Application of wavelets to adaptive optics and multiresolution Wiener filtering

Adaptive optical telescopes are designed to improve astronomical viewing through the turbulent atmosphere. A sensor measures the local slopes, or gradient, of the incoming wavefront, thereby characterizing the deformation introduced by the atmosphere. From the slope measurements, which are noisy, the wavefront reconstructor calculates an estimate of the wavefront itself. The wavefront information is then passed to a deformable mirror, which changes shape in real time to dynamically correct for the effects of the turbulence.; The problem of determining slope measurements in additive noise is addressed by introducing a new signal estimation technique called the multiresolution Wiener filter (MWF). The MWF is a linear, shift-variant filter based on the application of Wiener filters to the wavelet decomposition of a process. The MWF extends the range of stochastic processes to which Wiener filtering techniques are applicable to non-stationary processes like Kolmogorov and fractional Brownion motion processes. The performance of the FIR version of the MWF is compared to that of the Wiener filter for first-order Markov processes and shown to outperform the FIR and causal Wiener filter.; Slope measurements estimated by the MWF are passed to a new wavefront reconstruction algorithm called the analytical inverse gradient operator (AIGO) algorithm. The AIGO algorithm is an efficient method (the number of operations is approximately equal to the number of wavefront slope measurements) derived from an analytical solution to the gradient equation. When combined with the MWF, the AIGO algorithm is shown to be competitive with other wavefront reconstructors.; One of the principle difficulties with popular wavefront reconstruction techniques such as least-squares estimators is that they are computationally intensive. A least-squares reconstructor is represented in terms of a 4-D wavelet basis. This representation is called the multiresolution wavefront reconstructor (MWR). A thresholding operation is applied to the MWR in order to remove wavelet coefficients of negligible magnitude. The resulting thresholded reconstructor matrix is sparse, leading to an estimate calculated in {dollar}O(Nsp3){dollar} operations, as opposed to {dollar}O(Nsp4){dollar} operations for the standard least-squares wavefront reconstructor. The thresholded multiresolution wavefront reconstructor is compared with thresholded least-squares wavefront reconstructors, Zernike, and iterative techniques in terms of computational complexity.