In this thesis, we use multi-resolutional methods to firstly show how two relatively new transforms can be used to optimise wavefront sensor performance in terms of slope representation, and secondly, we explore how higher scales and orientations can capture wavefront curvature for optimal representation of optical aberrations.

Since wavefront distortions cannot be directly measured from an image, a wavefront sensor can use intensity variations from a point source to estimate slope or curvature of a wavefront. However, processing of measured aberration data from wavefront sensors can be computationally intensive, and this is a challenge for real-time image restoration. In essence, exploring the use of new signal processing transforms for possible use in optimising optical wavefront sensor performance is a key topic of this research.

To ensure the highest wavefront accuracy whilst maintaining low computational overhead with the highest noise immunity, wavelet analysis, which has been widely used in signal and image processing, was considered. The time-frequency localisation feature makes wavelet analysis ideal to model wavefront aberrations locally at different resolutions. However, wavelets are restricted to efficiently represent objects having anisotropic features such as lines or curvilinear structures. Following the success of wavelets, recent multi-scale representations such as ridgelets, and curvelets, have been developed and have been applied to astronomy for image processing, such as data filtering, deconvolution, and star and galaxy detection.

This work describes the implementation of two proposed, multi-resolution methods based on the ridgelet and curvelet transforms. Such transforms both optimised, and in some cases, enhanced, estimate atmospheric aberrations from an unknown aberrated phase screen. This was achieved using two defocused planes employed by the geometric and curvature wavefront sensors. The ridgelet method is employed with the slope- based geometric wavefront sensor, and in an open-loop configuration. Discrete ridgelet transforms are performed in the discrete Radon domain, where each Radon projection is computed over N angles, and where each is represented by a wavelet. The principle of the ridgelet transform is that line singularities along each projection are interpreted as point singularities by the Radon transform, for which a sparse representation is provided by the wavelet transform.

A comprehensive analysis of the proposed ridgelet method is presented, and this is compared to the geometric wavefront sensor using simulation. The proposed method also shows improved accuracy over a wide range of noise conditions.

Further investigation on the potential use of the curvelet transform is explored to estimate wavefront distortions from natural source beacons (stars) from distorted astronomical images. The simplest wavefront sensing device, the curvature wavefront sensor, which has a similar data acquisition system to the geometric sensor, is used. Wavefront sensor images are employed as the basis of an analysis where the curvelet transform is applied over different block sizes, but with a single transform. The curvelet transform is a multi-scale and multi-directional expansion that formulates an optimally sparse representation of functions with singularities on curves. The simulation results achieved from the multi-resolution curvelet method are promising and provide a solid basis for further research on extending the performance of the curvature sensor.

Deconvolution methods are known for post-processing perturbed images due to atmospheric turbulence using data from wavefront sensors. Ultimately, the optimisation of both geometric and curvature wavefront sensors for use in deconvolution from wavefront sensing was a subsequent motivation for this work. Securing on-sky data for image restoration from both slope and curvature wavefront sensors provided a practical basis for this research.