# Algorithms

## 3-step approach

### A fast and efficient reconstruction method

In AO systems with several guide stars, such as MCAO, GLAO, LTAO or MOAO, one needs to reconstruct deformable mirror (DM) commands from wavefront sensor (WFS) data. Due to steadily growing telescope sizes, there is a strong increase in the computational load for atmospheric reconstruction. The standard matrix-vector-multiplication (MVM) approach includes the inversion of a huge matrix, which is computationally prohibitive for the new generation of extremely large telescopes.

Instead of using one big matrix-vector system, one could solve three independent subproblems successively: First, the incoming wavefronts have to be obtained from the wavefront sensor measurements (wavefront reconstruction); second, a discretized atmosphere is reconstructed from the incoming wavefronts (atmospheric tomography problem); and finally, this reconstruction has to be fitted onto the available deformable mirrors (projection/fitting step).

For the first step – the wavefront reconstruction – we propose the CuReD algorithm in the case of Shack-Hartmann WFS. This linear algorithm is based on the principles of two-dimensional line integration and is described in more detail in subproject SCAO.

The second step – the reconstruction of discrete atmospheric layers from incoming wavefronts – is a limited angle tomography problem, i.e. an ill-posed inverse problem. For the atmospheric tomography problem in MCAO, LTAO and MOAO, several methods have been studied: the backprojection, CG and in more detail the Kaczmarz method and the Gradient–based method. These iterative methods scale linearly and can be implemented without any matrix-vector multiplication which implies a considerable decrease in computational effort. For noise due to spot elongation, an efficient preprocessing step has been developed.

The third step is the determination of the optimal mirror shapes. In the case of LTAO and MOAO, this is a projection through the atmosphere onto the DM.

In MCAO, the atmospheric reconstruction can be performed either on artificial layers at the altitudes where the deformable mirrors are conjugated to or on more layers. Then, by means of the Kaczmarz iteration or the conjugate gradients method, an efficient optimization of the mirror shapes into multiple directions can be performed.

The 3-step approach allows solving each of the three sub-problems independently and, thus, guarantees a fast and flexible reconstruction. Moreover, it can be easily used with the wavelet wavefront reconstructor instead of CuReD and also for the Pyramid WFS with the CuReD with preprocessing (P-CuReD). The 3-step approach scales linearly, as all three individual steps scale linearly, and is well parallelizable.

### References

[1] A. Neubauer. On the ill-posedness and convergence of the Shack-Hartmann based wavefront reconstruction. J. Inv. Ill-Posed Problems, 18:551-576, 2010.

[3] M. Zhariy, A. Neubauer, M. Rosensteiner, and R. Ramlau. Cumulative wavefront reconstructor for the Shack-Hartman sensor. Inverse Problems and Imaging, 5(4):893-913, Nov 2011.

[4] S. Kindermann, A. Neubauer, and R. Ramlau. A singular value decomposition for the Shack-Hartmann based wavefront reconstruction. J. Comp. Appl. Math, 236(1):2186-2199, 2012.

[5] A. Neubauer. On the Shack-Hartmann based wavefront reconstruction: stability and convergence rates of finite-dimensional approximations. J. Inv. Ill-Posed Problems, 20:591-614, 2012.

[9] D. Saxenhuber. On the Singular Value Decomposition for Shack-Hartmann based wavefront reconstructions. Master's thesis, Johannes Kepler University Linz, 2012.

[10] T. Helin and M. Yudytskiy. Wavelet methods in multi-conjugate adaptive optics. Inverse Problems, 29(8):085003, 2013.

[11] A. Neubauer. A new cumulative wavefront reconstructor for the Shack-Hartmann sensor. J. Inv. Ill-Posed Problems, 21:451-476, 2013.

[13] Iu. Shatokhina, A. Obereder, M. Rosensteiner, and R. Ramlau. Preprocessed cumulative reconstructor with domain decomposition: a fast wavefront reconstruction method for pyramid wavefront sensor. Applied Optics, 52(12):2640-2652, 2013.

[14] M. Yudytskiy, T. Helin, and R. Ramlau. A frequency dependent preconditioned wavelet method for atmospheric tomography. In Third AO4ELT Conference - Adaptive Optics for Extremely Large Telescopes, May 2013.

### Posters

### Presentations

AO Tomography Workshop, Leiden 2012

Adaptive Optics Tomography Workshop, Edinburgh 2014