Computation of 2D Fourier transforms and diffraction integrals using Gaussian radial basis functions

Martínez–Finkelshtein, Andrei; Ramos-López, Darío; Iskander, D. Robert

doi:10.1016/j.acha.2016.01.007

Cited by 4 publications

(9 citation statements)

References 44 publications

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“…In the modal form of the phase retrieval problem, also considered in [1] for extended Nijboer-Zernike (ENZ) basis functions, the GPF is assumed to be well approximated by a weighted sum of basis functions. We make use of real-valued radial basis functions [16] with complex coefficients to approximate the GPF. These are studied in the scope of wavefront estimation in [17] and an illustration of these basis function on a 4 × 4 grid in the pupil plane is given in Figure 1.…”

Section: Problem Formulation In Modal Formmentioning

confidence: 99%

Solving large-scale general phase retrieval problems via a sequence of convex relaxations

Doelman¹,

Thao²,

Verhaegen³

2018

J. Opt. Soc. Am. A

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We present a convex relaxation-based algorithm for large-scale general phase retrieval problems. General phase retrieval problems include, e.g., the estimation of the phase of the optical field in the pupil plane based on intensity measurements of a point source recorded in the image (focal) plane. The non-convex problem of finding the complex field that generates the correct intensity is reformulated into a rank constraint problem. The nuclear norm is used to obtain the convex relaxation of the phase retrieval problem. A new iterative method referred to as convex optimization-based phase retrieval (COPR) is presented, with each iteration consisting of solving a convex problem. In the noise-free case and for a class of phase retrieval problems, the solutions of the minimization problems converge linearly or faster towards a correct solution. Since the solutions to nuclear norm minimization problems can be computed using semidefinite programming, and this tends to be an expensive optimization in terms of scalability, we provide a fast algorithm called alternating direction method of multipliers (ADMM) that exploits the problem structure. The performance of the COPR algorithm is demonstrated in a realistic numerical simulation study, demonstrating its improvements in reliability and speed with respect to state-of-the-art methods.

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Section: Problem Formulation In Modal Formmentioning

confidence: 99%

Solving large-scale general phase retrieval problems via a sequence of convex relaxations

Doelman¹,

Thao²,

Verhaegen³

2018

J. Opt. Soc. Am. A

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“…Alternatively, the pupil function can be approximated by a linear combination of GRBFs [13]. The complex GPF is approximated by a real-valued, radially symmetric GRBF,…”

Section: B Gaussian Radial Basis Functionsmentioning

confidence: 99%

“…If they would be chosen independent from each other, they would introduce 2N α hyper-parameters to the PR problem. Estimating them is a highly non-linear problem usually solved by cross-validation [13]. In this section, by using their physical interpretation in the imaging application, we propose to reduce the number of parameters to one single shape parameter λ and a predefined node distribution.…”

Section: Fitting Accuracy Of Grbfsmentioning

confidence: 99%

“…2. All grids are defined over an area slightly larger than the unit disk (a disk with radius 1.05) to deal with the Gibbs phenomenon [13].…”

Section: A Node Distributionmentioning

confidence: 99%

“…Recently, GPF approximation based on Gaussian radial basis functions (GRBFs) was used for semi-analytic evaluation of the diffraction integral as an alternative to ENZ polynomials. An improvement in terms of complexity, accuracy, and execution time was achieved [13]. An important feature of the GRBF is the almost local character of each function.…”

Section: Introductionmentioning

confidence: 96%

See 2 more Smart Citations

Modal-based phase retrieval using Gaussian radial basis functions

2018

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In this paper, we propose the use of Gaussian radial basis functions (GRBFs) to model the generalized pupil function for phase retrieval. The selection of the GRBF hyper-parameters is analyzed to achieve an increased accuracy of approximation. The performance of the GRBF-based method is compared in a simulation study with another modal-based approach considering extended Nijboer-Zernike (ENZ) polynomials. The almost local character of the GRBFs makes them a much more flexible basis with respect to the pupil geometry. It has been shown that for aberrations containing higher spatial frequencies, the GRBFs outperform ENZ polynomials significantly, even on a circular pupil. Moreover, the flexibility has been demonstrated by considering the phase retrieval problem on an annular pupil.

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Enhancing thickness determination of nanoscale dielectric films in phase diffraction-based optical characterization systems with radial basis function neural networks

Ataç

Karatay

Dinleyici

2023

Meas. Sci. Technol.

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Accurate determination of the optical properties of ultra-thin dielectric films is an essential and challenging task in optical fiber sensor systems. However, nanoscale thickness identification of these films may be laborious due to insufficient and protracted classical curve matching algorithms. Therefore, this experimental study presents an application of a radial basis function (RBF) neural network in phase diffraction-based optical characterization systems to determine the thickness of nanoscale polymer films. The non-stationary measurement data with environmental and detector noise were subjected to a detailed analysis. The outcomes of this investigation are benchmarked against the linear discriminant analysis (LDA) method and further verified by means of scanning electron microscopy (SEM). The results show that the neural network has reached a remarkable accuracy of 98% and 82.5%, respectively, in tests with simulation and experimental data. In this way, rapid and precise thickness estimation may be realized within the tolerance range of 25 nm, offering a significant improvement over conventional measurement techniques.

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Computation of 2D Fourier transforms and diffraction integrals using Gaussian radial basis functions

Cited by 4 publications

References 44 publications

Solving large-scale general phase retrieval problems via a sequence of convex relaxations

Solving large-scale general phase retrieval problems via a sequence of convex relaxations

Modal-based phase retrieval using Gaussian radial basis functions

Enhancing thickness determination of nanoscale dielectric films in phase diffraction-based optical characterization systems with radial basis function neural networks

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