Adaptive Edge Detectors for Piecewise Smooth Data Based on the Minmod Limiter

Gelb, Anne; Tadmor, Eitan

doi:10.21236/ada448179

Cited by 4 publications

(3 citation statements)

References 7 publications

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“…We discuss an improved enhancement procedure based on the nonlinear limiting of low-and high-order concentration factors. The rationale, outlined in Gelb and Tadmor (2006), is as follows.…”

Section: Nonlinear Limiter: Minmod Edge Detectionmentioning

confidence: 99%

Filters, mollifiers and the computation of the Gibbs phenomenon

Tadmor¹

2007

Acta Numerica

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We are concerned here with processing discontinuous functions from their spectral information. We focus on two main aspects of processing such piecewise smooth data: detecting the edges of a piecewise smooth f, namely, the location and amplitudes of its discontinuities; and recovering with high accuracy the underlying function in between those edges. If f is a smooth function, say analytic, then classical Fourier projections recover f with exponential accuracy. However, if f contains one or more discontinuities, its global Fourier projections produce spurious Gibbs oscillations which spread throughout the smooth regions, enforcing local loss of resolution and global loss of accuracy. Our aim in the computation of the Gibbs phenomenon, is to detect edges and to reconstruct piecewise smooth f's, while regaining the high accuracy encoded in the spectral data.To detect edges, we utilize a general family of edge detectors based on concentration kernels. Each kernel forms an approximate derivative of the delta function, which detects edges by separation of scales. We show how such kernels can be adapted to detect edges with one-and two-dimensional discrete data, with noisy data, and with incomplete spectral information. The main feature is concentration kernels which enable us to convert global spectral moments into local information in physical space. To reconstruct f with high accuracy we discuss novel families of mollifiers and filters. The main feature here is making these mollifiers and filters adapted to the local region of smoothness while increasing their accuracy together with the dimension of the data. These mollifiers and filters form approximate delta functions which are properly parameterized to recover f with (root-) exponential accuracy. 306E. Tadmor

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“…We discuss an improved enhancement procedure based on the nonlinear limiting of low-and high-order concentration factors. The rationale, outlined in Gelb and Tadmor (2006), is as follows.…”

Section: Nonlinear Limiter: Minmod Edge Detectionmentioning

confidence: 99%

Filters, mollifiers and the computation of the Gibbs phenomenon

Tadmor¹

2007

Acta Numerica

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“…Early methods are based on the Markov models [18], [28], [25]. Currently, there are several approaches: applications of the Fourier transformation [5], [12], [4], [23], [17], [16], [11], wavelets [35], [45], Chebyshev series [31], triangulations [19]. Geometric measures of the curvature are described in [38], [22], [34], [1], [30], [36], the statistical-based methods in [8], [26].…”

Section: Singularity Detectionmentioning

confidence: 99%

Efficient plotting the functions with discontinuities based on combined sampling

Bayer¹

2018

Geoinformatics FCE CTU

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This article presents a new algorithm for interval plotting of the function y = f(x) based on combined sampling. The proposed method synthesizes the uniform and adaptive sampling approaches and provides a more compact and efficient function representation. During the combined sampling, the polygonal approximation with a given threshold α between the adjacent segments is constructed. The automated detection and treatment of the discontinuities based on the LR criterion are involved. Two implementations, the recursive-based and stack-based, are introduced. Finally, several tests of the proposed algorithms for the different functions involving the discontinuities and several map projection graticules are presented. The proposed method may be used for more efficient sampling the curves (map projection graticules, contour lines, or buffers) in geoinformatics.

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“…A sequence of GSD calculations was performed for both gas-dynamic and MHD shocks over a range of . Shock-shocks are detected as discontinuities in M by an edge-detection algorithm (Gelb & Tadmor 2000, 2006 with a minmod limiter giving a switch between an exponential concentration factor kernel (high order) away from shock-shocks and a first-order polynomial kernel close to shock-shocks, limiting spurious oscillations in the detector. A nonlinear enhancement technique was used to remove small jumps on the grid scale in the discretized solution while allowing O(M r ) jumps, which correspond to shock-shocks, using a tuning parameter chosen to reflect the expected minimum magnitude of the shock-shock when it forms.…”

Section: Stability Of Plane Fast Mhd Shocksmentioning

confidence: 99%

Geometrical shock dynamics for magnetohydrodynamic fast shocks

et al. 2016

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We describe a formulation of two-dimensional geometrical shock dynamics (GSD) suitable for ideal magnetohydrodynamic (MHD) fast shocks under magnetic fields of general strength and orientation. The resulting area–Mach-number–shock-angle relation is then incorporated into a numerical method using pseudospectral differentiation. The MHD-GSD model is verified by comparison with results from nonlinear finite-volume solution of the complete ideal MHD equations applied to a shock implosion flow in the presence of an oblique and spatially varying magnetic field ahead of the shock. Results from application of the MHD-GSD equations to the stability of fast MHD shocks in two dimensions are presented. It is shown that the time to formation of triple points for both perturbed MHD and gas-dynamic shocks increases as $\unicode[STIX]{x1D716}^{-1}$, where $\unicode[STIX]{x1D716}$ is a measure of the initial Mach-number perturbation. Symmetry breaking in the MHD case is demonstrated. In cylindrical converging geometry, in the presence of an azimuthal field produced by a line current, the MHD shock behaves in the mean as in Pullin et al. (Phys. Fluids, vol. 26, 2014, 097103), but suffers a greater relative pressure fluctuation along the shock than the gas-dynamic shock.

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Adaptive Edge Detectors for Piecewise Smooth Data Based on the Minmod Limiter

Cited by 4 publications

References 7 publications

Filters, mollifiers and the computation of the Gibbs phenomenon

Filters, mollifiers and the computation of the Gibbs phenomenon

Efficient plotting the functions with discontinuities based on combined sampling

Geometrical shock dynamics for magnetohydrodynamic fast shocks

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