Review on methods for wavefront reconstruction from pyramid wavefront sensor data

Pyramid wavefront sensors are planned to be a part of many instruments that are currently under development for the extremely large telescopes (ELT). The unprecedented scales of the upcoming ELT-era instruments are inevitably connected with serious challenges for wavefront reconstruction and control algorithms. Apart from the huge number of correcting elements to be controlled in real-time, real-life features such as the segmentation of the telescope pupil, the low wind effect, the nonlinearity of the pyramid sensor, and the noncommon path aberrations will have a significantly larger impact on the imaging quality in the ELT framework than they ever had before. We summarize various kinds of wavefront reconstruction algorithms for the pyramid wavefront sensor. Based on several forward models, different algorithms were developed in the last decades for linear and nonlinear wavefront correction. The core ideas of the algorithms are presented, and a detailed comparison of the presented methods with respect to underlying pyramid sensor models, computational complexities, and reconstruction qualities is given. In addition, we review the existing and possible solutions for the above-named real-life phenomena. At the same time, directions for further investigations are sketched.