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

In the present paper, optimal quadrature formulas in sense of Sard are constructed for numerical integration integral ?ba e2?i?x?(x)dx with ? R Sobolev space L(m)2 [a,b] complexvalued functions which square integrable m-th order derivative. Here, using discrete analogue differential operator d2m/dx2m, explicit coefficients obtained. The convergence obtained formula is O(hm). As an application, we implement filtered back-projection (FBP) algorithm, a well-known image reconstruction algorithm computed tomography (CT). By approximating Fourier transforms and its inversion proposed second third orders , observe that accuracy improved. experiments, compare quality reconstructed by conventional FBP, fast transform used calculation inversion. noise test, provides more reliable results against than FBP.