Wavefront sensing with critical sampling

Abstract

Different types of nonredundant sampling patterns are shown to guarantee completeness of the basis formed by the sampled partial derivatives of Zernike polynomials, commonly used to reconstruct the wavefront from its slopes (wavefront sensing). In the ideal noise-free case, this enables one to recover double the number of modes J than sampling points I (critical sampling J=2I). With real data, noise amplification makes the optimal number of modes lower I<J<2I. Our computer simulations show that optimized nonredundant sampling provides a significant improvement of wavefront reconstructions, with the number of modes recovered about 2.5 higher than with standard sampling patterns.

Complete Modal Representation with Discrete Zernike Polynomials - Critical Sampling in Non Redundant Grids

“…The above results support the relationship between redundancy, low efficiency of sampling and lack of completeness. Taking into account the symmetry of ZPs where radial and angular parts are separable, polar (or hexapolar) sampling schemes are expected to have the highest redundancy in the Z matrix, which is confirmed by the lower The same non redundant sampling patterns, which guarantee completeness of the ZPs, namely random, perturbed regular, and spirals (especially Fermat and quadratic ones), do also guarantee completeness of the D basis (Navarro et al, 2011). In other words, the 2 sampled partial derivatives of ZPs form a complete basis for the set of measurements m. The size of the matrix is 2IxJ with 2I = J.…”

“…It is worth remarking that critical sampling in this case means to recover double number of modes than sampling points, J=2I, simply applying c = D -1 m. This possibility is plausible since we have two measures (two partial derivatives in m i ) at each point, provided that there is no redundancy (Navarro et al, 2011). This would be similar to the Hermite interpolation, where one has the function and its first derivative at each point and recovers J=2I coefficients.…”

“…However, orthogonality is lost in both cases after sampling. One of the most important problems caused by the lack of orthogonality is a bad condition number of matrix Z (or D), which makes the inversion (Z -1 or D -1 ) to be numerically instable (Navarro et al, 2011) (Zou & Rolland, 2006). The consequence is noise amplification when one tries to estimate the coefficients, using either c = Z -1 w or c = D -1 m. The condition number (CN), ratio between the highest and lowest singular value of the matrix, is the main metric for the expected numerical instability, and also provides an initial prediction of the level of expected noise amplification when passing from the measures (samples) to the coefficients.…”

“…In order to correctly reference this scholarly work, feel free to copy and paste the following: Rafael Navarro and Justo Arines (2011). Complete Modal Representation with Discrete Zernike PolynomialsCritical Sampling in Non Redundant Grids, Numerical Simulations of Physical and Engineering Processes, Prof. Jan Awrejcewicz (Ed.…”

Hypatia: a 4m active space telescope concept and capabilities

“…It may therefore be considered the baseline, low-risk option. Novelties could include the use non-redundant sampling patterns in order to reduce the number of required sub-apertures [2]. Phase diversity allows wavefront aberrations to be estimated from a pair of images with a know phase difference.…”