@article {1144,
	title = {Inverse polynomial reconstruction of two dimensional Fourier images},
	journal = {Journal of Scientific Computing},
	volume = {25},
	number = {3},
	year = {2005},
	note = {ISI Document Delivery No.: 992GPTimes Cited: 10Cited Reference Count: 31},
	month = {Dec},
	pages = {367-399},
	type = {Article},
	abstract = {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},
	keywords = {Fourier approximation, Gibbs phenomenon, inverse polynomial, reconstruction method, RESOLUTION, Shepp-Logan phantom image, two-dimensional image reconstruction},
	isbn = {0885-7474},
	url = {<Go to ISI>://000233872700001},
	author = {Jung, J. H. and Shizgal, B. D.}
}
@article {900,
	title = {Generalization of the inverse polynomial reconstruction method in the resolution of the Gibbs phenomenon},
	journal = {Journal of Computational and Applied Mathematics},
	volume = {172},
	number = {1},
	year = {2004},
	note = {ISI Document Delivery No.: 856YKTimes Cited: 20Cited Reference Count: 17},
	month = {Nov},
	pages = {131-151},
	type = {Article},
	abstract = {The finite Fourier representation of a function f(x) exhibits oscillations where the function or its derivatives are nonsmooth. This is known as the Gibbs phenomenon. A robust and accurate reconstruction method that resolves the Gibbs oscillations was proposed in a previous paper (J. Comput. Appl. Math. 161 (2003) 41) based on the inversion of the transformation matrix which represents the projection of a set of basis functions onto the Fourier space. If the function is a polynomial, this inverse polynomial reconstruction method (IPRM) is exact. In this paper, we develop the IPRM by requiring that the proper error be orthogonal to the Fourier or polynomial space. The IPRM is generalized to any set of basis functions. The primitive basis polynomials, nonclassical orthogonal polynomials and the Gegenbauer polynomials are used to illustrate the wide validity of the IPRM. It is shown that the IPRM yields a unique reconstruction irrespective of the basis set for any analytic function and yields spectral convergence. The ill-posedness of the transformation matrix due to the exponential growth of the condition number of the matrix is also discussed. (C) 2004 Elsevier B.V. All rights reserved.},
	keywords = {FOURIER, Fourier approximation, Gibbs phenomenon, Ill-posedness, inverse polynomial, reconstruction method, spectral accuracy},
	isbn = {0377-0427},
	url = {<Go to ISI>://000224081800009},
	author = {Jung, J. H. and Shizgal, B. D.}
}
