Gabor's signal expansion and the Gabor transform based on a non-orthogonal sampling geometry

Bastiaans, M.J.; Leest, A.J. van

doi:10.1109/isspa.2001.949801

Cited by 4 publications

(3 citation statements)

References 3 publications

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“…We first write the field on the curved line as a sum of modulated and shifted Gaussian window functions and then find the coefficients of each Gaussian elementary signal according to Eq. (26). Afterward, we calculate the samples of the diffraction field on the observation line at z 0.1 m via Eq.…”

Section: Simulation Resultsmentioning

confidence: 99%

“…(22) is extremely redundant. By choosing the discrete set of evaluation points on S and shift steps in frequency properly, we can still decompose the field on S into a discrete set of modulated Gaussian window functions on the tangent planes of S. That is, it is possible to write ux as a sum of Gaussian beams that correspond to Gaussian window functions at discrete positions fr 1 ; r 2 ; …; r n g on S and having discrete modulation frequencies as multiples of F x and F y , as If the discrete evaluation points are taken on a regular grid, then, similar to the planar surface case, an analysis window function w r x r ; y r , which is valid for all r i ∈ fr 1 ; r 2 ; …; r n g, can be found [26]. However, because it is usually not possible to place a regular grid on the entire surface, the surface can be partitioned into small patches and each patch can be treated separately.…”

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confidence: 99%

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Scalar diffraction field calculation from curved surfaces via Gaussian beam decomposition

Şahin

Onural

2012

J. Opt. Soc. Am. A

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We introduce a local signal decomposition method for the analysis of three-dimensional (3D) diffraction fields involving curved surfaces. We decompose a given field on a two-dimensional curved surface into a sum of properly shifted and modulated Gaussian-shaped elementary signals. Then we write the 3D diffraction field as a sum of Gaussian beams, each of which corresponds to a modulated Gaussian window function on the curved surface. The Gaussian beams are propagated according to a derived approximate expression that is based on the RayleighSommerfeld diffraction model. We assume that the given curved surface is smooth enough that the Gaussian window functions on it can be treated as written on planar patches. For the surfaces that satisfy this assumption, the simulation results show that the proposed method produces quite accurate 3D field solutions.

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Section: Simulation Resultsmentioning

confidence: 99%

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confidence: 99%

Scalar diffraction field calculation from curved surfaces via Gaussian beam decomposition

Şahin

Onural

2012

J. Opt. Soc. Am. A

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“…Namely, the reference works our study is based upon concern symplectic geometry, 10 metaplectic group, 11 Gabor Frames and their relations to operators 2,3,9,12 and Gabor synthesis on non-separable lattices . 1,16 Note that since Linear Canonical Transformations constitute the subset of Canonical Transformations that model the paraxial optics, the illustrative virtue of this application example has its limitations; in practice it is likely that the some optical elements within the headsets will not suit to paraxial optics, because of the form factor that constraints the spacing between the elements. Also, the rotation of the headset is not a linear transformation, and its paraxial approximation is only relevant when the rotation angle is small.…”

Section: Motivation and Related Workmentioning

confidence: 99%

Linear canonical transformations in phase space: the Gabor frames approach

Gioia,

Gilles,

Lagrange

et al. 2024

Optics, Photonics, and Digital Technologies for Imaging Applications VIII

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Linear Canonical Transformations are phase space deformations that can be used to describe linear optical systems. In geometric optics they correspond to the transformations of the light rays during propagation in arbitrary media, while in wave optics they describe the evolution of the space-frequency components. Although their meaning is non-ambiguous in the former case, in the latter it depends on the specific space-frequency decomposition of the wavefield. In this paper we gather several results from theoretical time-frequency analysis with Gabor Frames and provide a preliminary analysis on the implementation of phase space deformation for linear optical systems.

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Efficient Algorithms for Discrete Gabor Transforms on a Nonseparable Lattice

Wiesmeyr¹,

Holighaus²,

Søndergaard³

2013

IEEE Trans. Signal Process.

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The Discrete Gabor Transform (DGT) is the most commonly used transform for signal analysis and synthesis using a linear frequency scale. It turns out that the involved operators are rich in structure if one samples the discrete phase space on a subgroup. Most of the literature focuses on separable subgroups, in this paper we will survey existing methods for a generalization to arbitrary groups, as well as present an improvement on existing methods. Comparisons are made with respect to the computational complexity, and the running time of optimized implementations in the C programming language. The new algorithms have the lowest known computational complexity for nonseparable lattices and the implementations are freely available for download. By summarizing general background information on the state of the art, this article can also be seen as a research survey, sharing with the readers experience in the numerical work in Gabor analysis.

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Gabor's signal expansion and the Gabor transform based on a non-orthogonal sampling geometry

Cited by 4 publications

References 3 publications

Scalar diffraction field calculation from curved surfaces via Gaussian beam decomposition

Scalar diffraction field calculation from curved surfaces via Gaussian beam decomposition

Linear canonical transformations in phase space: the Gabor frames approach

Efficient Algorithms for Discrete Gabor Transforms on a Nonseparable Lattice

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