The imaging of astronomical objects is limited by atmospheric turbulence, which consists of layers of varying refractive index surrounding the earth. These refractive index fluctuations are a direct consequence of the warming and cooling of air and water vapour in the atmosphere. Wavefronts entering the atmosphere acquire phase distortions, which when propagated result in amplitude fluctuations known as scintillation. Hence the practical manifestation of the atmosphere is a degradation of the signals passing through it, for example it severely limits the resolution of images captured by ground-based telescopes.

A variety of solutions, or inverse problems, have been proposed and trialed in the attempt to obtain the best possible images from astronomical telescopes. An orbiting telescope (for example the Hubble space telescope) is one solution. In this case light is captured before it is distorted by the atmosphere. Less expensive ground-based solutions include the post processing of short exposure images and real-time compensation using adaptive optics, both of which are investigated in this thesis.

However, the success of an inverse problem lies in the accurate modelling of the processes that give rise to the corresponding forward problem, in this case the random refractive index fluctuations that characterise the atmosphere. Numerical simulation of atmospheric turbulence is achieved using phase screens in which the assumption of Kolmogorov statistics is often made. A previously presented method for modelling Kolmogorov phase fluctuations over a finite aperture, the midpoint displacement method, is both formalised and improved.

This enables the accurate generation of atmospheric speckle images for the development and testing of post processing methods. Another aspect of the forward problem is the accurate simulation of scintillation, resulting from the propagation of phase distorted wavefronts. Commonly used simulation methods achieve this by assuming periodic boundary conditions. A technique for the accurate modelling and simulation of scintillation from an aperiodic Kolmogorov phase screen is presented. The more physically justifiable assumption of smoothness is shown to result in a propagation kernel of finite extent. This allows the phase screen dimensions for an accurate simulation to be determined and truncation can then be used to eliminate the unwanted spectral leakage and diffraction effects usually inherent in the use of finite apertures. Deconvolution methods are popular for the post processing of atmospheric speckle images to compensate for the effects of the atmosphere. Conventional deconvolution algorithms are applied when the distortion is known or well-characterised, whereas, blind deconvolution algorithms are used when the distortion is unknown. Conventional deconvolution techniques are not often directly applied to astronomical imaging problems as the distortion introduced by the atmosphere is unknown. However, their extension to blind deconvolution is straightforward and hence their development is valuable. The ill-conditioning of the deconvolution problem requires the addition of prior information, such as positivity, to enable its solution. It is shown that the conventional deconvolution problem can be reformulated as an equivalent quadratic programming problem. Consequently, an accelerated quadratic programming approach is applied and shown to be an improvement to an existing method used for enforcing positivity in deconvolution applications. The main algorithmic differences of the new method are implementation via the fast Fourier transform (FFT) and guaranteed convergence to the constrained minimum. Blind deconvolution is also an interesting problem that may arise in many fields of research. It is of particular relevance to imaging through turbulence where the point spread function can only be modelled statistically, and direct measurement may be difficult. The extension of the quadratic programming method to blind deconvolution, combined with Tikhonov-Miller regularisation (energy constraints), smoothness constraints, penalty terms and statistical priors produced a series of new algorithms. The performance of these algorithms is illustrated on simulated astronomical speckle images. Ground-based adaptive optics (AO) technologies are an alternative to post processing methods and aim to compensate for the distortion introduced by the atmosphere in real-time. Knowledge of the vertical structure of the atmosphere combined with AO provides the potential for compensation over a wide field of view. However, the continually changing nature of atmospheric turbulence places strict requirements on techniques for determining the turbulence structure. The remote sensing of scintillation data to estimate this information is known as scintillation detection and ranging (SCIDAR). Application of SCIDAR methods to the capture and analysis of experimental data, as demonstrated in this thesis, highlighted a number of problems with the technique. Methods for overcoming these difficulties are discussed and demonstrated. Finally, alternative approaches to the estimation of atmospheric turbulence profiles and a proposed new technique are investigated.