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A numerically efficient algorithm for expanding a function in a series of Zernike polynomials is presented. The algorithm evaluates the expansion coefficients through the standard 2-D integration formula derived from the Zernike polynomials’ orthogonal properties. Quadratic approximations are used along with the function to be expanded to eliminate...
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... Despite their straightforward nature, these techniques were often computationally expensive and prone to numerical instability. Over time, more sophisticated methods have emerged, such as the use of recurrence relations [15,16,17,18,19,20,21]. These studies successfully addressed instability issues for higher mode numbers and provided fast algorithms for calculating Zernike polynomials. ...
Zernike Polynomials serve as an orthogonal basis on the unit disc, and have been proven to be effective in optics simulations, astrophysics, and more recently in plasma simulations. Unlike Bessel functions, they maintain finite values at the disc center, ensuring inherent analyticity along the axis. We developed ZERNIPAX, an open-source Python package capable of utilizing CPU/GPUs, leveraging Google's JAX package and available on https://github.com/PlasmaControl/FastZernike.git as well as PyPI. Our implementation of the recursion relation between Jacobi polynomials significantly improves computation time compared to alternative methods by use of parallel computing while still preserving accuracy for mode numbers n>100.
... However, the stopping condition is unknown at that time and the formula is singular when k 1 = 0. In 1989, Prata and Rusch [15] proposed the following recursive scheme R m n (ρ) = ρL 1 R m−1 n−1 (ρ) + L 2 R m n−2 (ρ), n ≥ 2 (8) with the coefficients ...
... This m-recursive scheme is more efficient than the other recursive schemes for computing R m n (ρ). However, ρ = 0 is a singular point in (15) although R m n (ρ) is regular for all ρ ∈ [0, 1]. Thus the computation will be unstable if ρ is small enough. ...
... However, for the R m n (ρ) with two integers n and m as arguments, there is a lack of feasible method to convert the recursive formula to iterative versions. Although Kintner's n-recursive formula (6) can be reformulated as an iterative formula, it is limited for n ≥ 4; Chong's m-recursive scheme (15) can be converted to its iterative version, however the singular point ρ = 0 will still exist. For the coupled recursive formulae (8) (or (12) equivalently) and (13), their iterative implementations are still to be explored. ...
Zernike radial polynomials play a significant role in application areas such as optics design, imaging systems, and image processing systems. Currently, there are two kinds of numerical schemes for computing the Zernike radial polynomials automatically with computer programs: one is based on the definition in which the factorial operations may lead to the overflow problem and the high order derivatives are troublesome, and the other is based on recursion which is either unstable or with high computational complexity. In this paper, our emphasis is focused on exploring the balanced binary tree (BBT) schemes for computing Zernike radial polynomials: firstly we establish an elegant formulae for computation; secondly we propose the recursive and iterative algorithms based-on BBT; thirdly we analyze the computational complexity of the algorithms rigorously; finally we verify and validate the performance of BBT schemes by testing the running time. Theoretical analysis shows that the computational complexity of BBT recursive algorithm and iterative algorithm are exponential and quadratic respectively, which coincides with the running time test very well. Experiments show that the time consumption is about 1 ~ 10 microseconds with different computation platforms for the BBT iterative algorithm (BBTIA), which is stable and efficient for real-time applications.
... After solving the coefficient vector , Zernike polynomial can be used to describe the imaging wave surface of the optical system. [15] It is used as the aberration input of the pupil position of the optical system through data acquisition and data preprocessing. The image quality degradation will happen in different degrees due to simulated aberration. ...
Computational optical imaging is an interdisciplinary subject integrating optics, mathematics, and information technology. It introduces information processing into optical imaging and combines it with intelligent computing, subverting the imaging mechanism of traditional optical imaging which only relies on orderly information transmission. To meet the high-precision requirements of traditional optical imaging for optical processing and adjustment, as well as to solve its problems of being sensitive to gravity and temperature in use, we establish an optical imaging system model from the perspective of computational optical imaging and studies how to design and solve the imaging consistency problem of optical system under the influence of gravity, thermal effect, stress and other external environment to build a high robustness optical system. The results show that the high robustness interval of the optical system exists and can effectively reduce the sensitivity of the optical system to the disturbance of each link, thus realizing the high robustness of optical imaging.
... However, the stopping condition is unknown at that time and the formulae is singular when k 1 = 0. In 1989, Prata and Rusch [14] proposed the following recursive scheme R m n (ρ) = ρL 1 R m−1 n−1 (ρ) + L 2 R m n−2 (ρ), n ≥ 2; (6) with the coefficients ...
... Our novel recursive formulae is a combination of the recursive schemes in [5] and [14]. The trick of the exploring is to reduce the difference of the up-down scripts appearing the right hand side in (10) and (11) with a common constant so as to get a balanced result, see Table 1. ...
Zernike radial polynomials play a significant role in application areas such as optics design, imaging systems, and image processing systems. Currently, there are two kinds of numerical schemes for computing the Zernike radial polynomials automatically with computer programs: one is based on the definition in which the factorial operations may lead to the overflow problem and the high order derivatives are troublesome, and the other is based on recursion which is either unstable or with high computational complexity. In this paper, our emphasis is focused on exploring the \textit{balanced binary tree} (BBT) schemes for computing Zernike radial polynomials: firstly we established an elegant formulae for computation; secondly we proposed the recursive and iterative algorithms based-on BBT; thirdly we analyzed the computational complexity of the algorithms rigorously; finally we verified and validated the performance of BBT schemes by testing the running time. Theoretic analysis shows that the computational complexity of BBT recursive algorithm and iterative algorithm are exponential and quadratic respectively, which coincides with the running time test very well. Experiments show that the time consumption is about microseconds with different computation platforms for the BBT iterative algorithm (BBTIA), which is stable and efficient for realtime applications.
... To deal with these problems, various recurrence relations have been proposed for evaluating the radial polynomials [60][61][62][63][64]. Here we briefly review four widely-used recurrence methods, including the modified Kintner method [59,65], the Prata's method [66], the q-recursive method [59], and the Shakibaei and Paramesran method [62]. The modified Kintner method was first proposed by Kintner in 1976 [65] and improved by Chong et al. in 2003 [59] by adding recurrence relations for special cases when n − m = 0 and 2. The improved recurrence relation can be expressed as ...
... The modified Kintner method is a degree-varying (n-varying) approach that computes radial polynomials at higher order from those at lower order for a fixed value of m. The Prata method was proposed by Prata and Rusch in 1989 [66] and the recurrence relation can be written as ...
The Zernike polynomials are a complete set of continuous functions orthogonal over a unit circle. Since first developed by Zernike in 1934, they have been in widespread use in many fields ranging from optics, vision sciences, to image processing. However, due to the lack of a unified definition, many confusing indices have been used in the past decades and mathematical properties are scattered in the literature. This review provides a comprehensive account of Zernike circle polynomials and their noncircular derivatives, including history, definitions, mathematical properties, roles in wavefront fitting, relationships with optical aberrations, and connections with other polynomials. We also survey state-of-the-art applications of Zernike polynomials in a range of fields, including the diffraction theory of aberrations, optical design, optical testing, ophthalmic optics, adaptive optics, and image analysis. Owing to their elegant and rigorous mathematical properties, the range of scientific and industrial applications of Zernike polynomials is likely to expand. This review is expected to clear up the confusion of different indices, provide a self-contained reference guide for beginners as well as specialists, and facilitate further developments and applications of the Zernike polynomials.
... Prata and Rusch have derived relationships for similar integrals for the 2D Zernike radial polynomials [58]. We expand their results here. ...
The 3D Zernike polynomials form an orthonormal basis of the unit ball. The associated 3D Zernike moments have been successfully applied for 3D shape recognition; they are popular in structural biology for comparing protein structures and properties. Many algorithms have been proposed for computing those moments, starting from a voxel-based representation or from a surface based geometric mesh of the shape. As the order of the 3D Zernike moments increases, however, those algorithms suffer from decrease in computational efficiency and more importantly from numerical accuracy. In this paper, new algorithms are proposed to compute the 3D Zernike moments of a homogeneous shape defined by an unstructured triangulation of its surface that remove those numerical inaccuracies. These algorithms rely on the analytical integration of the moments on tetrahedra defined by the surface triangles and a central point and on a set of novel recurrent relationships between the corresponding integrals. The mathematical basis and implementation details of the algorithms are presented and their numerical stability is evaluated.
... The solution of (1.3) denoted by R m n (r) is known as the radial part of a Zernike polynomial and is given by (1.4) below when n, m are non-negative integers, and n − m is even and non-negative, see [20], [22] and [3]: ...
... An efficient algorithm for calculating the coefficients A nm and B nm is discussed in [22], see also [14]. An approximation of f of order (m, n) can then be calculated as ...
... Using the recurrence relation given in [21] and [22] ...
Integration operational matrix methods based on Zernike polynomials are used to determine approximate solutions of a class of non-homogeneous partial differential equations (PDEs) of first and second order. Due to the nature of the Zernike polynomials being described in the unit disk, this method is particularly effective in solving PDEs over a circular region. Further, the proposed method can solve PDEs with discontinuous Dirichlet and Neumann boundary conditions, and as these discontinuous functions cannot be defined at some of the Chebyshev or Gauss-Lobatto points, the much acclaimed pseudo-spectral methods are not directly applicable to such problems. Solving such PDEs is also a new application of Zernike polynomials as so far the main application of these polynomials seem to have been in the study of optical aberrations of circularly symmetric optical systems. In the present method, the given PDE is converted to a system of linear equations of the form Ax = b which may be solved by both l1 and l2 minimization methods among which the l1 method is found to be more accurate. Finally, in the expansion of a function in terms of Zernike polynomials, the rate of decay of the coefficients is given for certain classes of functions.
... The solution of (1.3) denoted by R m n (r) is known as the radial part of a Zernike polynomial and is given by (1.4) below when n, m are non-negative integers, and n − m is even and non-negative, see [20], [22] and [3]: ...
... They can therefore be normalized to form an orthonormal basis for the space L 2 (B(0, 1)), see Section 4. Let f (r, φ) be an arbitrary function defined on B(0, 1). In terms of the Zernike polynomials given in (1.5), f can be represented as [28,3] f (r, φ) = An efficient algorithm for calculating the coefficients A nm and B nm is discussed in [22], see also [14]. An approximation of f of order (m, n) can then be calculated as ...
... Using the recurrence relation given in [21] and [22] ...
Integration operational matrix methods based on Zernike polynomials are used to determine approximate solutions of a class of non-homogeneous partial differential equations (PDEs) of first and second order. Due to the nature of the Zernike polynomials being described in the unit disk, this method is particularly effective in solving PDEs over a circular region. Further, the proposed method can solve PDEs with discontinuous Dirichlet and Neumann boundary conditions, and as these discontinuous functions cannot be defined at some of the Chebyshev or Gauss-Lobatto points, the much acclaimed pseudo-spectral methods are not directly applicable to such problems. Solving such PDEs is also a new application of Zernike polynomials as so far the main application of these polynomials seem to have been in the study of optical aberrations of circularly symmetric optical systems. In the present method, the given PDE is converted to a system of linear equations of the form Ax = b which may be solved by both l1 and l2 minimization methods among which the l1 method is found to be more accurate. Finally, in the expansion of a function in terms of Zernike polynomials, the rate of decay of the coefficients is given for certain classes of functions.
... where r and θ represent radial and angular coordinates, respectively; Z q (r, θ ) is the q th Zernike polynomial evaluated at (r, θ ); q is the mode ordering number. The Zernike functions are given as [36] Z q (r, θ ) = R m n (r) cos(mθ ) m = 0 and even q R m n (r) sin(mθ ) m = 0 and odd q R m n (r) ...
A novel algorithm for closed fringe demodulation for an absolute phase estimation, to the best of our knowledge, is proposed. The two-dimensional phase is represented as a weighted linear combination of a certain number of Zernike polynomials (ZPs). Essentially, the problem of phase estimation is converted into the estimation of ZP coefficients. The task of ZP coefficient estimation is performed based on a state space model. Due to the nonlinear dependence of the fringe intensity measurement model on the ZP coefficients, the extended Kalman filter (EKF) is used for the state estimation. A pseudo-measurement model is considered based on the state vector sparsity constraint to improve the convergence performance of the EKF. Simulation and experimental results are provided to demonstrate the noise robustness and the practical applicability of the proposed method.
... Unlike GMs and CMs, ZMs values have a smaller dynamic range [23], which simplifies the process of feature matching in the database. Some authors claim that the presence of factorial terms in the radial polynomials increases the computation time needed to compute ZMs, especially for higher order moments, and methods to speed up the computation of the moments are proposed in [42], [43], [44]. However, as discussed in Section V, the optimal performance of the ZM-based pose estimation algorithm is obtained with moments up to the seventh and ninth order. ...
This article revisits methods based on global descriptors to estimate the pose of a known object using a monocular camera, in the context of space rendezvous between an autonomous spacecraft and a noncooperative target. These methods estimate the pose by detection, i.e., they do not require any prior information about the pose of the observed object, making them suitable for initial pose acquisition and the monitoring of faults in other on-board estimators. We consider here specifically methods that retrieve the pose of a known object using a precomputed set of invariants and geometric moments. Three classes of global invariant features are analyzed, based on complex moments, Zernike moments, and Fourier descriptors. The robustness, accuracy, and computational efficiency of the different invariants are tested and compared under various conditions. We also discuss certain implementation aspects of the method that lead to improved accuracy and efficiency over previously reported results. Overall, our results can be used to identify which variations of the method offer a sufficiently fast and robust solution for pose estimation by detection, with low computational requirements that are compatible with space-qualified processors.