A question which has bothered control researchers for some time is whether hybrid control offers any advantage over linear control for linear plants. We show in this paper, via several examples, that hybrid control can indeed overcome certain types of limitation which are unavoidable if linear feedback control is used.

Variable length stochastic gradient algorithm

The transversal variable-length stochastic gradient algorithm is described. It is a modification of the stochastic gradient algorithm that allows dynamic allocation of coefficients of an adaptive filter. The order of the filter and the adaptation step size are changed automatically when an appropriate level of performance is reached during the course of the adaptation process. In this way, the algorithm results in fast convergence, typical of low-order filters, and good steady-state performance, typical of high-order filters. >

Time delay estimation in continuous linear time-invariant systems

The use of a time delay in modeling LTI systems is quite common. However, attempts to estimate these time delays in continuous systems typically resorted to methods which increase the number of parameters in the system, in contradiction to the use of time delay in the model to begin with. We present here an attempt to estimate the time delays directly. The algorithm we present is supported both by analysis and simulations with very encouraging results. >