Optimal quadrature formulas for CT image reconstruction in the Sobolev space of non-periodic functions

In the present paper, optimal quadrature formulas in the sense of Sard are constructed for numerical integration of the integral $\int_a^b e^{2\pi i\omega x}\varphi(x)dx$ 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 differential operator $\frac{d^{2m}}{d x^{2m}}$, the explicit formulas for optimal coefficients are obtained. The order of convergence of the obtained optimal quadrature formula is $O(h^m)$. The optimal quadrature formulas of the cases $m=2$ and $m=3$ are applied for reconstruction of Computed Tomography (CT) images by approximating Fourier transforms in the filtered back-projection formula. In numerical experiments two kinds of phantoms are considered and numerical results are compared with the results of a built-in function of MATLAB 2019a, iradon. Numerical results show that the quality of the reconstructed images with optimal quadrature formula of the second and third orders is better than that of the results obtained by iradon.