Digital image processing (2nd ed.)
Digital image processing (2nd ed.)
Journal of Computational and Applied Mathematics - Orthogonal polynomials and numerical methods
Signals & systems (2nd ed.)
Matrix computations (3rd ed.)
On the Gibbs Phenomenon and Its Resolution
SIAM Review
Spectral methods for hyperbolic problems
Journal of Computational and Applied Mathematics - Special issue on numerical analysis 2000 Vol. VII: partial differential equations
Numerical harmonic analysis and image processing
Digital image analysis
Digital Image Processing: A Systems Approach
Digital Image Processing: A Systems Approach
Digital Image Processing Techniques
Digital Image Processing Techniques
Digital Image Processing of Remotely Sensed Data
Digital Image Processing of Remotely Sensed Data
Journal of Computational and Applied Mathematics
Digital Image Processing
Reducing the Effects of Noise in Image Reconstruction
Journal of Scientific Computing
Towards the resolution of the Gibbs phenomena
Journal of Computational and Applied Mathematics
Journal of Computational and Applied Mathematics
A note on the Gibbs phenomenon with multiquadric radial basis functions
Applied Numerical Mathematics
A wavelet-based method for overcoming the Gibbs phenomenon
MATH'08 Proceedings of the American Conference on Applied Mathematics
Journal of Computational Physics
Edge Detection Free Postprocessing for Pseudospectral Approximations
Journal of Scientific Computing
Algorithm 899: The Matlab postprocessing toolkit
ACM Transactions on Mathematical Software (TOMS)
Pseudospectral Fourier reconstruction with the modified Inverse Polynomial Reconstruction Method
Journal of Computational Physics
Two-Dimensional gibbs phenomenon for fractional fourier series and its resolution
AICI'12 Proceedings of the 4th international conference on Artificial Intelligence and Computational Intelligence
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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