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

Abstract

In the present paper, optimal quadrature formulas in the sense of Sard are constructed for numerical integration of the integral ?ba e2?i?x?(x)dx with ? ? R in the Sobolev space L(m)2 [a,b] of complexvalued functions which are square integrable with m-th order derivative. Here, using the discrete analogue of the differential operator d2m/dx2m, 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.

An Algorithm for Construction of Optimal Integration Formulas in Hilbert Spaces and Its Realization

“…It should be noted that the similar algorithm was applied for construction of the optimal numerical integration formulas for the Fourier coefficients, Fourier integrals and they used for reconstruction of Computed Tomography images in the works [3,4,[12][13][14].…”

Оптимальные квадратурные формулы в пространстве W2​​(m,m−1) периодических функций

“…Recently, in the works [6,7,8,25] authors constructed optimal quadrature formulas for numerical calculation of Fourier integrals in the Sobolev space L , and these formulas were applied to approximate reconstruction of Computed Tomography images. Compared with the optimal quadrature formulas in non-periodic case constructed in [6], the approximation formula for the periodic case constructed in the works [7,17,18] is much simpler, therefore it is easy to implement and it costs less computation even though both provide similar performances.…”