Robust reprojection methods for the resolution of the Gibbs phenomenon

Abstract

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 from numerical round-off error and the Runge phenomenon. In some cases the method fails to converge. Following the work in [D. Gottlieb, C.W. Shu, Atti Conv. Lincei 147 (1998) 39-48], a more general framework for reprojection methods is introduced here. From this insight we propose an additional requirement on the reprojection basis which ameliorates the limitations of the Gegenbauer reconstruction. The new robust Gibbs complementary basis yields a reliable exponentially accurate resolution of the Gibbs phenomenon up to the discontinuities.  2005 Elsevier Inc. All rights reserved.MSC: 41A25; 42A10; 42A16; 42A20; 42C25; 65B10; 65T40

Recovering Exponential Accuracy from Non-harmonic Fourier Data Through Spectral Reprojection

“…That is, the projection of the delta function onto the space spanned by a finite number of frame functions is related to the projection onto the space spanned by the same number of Fourier basis functions. This is particularly evident in the jittered sampling case, (12), seen in Figure 6. Because it is this projection that indicates how finite-dimensional Fourier approximations respond to jump discontinuities, we expect the behavior of non-uniform approximations near jump discontinuities to behave similarly to that of the uniform case.…”

“…Figures 12, 13, and 14 demonstrate the results of the Gegenbauer reconstruction method for frames, (24), when applied to f (x) = x, using log-spaced (11) and jittered sampling schemes, (12) and (13), respectively. In all examples, we choose the parameters µ and M in (24) to minimize the L ∞ error.…”

“…Suppose that we are given the Fourier coefficients at jittered points defined by (12). Figure 5 demonstrates that there is a Gibbs phenomenon type of behavior present in non-uniform (frame) reconstructions for piecewise-analytic functions.…”

“…However, this case is still of high interest, since it is the 1D analogue of a spiral, which is a commonly-studied MRI sampling scheme. Jittered sampling, (12) and (13), reflects the common situation when sensing equipment may err in collecting uniform samples. Figure 3 compares sampling schemes (11) and (13) to the uniform case.…”

Edge Detection from Non-Uniform Fourier Data Using the Convolutional Gridding Algorithm

“…It was used effectively in [14,20] as a robust inner product weight for spectral reprojection. Unfortunately, there is no explicit formula for φ (ω).…”