Application of optimal quadrature formulas for reconstruction of CT images

In the present paper, the construction process of the optimal quadrature formulas for weighted integrals is presented in the Sobolev space L̃2(m)(0,1] of complex-valued periodic functions which are square integrable with mth order derivative. In particular, optimal quadrature formulas are given for Fourier coefficients. Here, using these optimal quadrature formulas the approximation formulas for Fourier integrals ∫abe2πiωxf(x)dx with ω∈R are obtained. In the cases m=1,2 and 3, the obtained approximation formulas are applied for reconstruction of Computed Tomography (CT) images coming from the filtered back-projection method. Compared with the optimal quadrature formulas in non-periodic case, the approximation formula for the periodic case is much simpler, therefore it is easy to implement and costs less computation.