Inverse Polynomial Reconstruction of Two Dimensional Fourier Images

The Gibbs phenomenon is intrinsic to the Fourier representation for discontinous problems. The inverse polynomial reconstruction method (IPRM) was proposed for the resolution of the Gibbs phenomenon in previous papers Shizgal, B. D., and Jung, J.-H. (2003) and Jung, J.-H., and Shizgal, B. D. (2004)] providing spectral convergence for one dimensional global and local reconstructions. The inverse method involves the expansion of the unknown function in polynomials such that the residue between the Fourier representations of the final representation and the unknown function is orthogonal to the Fourier or polynomial spaces. The main goal of this work is to show that the one dimensional inverse method can be applied successfully to reconstruct two dimensional Fourier images. The two dimensional reconstruction is implemented globally with high accuracy when the function is analytic inside the given domain. If the function is piecewise analytic and the local reconstruction is sought, the inverse method is applied slice by slice. That is, the one dimensional inverse method is applied to remove the Gibbs oscillations in one direction and then it is applied in the other direction to remove the remaining Gibbs oscillations. It is shown that the inverse method is exact if the two-dimensional function to be reconstructed is a piecewise polynomial. The two-dimensional Shepp–Logan phantom image of the human brain is used as a preliminary study of the inverse method for two dimensional Fourier image reconstruction. The image is reconstructed with high accuracy with the inverse method