A note on speed comparison of wavefront reconstruction approaches

The two main reconstruction geometries are namely Hudgin and Fried. In the Hudgin geometry, gradients are the first differences between neighbouring actuators\cite{poyneer2003advanced}. In the Fried geometry, gradients are the average of the two first-differences on a square. Many adaptive optics systems use vector-matrix-multiply (VMM) reconstructors to convert gradient measurements to wavefront phase estimates. The computation time of the reconstruction using the VMM method scales as $O(n^2)$\cite{PoyneerFastFFTreconWFR}.

The reconstruction of the wavefront by means of the FFT was proposed by Freischlad and Koliopoulos\cite{freischlad1985wavefront} for square apertures on the Hudgin geometry. In a further paper\cite{FreischladFFTreconWFR} the authors derived methods for additional geometries, including the Fried geometry, which uses one Shack-Hartmann (SH) sensor. Freischlad also considered the case of small circular apertures\cite{freischladwfrfft} and the boundary problem was identified.


Computational speed comparison
The computational speed of Hudgin-FT and Fried-FT are limited only by the FFT\cite{poyneer2003advanced}. The extra processing to solve the boundary problem is of a lower order of growth computationally. Therefore FFT implementations have computational costs that scale as $O(n \log n)$. However, the implementation of Fried-FT requires potentially 2 times as much total computation as the Hudgin-FT. For the $64 \times 64$ grid, the FFT can be calculated on currently available systems in around 1 ms\cite{poyneer2003advanced}. If $N$ is a power of two the spatial filter, operations can be implemented with FFT's very efficiently. The computational requirements then scale as $O( N^2 log_2 N )$ rather than as $O( N^4 )$ in the direct vector-matrix multiplication approach. The modified Hudgin takes half as much computation as the Fried geometry model\cite{poyneer2003advanced}.


Comparison of FFT WFR and Zernike reconstruction speed

A very interesting paper appeared in the Journal of Refract Surgery\cite{dai2006comparison}. A comparison between Fourier and Zernike reconstructions was performed. In the paper\cite{dai2006comparison}, noise-free random wavefronts were simulated with up to the 15th order of Zernike polynomials. Fourier full reconstruction was more accurate than Zernike reconstruction from the 6th to the 10th orders for low-to-moderate noise levels. Fourier reconstruction was found to be approximately \textbf{100 times faster than Zernike reconstruction}. For Zernike reconstruction, however, the optimal number of orders must be chosen manually. The optimal Zernike order for Zernike reconstruction is lower for smaller pupils than larger pupils. The paper\cite{dai2006comparison} concludes that the FFT WFR is faster and more accurate than Zernike reconstruction, makes optimal use of slope information, and better represents ocular aberrations of highly aberrated eyes.


Noise propagation

Analysis and simulation show that for apertures just smaller than the square reconstruction grid (DFT case), the noise propagations of the FT methods are favourable. For the Hudgin geometry, the noise propagator grows with $O(\ln n)$. For the Fried geometry, the noise propagator is best-fit by a curve that is quadratic in the number of actuators, or $O(\ln^2 n)$. For fixed power-of-two sized grids (required to obtain the speed of the FFT for all aperture sizes) the noise propagator becomes worse when the aperture was much smaller than the grid\cite{poyneer2003advanced}.


Shack-Hartmann sensor gain

The Shack-Hartmann WFS produces a measurement which deviates from the exact wavefront slope. The exact shape of this curve depends on number of pixels used per sub-aperture and the centroid computation method (see \cite{hardyAObook} section 5.3.1 for a representative set of response curves). The most important feature of the response curve is that even within the linear response range, the gain of the sensor is not unity\cite{poyneer2003advanced}. This gain is important in the open-loop: in a closed loop this problem in mitigated by the overall control loop gain, which can be adjusted instead.

References:

\begin{thebibliography}{1}  \bibitem{poyneer2003advanced} L.A. Poyneer. \newblock {Advanced techniques for Fourier transform wavefront reconstruction}. \newblock In {\em Proceedings of SPIE}, volume 4839, page 1023, 2003.  \bibitem{PoyneerFastFFTreconWFR} Lisa~A. Poyneer, Donald~T. Gavel, and James~M. Brase. \newblock Fast wave-front reconstruction in large adaptive optics systems with   use of the fourier transform. \newblock {\em J. Opt. Soc. Am. A}, 19(10):2100--2111, Oct 2002.  \bibitem{freischlad1985wavefront} K.~Freischlad and C.L. Koliopoulos. \newblock {Wavefront reconstruction from noisy slope or difference data using   the discrete Fourier transform}. \newblock 551:74--80, 1985.  \bibitem{FreischladFFTreconWFR} Klaus~R. Freischlad and Chris~L. Koliopoulos. \newblock Modal estimation of a wave front from difference measurements using   the discrete fourier transform. \newblock {\em J. Opt. Soc. Am. A}, 3(11):1852--1861, Nov 1986.  \bibitem{freischladwfrfft} Klaus~R. Freischlad. \newblock Wave-front integration from difference data. \newblock {\em Interferometry: Techniques and Analysis}, 1755(1):212--218,   1993.  \bibitem{dai2006comparison} G.~Dai. \newblock {Comparison of wavefront reconstructions with Zernike polynomials and   Fourier transforms.} \newblock {\em Journal of refractive surgery}, 22(9):943--948, 2006.  \bibitem{hardyAObook} John~W. Hardy. \newblock {\em {Adaptive optics for astronomical telescopes}}. \newblock Oxford University Press, USA, 1998.  \end{thebibliography}