The problem is to evaluate the behavior of an object when primary sources of information about the object become unavailable, so that any information must be obtained from the intelligent use of available secondary sources. This evaluative process is reactive sensing. Reactive sensing is initially viewed in terms of spatial super-resolution. The theory of reactive sensing is based on two equivalent ideas, one physical and one mathematical. The physical idea models volume, e.g., engine volume in the case of analyzing engine health, and the sensitivity of sensors to such volume. The mathematical idea of multiplicative frames provides the factorization theory to compare quantitatively such volume and sensitivity. This equivalence is the foundation for reactive sensing theory and its implementation.

We propose a new kernel and frame based dimension reducing algorithm by exploiting the synergy between endmembers and kernel based classification schemes. This synergy becomes available by means of utilizing the notions of frames and frame representations.

Constant amplitude zero autocorrelation (off the dc component) waveforms are constructed. These are called CAZAC waveforms. In the d-dimensional case they consist of N vectors, where N is given, and N is generally greater than d. The constructions are algebraic and have been implemented in user friendly software. They have the added feature that they are a spanning set for all d-dimensional signals. As such, and for N large, they are numerically stable in the presence of machine imperfections and they give good signal reconstruction in the presence of various noises. The one dimensional case provides effective thresholding to compute Doppler shifts

The K-level Sigma-Delta (/spl Sigma//spl Delta/) scheme with step size /spl delta/ is introduced as a technique for quantizing finite frame expansions for /spl Ropf//sup d/. Error estimates for various quantized frame expansions are derived, and, in particular, it is shown that /spl Sigma//spl Delta/ quantization of a unit-norm finite frame expansion in /spl Ropf//sup d/ achieves approximation error where N is the frame size, and the frame variation /spl sigma/(F,p) is a quantity which reflects the dependence of the /spl Sigma//spl Delta/ scheme on the frame. Here /spl par//spl middot//spl par/ is the d-dimensional Euclidean 2-norm. Lower bounds and refined upper bounds are derived for certain specific cases. As a direct consequence of these error bounds one is able to bound the mean squared error (MSE) by an order of 1/N/sup 2/. When dealing with sufficiently redundant frame expansions, this represents a significant improvement over classical pulse-code modulation (PCM) quantization, which only has MSE of order 1/N under certain nonrigorous statistical assumptions. /spl Sigma//spl Delta/ also achieves the optimal MSE order for PCM with consistent reconstruction.

It is shown that sigma-delta (/spl Sigma//spl Delta/) algorithms can be used effectively to quantize finite frame expansions for R/sup d/. Error estimates for various quantized frame expansions are derived, and, in particular, it is shown that /spl Sigma//spl Delta/ quantizers outperform the standard PCM schemes.

For a given spiral, a bandwidth B can be chosen and a sequence S can be constructed on the spiral with the property that all finite energy signals having bandwidth B can be reconstructed from sampled values on S. The bandwidth can be expanded as desired, and reconstruction is attained by constructing sampling sets on interleaving spirals. This solves a problem in MRI; and the algorithm can be modified to deal with irregular sampling problems in SAR. The algorithm is a consequence of our theoretical results, which in turn were inspired by seminal work on balayage in the 1960s by Beurling (1966) and Landau (1967). Our results depend on d-dimensional Fourier frames and tiling properties of spectral synthesis sets.

In this paper we present a method for suppressing noise in signals with a "coherent" time-frequency behavior. The method fundamentally incorporates a continuous wavelet transform followed by a (possibly signal dependent) irregular sampling of the time-scale plane. To suppress the noncoherent portion of a noisy signal the resulting irregular wavelet coefficients are then thresholded and presented to an iterative algorithm for reconstruction. This method is illustrated numerically on a signal with a coherent time-frequency structure.>