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In the present paper, optimal quadrature formulas in the sense of Sard areconstructed for numerical integration of the integral $\int_a^be^{2\pi i\omegax}\varphi(x)d x$ with $\omega\in \mathbb{R}$ in the Sobolev space$L_2^{(m)}[a,b]$ of complex-valued functions which are square integrable with$m$-th order derivative. Here, using the discrete analogue of the differentialoperator $\frac{d^{2m}}{d x^{2m}}$, the explicit formulas for optimalcoefficients are obtained. The order of convergence of the obtained optimalquadrature formula is $O(h^m)$. As an application, we implement the filteredback-projection (FBP) algorithm, which is a well-known image reconstructionalgorithm for computed tomography (CT). By approximating Fourier transforms andits inversion using the proposed optimal quadrature formula of the second andthird orders, we observe that the accuracy of the reconstruction algorithm isimproved. In numerical experiments, we compare the quality of the reconstructedimage obtained by using the proposed optimal quadrature formulas with theconventional FBP, in which fast Fourier transform is used for the calculationof Fourier transform and its inversion. In the noise test, the proposedalgorithm provides more reliable results against the noise than theconventional FBP.

Optimal quadrature formulas for non-periodic functions in Sobolev space and its application to CT image reconstruction