Modified Hudgin geometry for Wavefront reconstruction using Fast Fourier Transform

One of the attractive abilities of the FFT WFR is that the filtering construct provides flexibility: reconstruction is accomplished by filtering in the frequency domain and one can modify this filter with negligible computational overhead. It is easy to incorporate filtering options into the
reconstruction filters like noise reduction, modal removal, misalignment, or DM geometry compensation.

Misalignment of the WFS data and the DM geometry.

For example, the misalignment can be compensated: the WFS grid and the DM actuators may be misaligned by shifts along x or y dimensions. If the amount is known, shift slope estimate by a fraction of an actuator spacing such as $(exp[-i 2\pi (\Delta_x k + \Delta_y l)/ N])\Phi$.


Modified Hudgin geometry
For Shack-Hartmann, best the reconstructor is modified Hudgin, for which the
slopes are: $s_x[m,n] = \phi[m+1n,n+0.5] - \phi[m,n+0.5]$.
The reconstruction of the wavefront is performed with the following procedure:
$\begin{matrix} \hat{\Phi}[k,l] & = & \left\{ \begin{matrix} 0 & \mbox{if } k,l = 0 \\ S_x[k,l]\cdot H_x[k,l] + S_y[k,l]\cdot H_y[k,l] & \mbox{otherwise } \end{matrix} \right.\end{matrix}$

The the spatial filters are different: we ahve to shift each slope signal half a
sample along the orthogonal direction:
$ H_x[k,l] = \frac{ (\exp\left[ - \frac{2\pi i \cdot k }{N} \right] -1) \exp(-i\pi
l / N) }{ 4\left( \sin^2\frac{\pi k}{N} + \sin^2\frac{\pi l}{N} \right)} \\ H_y[k,l] = \frac{ (\exp\left[ - \frac{2\pi i \cdot l }{N} \right] -1) \exp(-i\pi
k / N) }{ 4\left( \sin^2\frac{\pi k}{N} + \sin^2\frac{\pi l}{N} \right)} $

Taking the inverse transform $\mathcal{F}^{-1}\hat{\Phi}[k,l] $ produces the
estimate of the wavefront $\hat{\phi}[m, n]$. Such a geometry estimations (see Fig.~\ref{fig:ao2004_3670_modified_Hudgin}) are of
high quality, and it does not suffer from global or local waffle like Fried
geometry.

Modified Hudgin geometry.

That was validated\cite{poyneer2003experimental} in on-sky testing at
Palomar Observatory. In the experiments, the authors of
\cite{poyneer2003experimental} tried out several options for geometries and
filtering, and the modified Hudgin performed best. It is noteworthy that the
regular Hudgin geometry suffered from misalignment-like errors. Finally, the
Fried geometry had excessive local waffle.

Moreover, the modified Hudgin takes half as much computation as the Fried geometry model.

Limitations and Disadvantages of the FTR

Disadvantages
First, if the aperture size in sub-apertures is not a power-of-2, that can cause
performance problems: extensive padding to get to a power-of-2 leads to
increased noise. The FTR requires the square or the pseudo-hex DM geometry. The
non-integer ratio of sub-apertures size or actuator spacing
requires correct re-sampling of estimate.

Advantages
The FTR is fast enough for the ExAO systems and large simulation codes. It provides adaptability with filtering - one can compensate the misalignment and
other errors. The modified Hudgin method does not suffer significantly from the global or local waffle.


The problems in implementation of modified Hudgin
The modified Hudgin still requires the suppression of the piston mode:

if ((rownum == 0)&(colnum == 0))
wavefront(rownum+1, colnum+1) = 0;
else

The original formula for the spatial filters works queer. The additional shift must be like this: $\exp(-i2\pi l / N)$ instead of simply $\exp(-i\pi l / N)$.

Moreover, the code for the different shifts gives slightly different results. The results with the spatial filter:

$$H_x[k,l] = \frac{ (\exp\left[ - \frac{2\pi i \cdot k }{N} \right] -1) \exp(-i2\pi l / N) }{ 4\left( \sin^2\frac{\pi k}{N} + \sin^2\frac{\pi l}{N} \right)} $$

and implemented as a MATLAB code:

H_row = (exp(-(i*2*rownum*pi)/N_row) -1)*(exp(-(i*2*colnum*pi)/N_row)); %% spatial filters for X axis that include the complex conjugate of exponentials
H_column = (exp(-(i*2*colnum*pi)/N_col) -1)*(exp(-(i*2*rownum*pi)/N_col)); %% spatial filters for Y axis that include the complex conjugate of exponentials

gives the reconstruction, shown in Fig.~\ref{fig:wfr_modified_Hudgin_row-col}a. Contrary, the spatial filter of:

$$H_x[k,l] = \frac{ (\exp\left[ - \frac{2\pi i \cdot l }{N} \right] -1) \exp(-i2\pi
k / N) }{ 4\left( \sin^2\frac{\pi k}{N} + \sin^2\frac{\pi l}{N} \right)} $$

with code:

H_row = (exp(-(i*2*rownum*pi)/N_row) -1)*(exp(-(i*2*colnum*pi)/N_row)); %% spatial filters for X axis that include the complex conjugate of exponentials
H_column = (exp(-(i*2*colnum*pi)/N_col) -1)*(exp(-(i*2*rownum*pi)/N_col)); %% spatial filters for Y axis that include the complex conjugate of exponentials

gives better result in the reconstruction, as seen in Fig.~\ref{fig:wfr_modified_Hudgin_row-col}b.

The results of reconstruction for the modified Hudgin algorithm for the row spatial filter: a) row-column relationships with shifts, and b) column-row relationships.

The results of the modified Hudgin geometry are indeed better compared with the standard Hudgin geometry (see the reconstruction with conventional Hudgin FFT WFR (below)

The results of reconstruction with conventional FFT Hudgin.