The objective of the research is to design a general blind deconvolution framework that can effectively utilize all available information, tackle severe degradations and be applicable to a wide-range of applications, degradations, signal types and dimensionality with small adaptation. The particular application of greatest interest is the problem of autofocusing in synthetic aperture radar (SAR) and in inverse SAR (ISAR). The motivation comes from the awareness that most of the blind deconvolution schemes available in the literature can only deal with relatively mild degradations [1-8]. This limitation arises from the fact that most schemes cannot easily incorporate all of the information that is available to them. Furthermore, most can guarantee convergence only to a locally optimal solution. For more severe degradations, there are more unknowns and more locally optimal solutions exist; therefore, converging to a globally optimal solution becomes much more difficult. A common remedy to ease the problem is to incorporate known information about the point spread function causing the degradation. Unfortunately, in most cases only very limited information of the point spread function is available, which is not enough to steer the solution to the global optimum.
In this work, we have identified the potential of the Bussgang blind deconvolution framework [1-4] to converge to the globally optimal solution despite its other limitations. As shown by the Benveniste-Goursat-Ruget theorem , the Bussgang blind deconvolution framework converges to a globally optimal solution as long as the probability density function of the input signal is non-Gaussian and the support size of the equalization filter size tends to infinity. While it is not possible to have an infinitively long equalization filter support, intuitively, if the signal can be processed in frames instead of in an infinite stream, the support size of the equalization filter should not need to be greater than the frame size. In addition, in the multi-channel case, the attribute of the deconvolution noise on which the Bussgang blind deconvolution framework relies becomes easier to realize. Unlike the requirement of an infinite support size for the equalization filter, the multi-channel implementation is, in fact, both practical and feasible. For example, in the optical imaging case, multiple degraded shots can result from the motion of the targeted subject, or in the synthetic aperture radar (SAR) imaging case, multiple similar flight paths can result in similar, but differently degraded, SAR images. Furthermore, the Bussgang blind deconvolution framework also achieves the goal of having one framework that is applicable to multiple applications. A different application with a different probability density function (pdf) only changes the nonlinearity in the Bussgang blind deconvolution framework. Therefore, we utilize the multi-channel Bussgang blind deconvolution framework as our fundamental building block for the design of a blind deconvolution procedure that can cope with severe degradations and a wide variety of applications.
To achieve our goal, two obstacles associated with the Bussgang blind deconvolution procedure need to be overcome. These are the requirement that the probability density function (pdf) of the original signal be known and that the original signal be white, which can greatly limit the applicability of the technique.
In this research, we relax the iid requirement and modify the multi-channel Bussgang blind deconvolution framework to allow the pdf of the original signal to be estimated iteratively. We call our proposed modification of the multi-channel Bussgang blind deconvolution framework the self-correcting multi-channel Bussgang (SCMB) blind deconvolution framework. The modifications include a non-conventional feedback mechanism, parameterization of the pdf utilizing a Gaussian mixture model, and parameter estimation using the expectation maximization (EM) algorithm that iterates simultaneously with the original multi-channel Bussgang estimator.
In the dissertation, we demonstrate the effectiveness of the proposed SCMB blind deconvolution framework on two very different problem: the binary image restoration problem and the SAR/ISAR autofocus problem. In the binary image restoration case, our approach recovers severely blurred binary images flawlessly. In the SAR/ISAR autofocus case, our approach outperforms popular autofocus algorithms including phase gradient algorithm (PGA) and minimum entropy autofocus consistently, especially in the ground moving-target ISAR autofocus scenario with both significant translational and rotational motion.