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 Physics , 2015, DOI: 10.1364/OL.40.000143 Abstract: We use a theoretical frame-work to analytically assess temporal prediction error functions on von-Karman turbulence when a zonal representation of wave-fronts is assumed. Linear prediction models analysed include auto-regressive of order up to three, bilinear interpolation functions and a minimum mean square error predictor. This is an extension of the authors' previously published work (see ref. 2) in which the efficacy of various temporal prediction models was established. Here we examine the tolerance of these algorithms to specific forms of model errors, thus defining the expected change in behaviour of the previous results under less ideal conditions. Results show that +/- 100pc wind-speed error and +/- 50 deg are tolerable before the best linear predictor delivers poorer performance than the no-prediction case.
 Augustus J. E. M Janssen Physics , 2014, DOI: 10.1364/JOSAA.31.001604 Abstract: The partial derivatives and Laplacians of the Zernike circle polynomials occur in various places in the literature on computational optics. In a number of cases, the expansion of these derivatives and Laplacians in the circle polynomials are required. For the first-order partial derivatives, analytic results are scattered in the literature, starting as early as 1942 in Nijboer's thesis and continuing until present day, with some emphasis on recursive computation schemes. A brief historic account of these results is given in the present paper. By choosing the unnormalized version of the circle polynomials, with exponential rather than trigonometric azimuthal dependence, and by a proper combination of the two partial derivatives, a concise form of the series expressions emerges. This form is appropriate for the formulation and solution of a model wave-front sensing problem of reconstructing a wave-front on the level of its expansion coefficients from (measurements of the expansion coefficients of) the partial derivatives. It turns out that the least-squares estimation problem arising here decouples per azimuthal order $m$, and per $m$ the generalized inverse solution assumes a concise analytic form, thereby avoiding SVD-decompositions. The preferred version of the circle polynomials, with proper combination of the partial derivatives, also leads to a concise analytic result for the Zernike expansion of the Laplacian of the circle polynomials. From these expansions, the properties of the Laplacian as a mapping from the space of circle polynomials of maximal degree $N$, as required in the study of the Neumann problem associated with the Transport-of-Intensity equation, can be read off within a single glance. Furthermore, the inverse of the Laplacian on this space is shown to have a concise analytic form.
 计算机科学技术学报 , 2002, Abstract: The fast computation of Zernike moments from normalized geometric moments has been developed in this paper. The computation is multiplication free and only additions are needed to generate Zernike moments. Geometric moments are generated using Hatamian’s filter up to high orders by a very simple and straightforward computation scheme. Other kinds of moments (e.g., Legendre, pseudo Zernike) can be computed using the same algorithm after giving the proper transformations that state their relations to geometric moments. Proper normalizations of geometric moments are necessary so that the method can be used in the efficient computation of Zernike moments. To ensure fair comparisons, recursive algorithms are used to generate Zernike polynomials and other coefficients. The computational complexity model and test programs show that the speed-up factor of the proposed algorithm is superior with respect to other fast and/or direct computations. It perhaps is the first time that Zernike moments can be computed in real time rates, which encourages the use of Zernike moment features in different image retrieval systems that support huge databases such as the XM experimental model stated for the MPEG-7 experimental core. It is concluded that choosing direct computation would be impractical. Supported by the National Natural Science Foundation of China (No.30170274) and the National “863” High-Tech Programme of China (No. 863-306-ZB13-05-6). Al-Rawi Mohammed was born in 1966, in Iraq, and received his M.S. degree from the College of Sciences, Baghdad University, in 1993. Currently, he is a Ph.D. candidate of the Institute of Image Processing & Pattern Recognition, SJTU. His major research interests are invariant pattern recognition, recognition of color texture, image processing. YANG Jie was born in 1964, and received his Ph.D. degree from the Department of Computer Science, Hamburg University, Germany. Currently, he is vice director of the Institute of Image Processing & Pattern Recognition, SJTU. He has taken charge of many research projects (e.g. National Natural Science Foundation, National “863” High-Tech, Programme) and published a book in Germany and more than 70 journal papers. His major research interests are object detection and recognition, data fusion and data mining, intelligent systems and applications, medical image processing.
 中国物理 B , 2009, Abstract: The segmented mirror telescope is widely used. The aberrations of segmented mirror systems are different from single mirror systems. This paper uses the Fourier optics theory to analyse the Zernike aberrations of segmented mirror systems. It concludes that the Zernike aberrations of segmented mirror systems obey the linearity theorem. The design of a segmented space telescope and segmented schemes are discussed, and its optical model is constructed. The computer simulation experiment is performed with this optical model to verify the suppositions. The experimental results confirm the correctness of the model.
 Richard J. Mathar Mathematics , 2007, Abstract: The Zernike radial polynomials are a system of orthogonal polynomials over the unit interval with weight x. They are used as basis functions in optics to expand fields over the cross section of circular pupils. To calculate the roots of Zernike polynomials, we optimize the generic iterative numerical Newton's Method that iterates on zeros of functions with third order convergence. The technique is based on rewriting the polynomials as Gauss hypergeometric functions, reduction of second order derivatives to first order derivatives, and evaluation of some ratios of derivatives by terminating continued fractions. A PARI program and a short table of zeros complete up to polynomials of 20th order are included.
 Serbian Astronomical Journal , 2009, Abstract: The radial polynomials of the 2D (circular) and 3D (spherical) Zernike functions are tabulated as powers of the radial distance. The reciprocal tabulation of powers of the radial distance in series of radial polynomials is also given, based on projections that take advantage of the orthogonality of the polynomials over the unit interval. They play a role in the expansion of products of the polynomials into sums, which is demonstrated by some examples. Multiplication of the polynomials by the angular bases (azimuth, polar angle) defines the Zernike functions, for which we derive transformations to and from the Cartesian coordinate system centered at the middle of the circle or sphere.
 Richard J. Mathar Physics , 2008, DOI: 10.2298/SAJ0979107M Abstract: The radial polynomials of the 2D (circular) and 3D (spherical) Zernike functions are tabulated as powers of the radial distance. The reciprocal tabulation of powers of the radial distance in series of radial polynomials is also given, based on projections that take advantage of the orthogonality of the polynomials over the unit interval. They may play a role in the expansion of products of the polynomials into sums, which is demonstrated by some examples. Multiplication of the polynomials by the angular bases (azimuth, polar angle) defines the Zernike functions, for which we derive and tabulate transformations to and from the Cartesian coordinate system centered at the middle of the circle or sphere.
 Mathematics , 2015, Abstract: A pattern of interpolation nodes on the disk is studied, for which the interpolation problem is theoretically unisolvent, and which renders a minimal numerical condition for the collocation matrix when the standard basis of Zernike polynomials is used. It is shown that these nodes have an excellent performance also from several alternative points of view, providing a numerically stable surface reconstruction, starting from both the elevation and the slope data. Sampling at these nodes allows for a more precise recovery of the coefficients in the Zernike expansion of a wavefront or of an optical surface.
 Richard J. Mathar Mathematics , 2015, Abstract: An explicit C++ library is provided which deals with Zernike Functions over the unit circle as the main subject. The implementation includes basic means to evaluate the functions at points inside the unit circle and to convert the radial and azimuthal parameters to Noll's index and vice versa. Advanced methods allow to expand products of Zernike Functions into sums of Zernike Functions, and to convert Zernike Functions to polynomials over the two Cartesian coordinates and vice versa.
 Augustus J. E. M. Janssen Mathematics , 2015, Abstract: We consider 3D versions of the Zernike polynomials that are commonly used in 2D in optics and lithography. We generalize the 3D Zernike polynomials to functions that vanish to a prescribed degree $\alpha\geq0$ at the rim of their supporting ball $\rho\leq1$. The analytic theory of the 3D generalized Zernike functions is developed, with attention for computational results for their Fourier transform, Funk and Radon transform, and scaling operations. The Fourier transform of generalized 3D Zernike functions shows less oscillatory behaviour and more rapid decay at infinity, compared to the standard case $\alpha=0$, when the smoothness parameter $\alpha$ is increased beyond 0. The 3D generalized Zernike functions can be used to expand smooth functions, supported by the unit ball and vanishing at the rim and the origin of the unit ball, whose radial and angular dependence is separated. Particular instances of the latter functions (prewavelets) yield, via the Funk transform and the Fourier transform, an anisotropic function that can be used for a band-limited line-detecting wavelet transform, appropriate for analysis of 3D medical data containing elongated structures. We present instances of prewavelets, with relevant radial functions, that allow analytic computation of Funk and Fourier transform. A key step here is to identify the special form that is assumed by the expansion coefficients of a separable function on the unit ball with respect to generalized 3D Zernike functions. A further issue is how to scale a function on the unit ball while maintaining its supporting set, and this issue is solved in a particular form.
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