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We are interested in the detection of jump discontinuities in piecewise smooth functions which are realized by their spectral data. Specifically, given the Fourier coefficients, { f ˆ k ϭ a k ϩ ib k } kϭ1 N , we form the generalized conjugate partial sum S ˜ N ͓ f ͔͑x͒ ϭ ¥ kϭ1 N ͩ k N ͪ ͑a k sin kx Ϫ b k cos kx͒. The classical conjugate partial sum, S ˜… (More)

- Anne Gelb, Eitan Tadmor, A Gelb, E Tadmor
- 1998

In this paper we construct, analyze and implement a new procedure for the spectral approximations of nonlinear conservation laws. It is well known that using spectral methods for nonlinear conservation laws will result in the formation of the Gibbs phenomenon once spontaneous shock discontinuities appear in the solution. These spurious oscillations will in… (More)

- Rick Archibald, Anne Gelb, Jungho Yoon
- SIAM J. Numerical Analysis
- 2005

We propose a new edge detection method that is effective on multivariate irregular data in any domain. The method is based on a local polynomial annihilation technique and can be characterized by its convergence to zero for any value away from discontinuities. The method is numerically cost efficient and entirely independent of any specific shape or… (More)

- Anne Gelb, Eitan Tadmor
- SIAM J. Numerical Analysis
- 2000

We discuss a general framework for recovering edges in piecewise smooth functions with finitely many jump discontinuities, where [f ](x) := f (x+) − f (x−) = 0. Our approach is based on two main aspects—localization using appropriate concentration kernels and separation of scales by nonlinear enhancement. To detect such edges, one employs concentration… (More)

The classical Gibbs phenomenon exhibited by global Fourier projections and interpolants can be resolved in smooth regions by reprojecting in a truncated Gegenbauer series, achieving high resolution recovery of the function up to the point of discontinuity. Unfortunately, due to the poor conditioning of the Gegenbauer polynomials, the method suffers both… (More)

- Rick Archibald, Anne Gelb
- IEEE Trans. Med. Imaging
- 2002

Gibbs ringing is a well known artifact that effects reconstruction of images having discontinuities. This is a problem in the reconstruction of magnetic resonance imaging (MRI) data due to the many different tissues normally present in each scan. The Gibbs ringing artifact manifests itself at the boundaries of the tissues, making it difficult to determine… (More)

- Anne Gelb
- Math. Comput.
- 1997

Spherical harmonics have been important tools for solving geo-physical and astrophysical problems. Methods have been developed to effectively implement spherical harmonic expansion approximations. However, the Gibbs phenomenon was already observed by Weyl for spherical harmonic expansion approximations to functions with discontinuities, causing undesirable… (More)

- Rick Archibald, Kewei Chen, Anne Gelb, Rosemary Renaut
- NeuroImage
- 2003

The Gegenbauer image reconstruction method, previously shown to improve the quality of magnetic resonance images, is utilized in this study as a segmentation preprocessing step. It is demonstrated that, for all simulated and real magnetic resonance images used in this study, the Gegenbauer reconstruction method improves the accuracy of segmentation.… (More)

- Anne Gelb
- J. Sci. Comput.
- 2004

- Anne Gelb, Zdzislaw Jackiewicz
- SIAM J. Scientific Computing
- 2005

The Gegenbauer reconstruction method effectively eliminates the Gibbs phenomenon and restores exponential accuracy to the approximations of piecewise smooth functions. Recent investigations show that its success depends upon choosing parameters in such a way that the reg-ularization and the truncation error estimates are equally considered. This paper shows… (More)