Reconstruction of a discontinuous function from a few fourier coefficients using bayesian estimation

Alex Solomonoff

doi:10.1007/bf02087960

Reconstruction of a discontinuous function from a few fourier coefficients using bayesian estimation

Journal of Scientific Computing ◽

10.1007/bf02087960 ◽

1995 ◽

Vol 10 (1) ◽

pp. 29-80 ◽

Cited By ~ 8

Author(s):

Alex Solomonoff

Keyword(s):

Bayesian Estimation ◽

Fourier Coefficients ◽

Discontinuous Function

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Analysis of the results of a computational experiment to restore the discontinuous functions of two variables using projections

Physico-mathematical modelling and informational technologies ◽

10.15407/fmmit2021.33.012 ◽

2021 ◽

pp. 12-17

Author(s):

Oleg Lytvyn ◽

Oleksandra Lytvyn

Keyword(s):

Differentiable Function ◽

Fourier Coefficients ◽

Gibbs Phenomenon ◽

Discontinuous Function ◽

Projection Data ◽

Approximation Accuracy ◽

Discontinuous Functions ◽

Functions Of Two Variables ◽

Fourier Sums ◽

The Difference

This article presents the main statements of the method of approximation of discontinuous functions of two variables, describing an image of the surface of a 2D body or an image of the internal structure of a 3D body in a certain plane, using projections that come from a computer tomograph. The method is based on the use of discontinuous splines of two variables and finite Fourier sums, in which the Fourier coefficients are found using projection data. The method is based on the following idea: an approximated discontinuous function is replaced by the sum of two functions – a discontinuous spline and a continuous or differentiable function. A method is proposed for constructing a spline function, which has on the indicated lines the same discontinuities of the first kind as the approximated discontinuous function, and a method for finding the Fourier coefficients of the indicated continuous or differentiable function. That is, the difference between the function being approximated and the specified discontinuous spline is a function that can be approximated by finite Fourier sums without the Gibbs phenomenon. In the numerical experiment, it was assumed that the approximated function has discontinuities of the first kind on a given system of circles and ellipses nested into each other. The analysis of the calculation results showed their correspondence to the theoretical statements of the work. The proposed method makes it possible to obtain a given approximation accuracy with a smaller number of projections, that is, with less irradiation.

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On the Fourier coefficients of a discontinuous function

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500027206 ◽

1941 ◽

Vol 6 (4) ◽

pp. 231-256 ◽

Cited By ~ 1

Author(s):

S. P. Bhatnagar

Keyword(s):

Fourier Series ◽

Fourier Coefficients ◽

Discontinuous Function ◽

Image Position

We suppose throughout that f(t) is periodic with period 2π, and Lebesgue-integrable in (− π, π).We writeand suppose that the Fourier series of φ(t) and ψ(t) are respectively cos nt and sin nt. Then the Fourier series and allied series of f(t) at the point t = x are respectively and , where A0 = ½a0, An = ancos nx + bnsin nx, Bn = bncos nx − ansin nx and an, bn are the Fourier coefficients of f(t).

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