Blind deconvolution with an application to speckle imaging.

It is well known that the quality of an image that can be obtained from a groundbased telescope is limited by the wavefront distortion associated with the atmospheric turbulence. The turbulence is present as a result of the mixing of the warm and the cold air. Although the turbulence affects both the amplitude and the phase of the wavefront, the random phase aberration is found to be more important in the distortion of an image than the magnitude distortion. As a result, the available light spread over a wide area and the observed image is blurred. This effect has hindered astronomers since the invention of the telescope in the 17t h century. In order to reduce the effect of the atmospheric turbulence and improve the quality of the image obtained at the ground-based telescopes, large collection aperture telescopes have been built at high altitude places. Although the light gathering properties of these modern telescopes are remarkable, the problem that originated from the atmospheric turbulence has not been solved completely. The aim of this study is to find ways that can compensate the effects of atmospheric turbulence and provide a good quality image. There are two approaches proposed to deal with this problem, namely the adaptive optics and the post-detection processing approaches. The first half of the thesis is concerned with an adaptive optics approach while the second half is concerned with a post-detection processing approach. The adaptive optics approach compensates the wavefront distortion in real time. It uses hardware designed to physically alter the optical path length of the distorted wavefront. Thus a disturbance which is in the optical path just opposite to the disturbance associated with the atmospheric turbulence is introduced so that both disturbances cancel. Therefore, an accurate estimation of the disturbance introduced by the atmospheric turbulence is crucial in this approach. While the problem can be reformulated as a standard least square problem from the measurements from Shack Hartmann sensor, the least square problem is itself under-determined. This means that the quality of distortion is limited by the number of measurements available, which in turn is limited by the available light. In order to solve this under-determined problem, a priori information needs to be applied so as to put more constraints into the estimation. One piece of a priori information that can be obtained is the statistics of the turbulence. This can be derived theoretically using the Kolmogorov turbulence model. By incorporating this information into the estimation, the problem becomes well-posed. As the simulation results show, the accuracy in estimating the distorted wavefront is improved by incorporating the a priori information. The second approach to compensate the disturbance is the post-detection processing approach. This is founded upon the insight that short exposure images preserve frequencies up to the diffraction limit. A variety of techniques have been proposed to recover this diffraction-limited information, for example, the correlation-based techniques of Knox-Thompson and bispectrum approaches. However, these techniques are required to average the effects of atmospheric turbulence before the algorithm can be applied to recover the object of interest. Another approach is to consider the problem as a deconvolution problem in which the object of interest is reconstructed based on the disturbance information and the short exposure image. In this case the disturbance is unknown. It is a blind deconvolution problem, that means we need to reconstruct the object from the short exposure image without a statistical ensemble of distorted images. Three methods are discussed and compared, they are the conjugate gradient approach, the projection-based approach and the maximum likelihood approach. All these three methods can be formulated using the Bayes' theorem but with different assumptions made on the noise statistics and with different application of the a priori knowledge. They are compared in terms of the convergence rate, the quality of the reconstructed image and also in the presence of different noise levels. As simulation results show, all these three algorithms can blindly deconvolve a blurred image. In general, the projection-based algorithm and the maximum likelihood algo-rithm produce clean reconstructed images compared to the conjugate gradient algorithm. However, all three algorithms experience the so-called superresolution effect in the presence of noise. A way to prevent suprereolving a blurred image is suggested.