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

2005 ◽  
Author(s):  
Anne Gelb ◽  
Eitan Tadmor
Keyword(s):  
Piecewise Smooth ◽  
Edge Detectors ◽  
Smooth Data

scholarly journals Local edge detectors using a sigmoidal transformation for piecewise smooth data

2013 ◽  
Vol 26 (2) ◽  
pp. 270-276
Author(s):  
Beong In Yun ◽  
Kyung Soo Rim
Keyword(s):  
Piecewise Smooth ◽  
Edge Detectors ◽  
Local Edge ◽  
Smooth Data

Filters, mollifiers and the computation of the Gibbs phenomenon

Acta Numerica ◽  
2007 ◽  
Vol 16 ◽  
pp. 305-378 ◽  
Author(s):  
Eitan Tadmor
Keyword(s):  
Physical Space ◽  
Gibbs Phenomenon ◽  
Delta Function ◽  
High Accuracy ◽  
Piecewise Smooth ◽  
Spectral Information ◽  
Smooth Functions ◽  
Exponential Accuracy ◽  
Local Loss ◽  
Edge Detectors

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 functions, 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 parametrized to recover f with (root-) exponential accuracy.

Approximation properties of repeated de la Vallée-Poussin means for piecewise smooth functions

2017 ◽  
Vol 60 (3) ◽  
pp. 695-713
Author(s):  
I. I. Sharapudinov ◽  
T. I. Sharapudinov ◽  
M. G. Magomed-Kasumov
Keyword(s):  
Piecewise Smooth ◽  
Approximation Properties ◽  
Smooth Functions ◽  
Piecewise Smooth Functions

scholarly journals Discontinuity-induced bifurcations in a piecewise-smooth capsule system with bidirectional drifts

2021 ◽  
pp. 105909
Author(s):  
Bingyong Guo ◽  
Joseph Páez Chávez ◽  
Yang Liu ◽  
Caishan Liu
Keyword(s):  
Piecewise Smooth

Local and global bifurcations in 3D piecewise smooth discontinuous maps

2021 ◽  
Vol 31 (1) ◽  
pp. 013126
Author(s):  
Mahashweta Patra ◽  
Sayan Gupta ◽  
Soumitro Banerjee
Keyword(s):  
Piecewise Smooth ◽  
Global Bifurcations ◽  
Local And Global Bifurcations

scholarly journals Singularly Perturbed Oscillators with Exponential Nonlinearities

2021 ◽  
Author(s):  
S. Jelbart ◽  
K. U. Kristiansen ◽  
P. Szmolyan ◽  
M. Wechselberger
Keyword(s):  
Limit Cycles ◽  
Space Dimension ◽  
Blow Up ◽  
Exponential Convergence ◽  
Model System ◽  
Piecewise Smooth ◽  
Singularly Perturbed ◽  
Model Systems ◽  
Smooth System ◽  
Unique Limit

AbstractSingular exponential nonlinearities of the form $$e^{h(x)\epsilon ^{-1}}$$ e h ( x ) ϵ - 1 with $$\epsilon >0$$ ϵ > 0 small occur in many different applications. These terms have essential singularities for $$\epsilon =0$$ ϵ = 0 leading to very different behaviour depending on the sign of h. In this paper, we consider two prototypical singularly perturbed oscillators with such exponential nonlinearities. We apply a suitable normalization for both systems such that the $$\epsilon \rightarrow 0$$ ϵ → 0 limit is a piecewise smooth system. The convergence to this nonsmooth system is exponential due to the nonlinearities we study. By working on the two model systems we use a blow-up approach to demonstrate that this exponential convergence can be harmless in some cases while in other scenarios it can lead to further degeneracies. For our second model system, we deal with such degeneracies due to exponentially small terms by extending the space dimension, following the approach in Kristiansen (Nonlinearity 30(5): 2138–2184, 2017), and prove—for both systems—existence of (unique) limit cycles by perturbing away from singular cycles having desirable hyperbolicity properties.

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