Kaczmarz and Cimmino: iterative and layer-oriented approaches to atmospheric tomography

Abstract

Multi Conjugated Adaptive Optics is based upon tomographic reconstruction of the atmospheric turbulence over the line of sight of a telescope, achieved by combining measurements from different directions in the sky. Using deformable mirrors optically conjugated to different altitudes, a correction can be performed directly on the reconstructed turbulence layers. Different approaches have been developed so far, notably the so called layer-oriented one, experienced with success at the VLT (Very Large Telescope) through MAD (Multi Conjugate Adaptive Optics Demonstrator). It was later shown that the tomography problem, once posed in terms of solving a set of linear equations describing the turbulence layers with respect to the observables, can be solved in an iterative manner through a technique first proposed by Kaczmarz in 1937. It was then speculated that a layer-oriented iteration would asymptotically converge to the same solution. In this paper, we placed the two approaches in the same theoretical framework, identifying them as two different iterative methods to solve the same system of linear equations. We found that the layer-oriented approach can be seen as a weighted form of the iterative method proposed by Cimmino in 1938. By using the known mathematical results relative to Kaczmarz's and the weighted Cimmino methods, we were able to demonstrate the validity of the initial speculation.