We study the recovery of operators with a bandlimited Kohn-Nirenberg symbol from the action of such operators on a weighted impulse train, a procedure we refer to as sampling of operators. Kailath, Bello, and later Kozek and the authors have shown that the sampling of operators is possible if the symbol of the operator is bandlimited to a set with area less than one. In this paper, we develop the theory of the sampling of operators in analogy with the classical theory of sampling of bandlimited functions. We define the notions of sampling set and sampling rate for operators and give necessary and sufficient conditions on the sampling rate that depend on the size and geometry of the bandlimiting set. We develop explicit reconstruction formulas for operator sampling that generalize reconstruction formulas for bandlimited functions. We give necessary and sufficient conditions on the bandlimiting set under which sampling of operators is possible by their action on a given periodically weighted delta train. We show that under mild geometric conditions on the bandlimiting set, classes of operators are bandlimited to an unknown set of area less than one-half permit sampling and reconstruction. We generalize two results of the Heckel and Bölcskei concerning sampling of operators with area not greater than one-half and less than one, respectively, by finding a larger class of operators to which they apply. Operators with bandlimited symbols have been used to model doubly dispersive communication channels with slowly time-varying impulse response. The results in this paper are rooted in work by Bello and Kailath in the 1960s.

The goal of channel measurement or operator identification is to obtain complete knowledge of a channel operator by observing the image of a finite number of input signals. In this paper, it is shown that the spreading support of the operator (that is, the support of the symplectic Fourier transform of the Kohn-Nirenberg symbol of the operator) has area less than one then the operator is identifiable. If the spreading support is larger than one, then the operator is not identifiable. The shape of the support region is essentially arbitrary thereby proving a conjecture of Bello. The input signal considered is a weighted delta train where the weights are the window function of a finite Gabor system whose elements satisfy a certain robust completeness property

In an earlier paper we utilized techniques from the theory of nonlinear dynamical systems to define a notion of embedding threshold estimators. These estimators were based on the analysis of delay-coordinates embeddings of sets of coefficients of the measured signal in some chosen frame. One of the motivations behind the method was the desire of building an estimator that would perform equally well regardless of the type of white noise contamination. In this paper we explicitly write the structure of one particular algorithm, the attenuated embedding threshold estimator, and we show the performance of the algorithm when different types of white noise are added to speech time series.

We show that recovery of a function from its averages over squares in the plane is closely related to a problem of recovery of bandlimited functions from samples on unions of regular lattices. We use this observation to construct explicit solutions to the Bezout equation which can be easily implemented in software. We also show that these sampling results give a new proof of the "three squares theorem" which says that a function in the plane can be recovered from its averages on translates of three squares oriented parallel to the coordinate axes whose sidelengths are pairwise irrationally related. Other proofs of this theorem and construction of solution to the Bezout equation rely on interpolation methods in the theory of functions of one and several complex variables. Our sampling technique gives much simpler solutions especially in higher dimensions.

A novel multisensor approach to deconvolution is developed. This theory circumvents the ill-posedness inherent in convolution equations by overdetermining the input signal by a multichannel system of convolvers {/spl mu//sub i/}, chosen so that any information lost by one channel is retained by another. The deconvolution problem is then solved by constructing "deconvolvers" that allow us to construct the Dirac /spl delta/ by filtering each /spl mu//sub i/ by its deconvolver, and then adding the filtered channels together. This in turn allows us to reconstruct the original signal f. The process is linear and stable with respect to noise. The general multichannel theory is discussed. The deconvolution theory in radially symmetric domains is then developed in greater detail.

We develop an algorithm to reconstruct the wavelet coefficients of an image from the Radon transform data. The proposed method uses the properties of wavelets to localize the Radon transform and can be used to reconstruct a local region of the cross section of a body, using almost completely local data that significantly reduces the amount of exposure and computations in X-ray tomography. The property that distinguishes our algorithm from the previous algorithms is based on the observation that for some wavelet bases with sufficiently many vanishing moments, the ramp-filtered version of the scaling function as well as the wavelet function has extremely rapid decay. We show that the variance of the elements of the null-space is negligible in the locally reconstructed image. Also, we find an upper bound for the reconstruction error in terms of the amount of data used in the algorithm. To reconstruct a local region 16 pixels in radius in a 256x256 image, we require 22% of full exposure data.

We develop an algorithm to reconstruct the wavelet coefficients of an image from the Radon transform data. The proposed method uses the properties of wavelets to localize the Radon transform and can be used to reconstruct a local region of the cross section of a body, using almost completely local data which significantly reduces the amount of exposure and computations in X-ray tomography. For example, for a local region of radius 20 pixels in a 256/spl times/256 image the proposed method can reduce the exposure to 12.5% of the conventional filtered backprojection method. Compared to the existing schemes, which can only reduce to 40%.