Optimal Quadrature Formulas for Non-periodic Functions in Sobolev Space and Its Application to CT Image Reconstruction

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In the present paper, optimal quadrature formulas in the sense of Sard are constructed for numerical integration of the integral integral(b)(a) e(2 pi i omega x)phi(x) dx with omega epsilon R in the Sobolev space L-2((m))[a, b] of complexvalued functions which are square integrable with m-th order derivative. Here, using the discrete analogue of the differential operator d(2m)/dx(2m), the explicit formulas for optimal coefficients are obtained. The order of convergence of the obtained optimal quadrature formula is O(h(m)). As an application, we implement the filtered back-projection (FBP) algorithm, which is a well-known image reconstruction algorithm for computed tomography (CT). By approximating Fourier transforms and its inversion using the proposed optimal quadrature formula of the second and third orders, we observe that the accuracy of the reconstruction algorithm is improved. In numerical experiments, we compare the quality of the reconstructed image obtained by using the proposed optimal quadrature formulas with the conventional FBP, in which fast Fourier transform is used for the calculation of Fourier transform and its inversion. In the noise test, the proposed algorithm provides more reliable results against the noise than the conventional FBP.
Publisher
UNIV NIS, FAC SCI MATH
Issue Date
2021-12
Language
English
Article Type
Article
Citation

FILOMAT, v.35, no.12, pp.4177 - 4195

ISSN
0354-5180
DOI
10.2298/FIL2112177H
URI
http://hdl.handle.net/10203/291858
Appears in Collection
MA-Journal Papers(저널논문)
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