Abstract
Abstract We study the problem of reconstructing a wavefront from measurements by Shack–Hartmann-type sensors. Mathematically, this leads to the problem of reconstructing a function from a discrete set of averages of the gradient. We investigate this problem in a scale of Hilbert spaces and show that the best approximate solutions are asymptotically ill-posed depending on the scale of the Hilbert spaces. Moreover, we prove convergence of the best approximate solutions and of approximations obtained by the truncated singular value decomposition for increasing dimension of measurements.
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