In this dissertation, a new method for the numerical integration of two-dimensional partial differences is presented. The approach is based on obtaining an estimate of the 2-D Haar wavelet decomposition of the integrated differences by filtering and down-sampling the partial difference measurement data as an intermediate step. Then, this decomposition estimate is synthesized into an estimate of the integrated differences.

The filterbanks required for estimating this decomposition are derived directly from the 2-D Haar Wavelet Analysis Filterbank. The order of operations of this process is manipulated in a novel way so that gradient or partial difference data can be used as input to the filterbank instead of the image data. The original data can then be obtained from this decomposition estimate using unmodified 2-D Haar Wavelet Synthesis Filterbanks. This use of the wavelet decomposition brings a reduction in computation complexity to less than 10 operations per pixel of the result.

This dissertation shows that the data used for this algorithm may be calculated partial differences or discretely sampled gradient data measurements. This data set may have any-sized convex area of support as long as it is on a Cartesian grid. The method is stable as a component of a closed loop system as shown by simulations of a recently developed woofer-tweeter adaptive optics control system.