Wavefronts reconstructed from measured gradients are composed of a straightforward integration of the measured data, plus a correction term that disappears when there are no measurement errors. For regions of any shape, this term is a solution of Poisson's equation with Dirichlet conditions (V=0 on the boundaries). We show that for rectangular regions, the correct solution is not a periodic one, but one expressed with Fourier cosine series. The correct solution has a lower variance than the periodic Fourier transform solution. Similar formulas exist for a circular region with obscuration. We present a near-optimal solution that is much faster than fast-Fourier-transform methods. By use of diagonal multigrid methods, a single iteration brings the correction term to within a standard deviation of 0.08, two iterations, to within 0.0064, etc.
© 2006 Optical Society of America
Original Manuscript: April 22, 2005
Manuscript Accepted: June 5, 2005
Amos Talmi and Erez N. Ribak, "Wavefront reconstruction from its gradients," J. Opt. Soc. Am. A 23, 288-297 (2006)