Geodesic
Optimal quadrature formulas for non-periodic functions in Sobolev space and its application to CT image reconstruction
Filomat, Tome 35 (2021) no. 12, p. 4177

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In the present paper, optimal quadrature formulas in the sense of Sard are constructed for numerical integration of the integral b a e 2πiωx ϕ(x) dx with ω ∈ R in the Sobolev space L (m) 2 [a, b] of complex-valued 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
DOI : 10.2298/FIL2112177H
Classification : 65D30, 65D32, 41A05, 41A15
Keywords: Optimal quadrature formula, square integrable function, error functional, Fourier transform, Radon transform, the filtered back-projection, CT image reconstruction 
A R Hayotov; S Jeon; C.-O Lee; Kh M Shadimetov. Optimal quadrature formulas for non-periodic functions in Sobolev space and its application to CT image reconstruction. Filomat, Tome 35 (2021) no. 12, p. 4177 . doi: 10.2298/FIL2112177H
@article{10_2298_FIL2112177H,
     author = {A R Hayotov and S Jeon and C.-O Lee and Kh M Shadimetov},
     title = {Optimal quadrature formulas for non-periodic functions in {Sobolev} space and its application to {CT} image reconstruction},
     journal = {Filomat},
     pages = {4177 },
     year = {2021},
     volume = {35},
     number = {12},
     doi = {10.2298/FIL2112177H},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.2298/FIL2112177H/}
}
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