Fast Evaluation of Equal-spaced Zernike Polynomial Expansion Samples

Phase-shifting interferometry with a genetic algorithm is proposed. The correction of unknown phase-shifting error is an important task in general phase-shifting interferometry. Since phase-shifting errors generate twin image noise in a reconstructed image, we can reduce the phase-shifting errors indirectly by trying to eliminate the twin-image noise in the reconstructed image. By Zernike polynomial expansion, the reconstructed image is represented as the evenness and oddness, where the ratio of the evenness and oddness is a measure of the amount of the twin image noise. We employ the genetic algorithm for finding the fittest phase shifts of interferograms by reducing the evenness of the reconstructed image, which leads to reduction of phase-shifting errors. This phase-shifting interferometry with a genetic algorithm is confirmed experimentally.

Various authors have presented the aberration function of an optical system as a power series expansion with respect to the ray coordinates in the exit pupil and the coordinates of the intersection point with the image field of the optical system. In practical applications, for reasons of efficiency and accuracy, an expansion with the aid of orthogonal polynomials is preferred for which, since the 1980s, orthogonal Zernike polynomials have become the reference. In the literature, some conversion schemes of power series coefficients to coefficients for the corresponding Zernike polynomial expansion have been given. In this paper we present an analytic solution for the conversion problem from a power series expansion in three or four dimensions to a double Zernike polynomial expansion. The solution pertains to a general optical system with four independent pupil and field coordinates and to a system with rotational symmetry in which case three independent coordinate combinations have to be considered. The conversion of the coefficients is analytically in closed form and the result is independent of a specific sampling scheme or sampling density as this is the case for the commonly used least squares fitting techniques. Computation schemes are given that allow the evaluation of coefficients of arbitrarily high order in pupil and field coordinates.

To determine the theoretical relationships between the changes in corneal paraxial power, asphericity, and the corresponding Zernike polynomial expansion after conventional and customized excimer laser correction of hyperopia. The initial corneal profile was modeled as a conic section of apical radius of curvature R1 and asphericity Q1. The theoretical value of the postoperative apical radius of curvature R2 was computed by using a paraxial formula from the value of R1 and hyperopic defocus D. The postoperative asphericity Q2 of the corneal surface was computed within the optical zone of diameter S after the delivery of a Munnerlyn-based profile of ablation for hyperopia using conic section-fitting and minimization of the squared residuals. These calculations were repeated for different values of defocus, initial apical radius of curvature, and asphericity. Taylor series expansions were also used to provide an approximation aimed at predicting change in asphericity. The coefficients of a Zernike polynomial expansion of the rotationally symmetrical corneal profile (defocus C2(0), spherical aberration C4(0), secondary spherical aberration C6(0)) were also computed, by using scalar products applied to the considered corneal profile modeled as a conic section and were expressed as a function of both its apical radius and asphericity. This allowed approximation of the variations of the Zernike polynomial expansion of the corneal profiles by subtracting the postoperative coefficient weighting a particular aberration from that of the preoperative one in different theoretical situations, after both conventional and customized hyperopia treatments aimed at controlling the postoperative corneal asphericity and delivered over a normalized pupil diameter. Conical least-squares fitting was unambiguous, allowing approximation of the postoperative corneal profile as a conic section of apical radius R2. After a Munnerlyn-based hyperopia treatment, the sign of the asphericity of this profile remains theoretically unchanged, but its value decreased for initially oblate and increased for initially prolate corneas, respectively. A similar trend was noted with the approximation obtained by the Taylor series expansion. The alteration of the apical radius and/or of the asphericity of the corneal surface resulted in variations of both the corneal profile Zernike coefficients C2(0) and C4(0). The former was essentially dependent on the variation of the apical radius and the latter essentially on the variation of both apical radius and asphericity. Conventional and customized profiles of ablation for hyperopia alter the postoperative corneal asphericity and the Zernike coefficients of the corneal profile. The results of this study may be useful in the interpretation of the postoperative variations of the corneal profile and their impact on corneal wavefront expansion variations after both conventional and customized profiles of ablation.

A novel transformation free nonuniform higher order compact finite difference scheme for solving incompressible flows on circular geometries

The use of transform techniques can greatly enhance The task of modeling cash flows. Considerable work has been done since Crubbatrom's on the application of transform technique to cash-flow analysis. However, little attention is given to transform techniques by the engineering economics community at large, as evidenced by the lack of ita coverage In engineering economy textbooks. This paper represents the continual effort of the authors to suggest the use of transform techniques for engineering economy problems. A loan problem is formulated using the traditional method, the Zeta-transform method, and a difference equation approach using matrix methods. Simplified results are obtained using closed-form solutions from both the transform and difference equation methods. The loan problem is extended to an equipment financing situation, which is formulated using the Zeta-transform method and the traditional method as well. An illustrative example is included.

Cross-impact analysis relies on decision makers to provide marginal probability estimates of interdependent events. Generally, these have to be revised in order to ensure overall system coherency. This paper describes cross-impact analysis experimentation in which a Monte Carlo based approach and a difference equation approach, respectively, were used to revise these marginal probabilities. The objective of the study was to determine the consequences of such revisions on the expected impact rankings of these events. A cross-impact analysis system was developed and used to conduct the experiments. The experiments show that the impact ranking of interdependent events may indeed depend on the technique used for revising event marginal probabilities. Moreover, the Monte Carlo technique generates a world view closer to the one of the decision makers, while the world view generated by the difference equation technique differs from that of the decision makers.

The combination of High-order methods and Large-Eddy Simulation (LES) is an ongoing research focus in turbulence due to the attractive dissipation characteristics of high-order methods. Whilst numerically speaking these methodologies are advantageous, their application is inhibited on industrial cases due to the inherent geometric complexities of such problems. Spectral/hp Element (SEM) solvers such as Nektar++, have potential to be bridge the gap between high-order methods and industrial geometric complexity. This study focuses on the intersection of the application of the SEM solver Nektar++ to an automotive geometry as well as the presentation of high-order mean flow characteristics for the SAE Notchback body. Using a 5th order polynomial expansion at ReL = 2.3 × 106 on a curvilinear grid, results are compared with those empirically achieved in other works. Implicit Sub-Grid scale modelling along with a novel Spectral-Vanishing Viscosity (SVV) approach is employed acting as an artificial diffusion operator preventing high-frequency instabilities and spurious oscillations. Suitable qualitative agreement between PIV and CFD methods is obtained, and quantitative agreement is demonstrated on CD with 9% difference. More extensive backlight separation and subsequent bootlid impingement is observed in CFD than presented in the literature. This might be caused due to differing inflow characteristics, resulting in CM and CL variance to experimental values. Along with the mean flow field characteristics, the methodology and the pipeline used to achieve such results and agreement is presented. The use of a wall-conforming unstructured curvilinear grid allows for significantly greater geometric flexibility whilst retaining the advantages of the high-order polynomial expansion.

Numerous applied problems of two-dimensional (2-D) and 3-D imaging are formulated in continuous domain. They place great emphasis on obtaining and manipulating the Fourier transform in polar and spherical coordinates. However, the translation of continuum ideas with the discrete sampled data on a Cartesian grid is problematic. There exists no exact and fast solution to the problem of obtaining discrete Fourier transform for polar and spherical grids in the literature. In this paper, we develop exact algorithms to the above problem for 2-D and 3-D, which involve only 1-D equispaced fast Fourier transform with no interpolation or approximation at any stage. The result of the proposed approach leads to a fast solution with very high accuracy. We describe the computational procedure to obtain the solution in both 2-D and 3-D, which includes fast forward and inverse transforms. We find the nested multilevel matrix structure of the inverse process, and we propose a hybrid grid and use a preconditioned conjugate gradient method that exhibits a drastic improvement in the condition number.

To compare experimental optical performance in eyes implanted with spherical and aspheric intraocular lenses (IOLs). Corneal, total, and internal aberrations were measured in 19 eyes implanted with spherical (n=9) and aspheric (n=10) IOLs. Corneal aberrations were estimated by virtual ray tracing on corneal elevation maps, and total aberrations were measured using a second-generation laser ray tracing system. Corneal and total wave aberrations were fit to a Zernike polynomial expansion. Internal aberrations were measured by subtracting corneal from total wave aberrations. Optical performance was evaluated in terms of root-mean-square (RMS) wavefront error and Strehl ratio (estimated from the modulation transfer function). Depth-of-field was obtained from through-focus Strehl estimates from each individual eye. Corneal aberrations increased after IOL implantation, particularly astigmatism and trefoil terms. Third and higher order RMS (and the corresponding Strehl ratio) were significantly better in eyes with aspheric IOLs than with spherical IOLs; however, this tendency was reversed when astigmatism was included. Spherical aberration was not significantly different in eyes with aspheric IOLs, whereas it was significantly positive in eyes with spherical IOLs. Third order aberrations were not significantly different across groups. Depth-of-field was significantly larger in eyes with spherical IOLs. Spherical IOLs showed better absolute optical quality in the presence of negative defocus >1.00 D. Our study shows a good degree of compensation of the corneal spherical aberration in eyes implanted with aspheric IOLs, as opposed to eyes implanted with spherical IOLs. Other sources of optical degradation, both with aspheric and spherical IOLs, are non-symmetric preoperative corneal aberrations, incision-induced aberrations, and third order internal aberrations. Although best corrected optical quality is significantly better with aspheric IOLs, tolerance to defocus tended to be lower.

Context. With the contemporary development of topological optimization, and parametric and AI-guided design, the problem of implicit surface representation became prominent in additive manufacturing. Although more and more software packages use implicit modeling for design, there is no common standard way of writing, storing, or passing a set of implicit surfaces or curves over the network. The object of the study is one of the possible ways of such representation, specifically: modeling implicit curves and surfaces using pseudo-Gaussian interpolation.Objective. The goal of the work is the development of a modeling method that improved the accuracy of the implicit object representation wothout significant increase in memory used or processing time spent.Method. One of the conventional ways to model an implicit surface would be to represent its signed distance function (SDF) with its values defined on a regular grid. Then a continuous SDF could be obtained from the grid values by means of interpolation.What we propose instead is to store not SDF values but the coefficients of a pseudo-Gaussian interpolating function in the grid, which would enable picking the exact interpolation points before the SDF model is written. In this way we achieve better accuracy in the regions we’re interested the most in with no additional memory overhead.Results. The developed method was implemented in software for curves in 2D and validated against several primitive implicit curves of different nanture: circles, sqaures, rectangles with different parameters of the model. The method has shown improved accuaracy in general, but there were several classes of corner cases found for which it deserves further development.Conclusions. Pseudo-Gaussian interpolation defined as a sum of radial basis functions on a regular grid with points of interpolation defined in the proximity of the grid points generally allows to model an implicit surface more accurately than a voxel model interpolation does. The memory intake or computational toll isn’t much different in these two approaches. However, the interpolating points selection strategy and the choice of the best modeling parameters for each particular modeling problem remain an open quesition.

Let $\Phi (z) = \Sigma _0^\infty {\beta _j}{z^j}$ have radius of convergence $r\;(0 < r < \infty )$ and no singularities other than poles on the circle $|z| = r$. A complete solution is obtained for the infinite order differential equation $( \ast )\;\Sigma _0^\infty {\beta _j}{u^{(j)}}(z) = g(z)$. It is shown that $(\ast )$ possesses a solution if and only if the function g has a polynomial expansion in terms of the Appell polynomials generated by $\Phi$. The solutions of $( \ast )$ are expressed in terms of the coefficients which appear in the Appell polynomial expansions of g. An alternate method of solution is obtained, in which the problem of solving $( \ast )$ is reduced to the problem of finding a solution, within a certain space of entire functions, of a finite order linear differential equation with constant coefficients. Additionally, differential operator techniques are used to study Appell polynomial expansions.

The iterative behaviour of the general bilinear transformation is analysed via a difference equation approach. This reveals the following possible behaviours: convergence, divergence, exact cycling, or approximate cycling. Conditions are given under which each of these possibilities occur.

Efficient and accurate mathematical codes for the prediction of underwater sound propagation are a critical component of SONAR system development and operation. The goal of the research presented herein is to develop, implement, and verify an efficient and rigorous coupled-mode solution for acoustic wave propagation in shallow water range-dependent environments. A theoretical framework involving a range-expanded inner product for capturing the coupling between modes as they propagate through a horizontally-variable medium is presented. This framework includes a novel discretization of the range-dependent acoustic medium. A difference equations approach, which implements the inner product to account for non-adiabatic energy transfer between modes, is used to recursively compute reflection and transmission coefficients throughout the discretized environment. Increased efficiency is gained in the method due to the ability to compute coupling via closed-form algebraic expressions and in the application of asymptotic analysis to simplify the transmission and reflection coefficients. Results will be shown for a sample of waveguides with horizontally-variable bottom depth, along with comparable results from R. Evans’ COUPLE model and an FEM solution produced by ARL-UT. [This work was funded by the Naval Undersea Warfare Center, Newport Rhode Island.]

: Nonlinear models leading to Michaelis-Menten scheme result from the study compartment systems as well as in other contexts. Using difference equation approach, rate constants are estimated in some simple models. Simulation methods are used to derive the distributions of these estimates. It is shown that the assumptions of normality of errors leads to the normal distribution of the estimated rate constants. (Author)

Photonic crystal and split ring resonator (SRR) metamaterial waveguides with Kerr nonlinear dielectric impurities are studied. The transmission coefficients for two guided modes of different frequencies scattering from the Kerr impurities are computed. The systems are shown to exhibit multiple transmission coefficient solutions arising from the Kerr nonlinearity. Multiple transmission coefficients occur when different input intensities into a waveguide result in the same transmitted output intensities past its nonlinear impurities. (In the case of a single incident guided mode the multiplicity of transmission coefficients is known as optical bistability.) The analytical conditions under which the transmission coefficients are single and multiple valued are determined, and specific examples of both single and multiple valued transmission coefficient scattering are presented. Both photonic crystal and split ring resonator systems are studied as the Kerr nonlinearity enters the photonic crystal and SRR systems in different ways. This allows for an interesting comparison of the differences in behaviors of these two types of system which are described by distinctly different mathematical structures. Both the photonic crystal and SRR models used in the calculations are based on a difference equation approach to the system dynamics. The difference equation approach has been extensively employed in previous papers to model the basic properties of these systems. The paper is a continuation of work on the optical bistability of single guided modes interacting with Kerr impurities in photonic crystals originally considered by McGurn [Chaos 13, 754 (2003)] and work on the resonant scattering from Kerr impurities in photonic crystal waveguides considered by McGurn [J. Phys.: Condens. Matter 16, S5243 (2004)]. It generalizes this work making the extension to the more complex interaction of two guided modes at different frequencies. It extends the two guided mode treatment by McGurn [Organ. Electron. 8, 227 (2007)] which was limited to a special case of one of the photonic crystal systems considered here.