Structured linear algebra problems in adaptive optics imaging
Advances in Computational Mathematics
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The earth's atmosphere is not a perfect media through which to view objects in outer-space; turbulence in the atmospheric temperature distribution results in refractive index variations that interfere with the propagation of light. As a result, wavefronts are nonplanar when they reach the ground. The deviation from planarity of a wavefront is known as phase error, and it is phase error that causes the refractive blurring of images. Adaptive optics systems seek to remove phase error from incoming wavefronts. In ground-based astronomy, an estimate of the phase error in a wavefront is typically obtained from wavefront gradient measurements collected by a Shack-Hartmann sensor. The estimate is then used to create a counter wavefront, e.g., using a deformable mirror that (approximately) removes the phase error from the incoming wavefronts. The problem of reconstructing the phase error from Shack-Hartmann gradient measurements requires the solution of a large linear system whose form is defined by the configuration of the sensor. We derive this system and present both the regular least squares and minimum variance approaches to its solution. The most effective existing approaches are then presented alongside new computational methods, and comparisons are made.