2020/01/07 by A.R. Hayotov, Soomin Jeon, Hayotov, Abdullo R. +5
Mathematics · Medicine · Physics and Astronomy · #41A05 #41A15 #65D30 #65D32 #FOS: Mathematics #Medical Imaging Techniques and Applications #Nuclear Physics and Applications #Numerical Analysis (math.NA) #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.2001.02636
openalex publication_date 2020/01/07 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28
In the present paper, optimal quadrature formulas in the sense of Sard are constructed for numerical integration of the integral ∫abe2πiωxφ(x)d x with ω∈ ℝ in the Sobolev space L2(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 \fracd2md x2m, the explicit formulas for optimal coefficients are obtained. The order of convergence of the obtained optimal quadrature formula is O(hm). 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.