Weighted nonnegative tensor factorization for atmospheric tomography reconstruction

Abstract

Context. Increasing the area on the sky over which atmospheric turbulences can be corrected is a matter of wide interest in astrophysics, especially when a new generation of extremely large telescopes (ELT) is to come in the near future. Aims. In this study we tested if a method for visual representation in three-dimensional displays, the weighted nonnegative tensor factorization (WNTF), is able to improve the quality of the atmospheric tomography (AT) reconstruction as compared to a more standardized method like a randomized Kaczmarz algorithm. Methods. A total of 1000 different atmospheres were simulated and recovered by both methods. Recovering was computed for two and three layers and for four different constellations of laser guiding stars (LGS). The goodness of both methods was tested by means of the radial average of the Strehl ratio across the field of view of a telescope of 8m diameter with a sky coverage of 97.8 arcsec. Results. The proposed method significantly outperformed the Kaczmarz in all tested cases (p ≤ 0.05). In WNTF, three-layers configuration provided better outcomes, but there was no clear relation between different LGS constellations and the quality of Strehl ratio maps. Conclusions. The WNTF method is a novel technique in astronomy and its use to recover atmospheric turbulence profiles was proposed and tested. It showed better quality of reconstruction than a conventional Kaczmarz algorithm independently of the number and height of recovered atmospheric layers and of the constellation of laser guide star used. The WNTF method was shown to be a useful tool in highly ill-posed AT problems, where the difficulty of classical algorithms produce high Strehl value maps.

A Kaczmarz Method with Simple Random Sampling for Solving Large Linear Systems

“…Consequently, Algorithm 5 is often superior to the others, and has more suitable to large-scale and dense problems. 7 . We run the VRK method [18], the VRGS method [18], the VGRK method [12], the AVGRK (ω) method [12], as well as Algorithm 6 and Algorithm 7 on this problem.…”

“…where A ∈ C m×n and b ∈ C m , with m n. This method has been applied to many important fields such as reconstruction of CT scanned images [15], biological calculation [10], computerized tomography [11,14,23], digital signal processing [5,14,23], image reconstruction [7,26,35,37], distributed computing [18,34]; and so on [28,13,19,22,29,42]. For the linear system (1.1), the Kaczmarz method cycles through the rows of the matrix in question, and each iteration is formed by projecting the current point to the hyperplane formed by the active row.…”

Greedy Randomized and Maximal Weighted Residual Kaczmarz Methods with Oblique Projection

“…The Kaczmarz method [25] selects the rows of the matrix A by using the cyclic rule, and in each iteration, the current iteration point is orthogonally projected onto the corresponding hyperplane. Due to its simplicity and performance, the Kaczmarz method has been applied to many fields, such as computerized tomography [1,2], image reconstruction [3,4,5,6], distributed computing [7], and signal processing [8,1,2]; and so on [9,10,11,12]. Since the Kaczmarz method cycles through the rows of A, the performance may depend heavily on the ordering of these rows.…”

Greedy Randomized and Maximal Weighted Residual Kaczmarz Methods with Oblique Projection

“…The Kaczmarz method [19] selects the rows of the matrix A by using the cyclic rule, and in each iteration, the current iteration point is orthogonally projected onto the corresponding hyperplane. Due to its simplicity and performance, the Kaczmarz method has been applied to many fields, such as computerized tomography [15,20], image reconstruction [7,22,39,43], distributed computing [17], and signal processing [6,15,20]; and so on [26,13,18,46]. Since the Kaczmarz method cycles through the rows of A, the performance may depend heavily on the ordering of these rows.…”