We present recent results from the initial testing of an Artificial Neural Network (ANN) based tomographic reconstructor Complex Atmospheric Reconstructor based on Machine lEarNing (CARMEN) on Canary, an Adaptive Optics demonstrator operated on the 4
.2m William Herschel Telescope, La Palma. The reconstructor was compared with contemporaneous data using the Learn and Apply (L&A) tomographic reconstructor. We find that the fully optimised L&A tomographic reconstructor outperforms CARMEN by approximately 5% in Strehl ratio or 15nm rms in wavefront error. We also present results for Canary in Ground Layer Adaptive Optics mode to show that the reconstructors are tomographic. The results are comparable and this small deficit is attributed to limitations in the training data used to build the ANN. Laboratory bench tests show that the ANN can out perform L&A under certain conditions, e.g. if the higher layer of a model two layer atmosphere was to change in altitude by ~300~m (equivalent to a shift of approximately one tenth of a subaperture).
We prove a constant term theorem which is useful for finding weight polynomials for Ballot/Motzkin paths in a strip with a fixed number of arbitrary `decorated weights as well as an arbitrary `background weight. Our CT theorem, like Viennots lattice
path theorem from which it is derived primarily by a change of variable lemma, is expressed in terms of orthogonal polynomials which in our applications of interest often turn out to be non-classical. Hence we also present an efficient method for finding explicit closed form polynomial expressions for these non-classical orthogonal polynomials. Our method for finding the closed form polynomial expressions relies on simple combinatorial manipulations of Viennots diagrammatic representation for orthogonal polynomials. In the course of the paper we also provide a new proof of Viennots original orthogonal polynomial lattice path theorem. The new proof is of interest because it uses diagonalization of the transfer matrix, but gets around difficulties that have arisen in past attempts to use this approach. In particular we show how to sum over a set of implicitly defined zeros of a given orthogonal polynomial, either by using properties of residues or by using partial fractions. We conclude by applying the method to two lattice path problems important in the study of polymer physics as models of steric stabilization and sensitized flocculation.
