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Sampling patterns for Zernike-like bases in non-standard geometries

Sergio Díaz-Elbal, Andrei Martínez-Finkelshtein, Darío Ramos-López

TL;DR

This paper addresses extending Zernike polynomials from the unit disk to non-circular domains by mapping via a diffeomorphism $\varphi: \mathbb{D} \to M$ and constructing orthonormal Zernike-like bases on $M$ (e.g., $K_j$, $H_j$, $E_j$, $O_j$, $C_j$). It analyzes how sampling patterns transfer under $\varphi$ to preserve numerical conditioning of the collocation matrices, with explicit results such as $\kappa_p(H_N)\le\frac{2\sqrt{3}}{3}\,\kappa_p(Z_N)$ for hexagonal domains. The work identifies optimal sampling strategies (OCS and Lebesgue points) and validates them through extensive numerical experiments on disks, hexagons, ellipses, and annuli, including a practical application to wavefront interpolation in hexagonal segmented mirrors, achieving sub-percent to low-percent relative errors. Overall, the approach enables stable, high-order interpolation and accurate wavefront modeling in complex optical geometries, with the OCS method providing an explicit and practically useful sampling rule.

Abstract

Zernike polynomials are widely used in optics and ophthalmology due to their direct connection to classical optical aberrations. While orthogonal on the unit disk, their application to discrete data or non-circular domains--such as ellipses, annuli, and hexagons--presents challenges in terms of numerical stability and accuracy. In this work, we extend Zernike-like orthogonal functions to these non-standard geometries using diffeomorphic mappings and construct sampling patterns that preserve favorable numerical conditioning. We provide theoretical bounds for the condition numbers of the resulting collocation matrices and validate them through extensive numerical experiments. As a practical application, we demonstrate accurate wavefront interpolation and reconstruction in segmented mirror telescopes composed of hexagonal facets. Our results show that appropriately transferred sampling configurations, especially Optimal Concentric Sampling and Lebesgue points, allow stable high-order interpolation and effective wavefront modeling in complex optical systems. Moreover, the Optimal Concentric Samplings can be computed with an explicit expression, which is a significant advantage in practice.

Sampling patterns for Zernike-like bases in non-standard geometries

TL;DR

This paper addresses extending Zernike polynomials from the unit disk to non-circular domains by mapping via a diffeomorphism and constructing orthonormal Zernike-like bases on (e.g., , , , , ). It analyzes how sampling patterns transfer under to preserve numerical conditioning of the collocation matrices, with explicit results such as for hexagonal domains. The work identifies optimal sampling strategies (OCS and Lebesgue points) and validates them through extensive numerical experiments on disks, hexagons, ellipses, and annuli, including a practical application to wavefront interpolation in hexagonal segmented mirrors, achieving sub-percent to low-percent relative errors. Overall, the approach enables stable, high-order interpolation and accurate wavefront modeling in complex optical geometries, with the OCS method providing an explicit and practically useful sampling rule.

Abstract

Zernike polynomials are widely used in optics and ophthalmology due to their direct connection to classical optical aberrations. While orthogonal on the unit disk, their application to discrete data or non-circular domains--such as ellipses, annuli, and hexagons--presents challenges in terms of numerical stability and accuracy. In this work, we extend Zernike-like orthogonal functions to these non-standard geometries using diffeomorphic mappings and construct sampling patterns that preserve favorable numerical conditioning. We provide theoretical bounds for the condition numbers of the resulting collocation matrices and validate them through extensive numerical experiments. As a practical application, we demonstrate accurate wavefront interpolation and reconstruction in segmented mirror telescopes composed of hexagonal facets. Our results show that appropriately transferred sampling configurations, especially Optimal Concentric Sampling and Lebesgue points, allow stable high-order interpolation and effective wavefront modeling in complex optical systems. Moreover, the Optimal Concentric Samplings can be computed with an explicit expression, which is a significant advantage in practice.

Paper Structure

This paper contains 7 sections, 2 theorems, 42 equations, 6 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Interpolation with Zernike circle polynomials
  3. Extension of Zernike polynomials to non-standard geometries
  4. Good sampling patterns in non-standard geometries
  5. Numerical experiments
  6. Wavefront fitting of atmospheric turbulence
  7. Conclusions

Key Result

Proposition 4.1

Consider a diffeomorphism $\varphi: \mathbb{D} \rightarrow M \subset \mathbb{R}^2$, and the functions $P_j=Z_j\circ \varphi^{-1}$ defined on $M$, where $Z_j$ are the regular Zernike polynomials on the disk. Then, the collocation matrix of functions $P_j$ at the transferred nodes $\varphi(\mathcal{S} In particular, the condition number $\kappa(\cdot)$ remains invariant for any underlying matrix $p$

Figures (6)

  • Figure 1: General parametrization of an elliptical aperture $G$.
  • Figure 2: Lebesgue points of degree $n=12$ (left) and OCS of degree $n=15$ (right) transferred to the hexagon.
  • Figure 3: Fekete points of degree $n=8$ and Carnicer points of degree $n=12$ transferred to the ellipse.
  • Figure 4: Cuyt points of degree $n=10$ and OCS of degree $n=14$, transferred to the annulus.
  • Figure 5: Random atmospheric turbulence following Kolmogorov model (left) and configuration of nodes from Lebesgue points of order $n=6$ in each of the 36 hexagons.
  • ...and 1 more figures

Theorems & Definitions (2)

  • Proposition 4.1
  • Proposition 4.2