doi: 10.3934/mfc.2021038
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Analysis of directional higher order jump discontinuities with trigonometric shearlets

Institute of Mathematics, University of Lübeck, Ratzeburger Allee 160, D-23562 Lübeck, Germany

*Corresponding author: Kevin Schober

Received  August 2021 Revised  October 2021 Early access December 2021

In a recent article, we showed that trigonometric shearlets are able to detect directional step discontinuities along edges of periodic characteristic functions. In this paper, we extend these results to bivariate periodic functions which have jump discontinuities in higher order directional derivatives along edges. In order to prove suitable upper and lower bounds for the shearlet coefficients, we need to generalize the results about localization- and orientation-dependent decay properties of the corresponding inner products of trigonometric shearlets and the underlying periodic functions.

Citation: Kevin Schober, Jürgen Prestin. Analysis of directional higher order jump discontinuities with trigonometric shearlets. Mathematical Foundations of Computing, doi: 10.3934/mfc.2021038
References:
[1]

R. Bergmann and J. Prestin, Multivariate periodic wavelets of de la Vallée Poussin-type, J. Fourier Anal. Appl., 21 (2015), 342-369.  doi: 10.1007/s00041-014-9372-z.  Google Scholar

[2]

E. J. Candès and D. L. Donoho, New tight frames of curvelets and optimal representations of objects with piecewise $ C^2$ singularities, Commun. Pure Appl. Math., 57 (2004), 219-266.  doi: 10.1002/cpa.10116.  Google Scholar

[3]

E. J. Candès and D. L. Donoho, Ridgelets: A key to higher–dimensional intermittency?, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 357 (1999), 2495-2509.  doi: 10.1098/rsta.1999.0444.  Google Scholar

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F. J. Canny, A computational approach to edge detection, Readings in Computer Vision, (1987), 184-203.  doi: 10.1016/B978-0-08-051581-6.50024-6.  Google Scholar

[5]

D. L. Donoho, Wedgelets: Nearly minimax estimation of edges, Ann. Statist., 27 (1999), 859-897.  doi: 10.1214/aos/1018031261.  Google Scholar

[6]

D. L. Donoho, Sparse components of images and optimal atomic decompositions, Constr. Approx., 17 (2001), 353-382.  doi: 10.1007/s003650010032.  Google Scholar

[7]

P. Grohs and G. Kutyniok, Parabolic molecules, Found. Comput. Math., 14 (2014), 299-337.  doi: 10.1007/s10208-013-9170-z.  Google Scholar

[8]

P. GrohsS. KeiperG. Kutyniok and M. Schäfer, $ \alpha$-molecules, Appl. Comput. Harmon. Anal., 41 (2016), 297-336.  doi: 10.1016/j.acha.2015.10.009.  Google Scholar

[9]

P. Grohs and Z. Kereta, Analysis of edge and corner points using parabolic dictionaries, Appl. Comput. Harmon. Anal., 48 (2020), 655-681.  doi: 10.1016/j.acha.2018.08.005.  Google Scholar

[10]

K. Guo and D. Labate, Characterization and analysis of edges using the continuous shearlet transform, SIAM J. Imaging Sci., 2 (2009), 959-986.  doi: 10.1137/080741537.  Google Scholar

[11]

K. Guo and D. Labate, Detection of singularities by discrete multiscale directional representations, J. Geom. Anal., 28 (2018), 2102-2128.  doi: 10.1007/s12220-017-9897-x.  Google Scholar

[12]

K. Guo and D. Labate, Characterization and analysis of edges in piecewise smooth functions, Appl. Comput. Harmon. Anal., 41 (2016), 139-163.  doi: 10.1016/j.acha.2015.10.007.  Google Scholar

[13]

G. Kutyniok and D. Labate, Introduction to shearlets, Appl. Numer. Harmon. Anal. Birkhäuser, (2012), 1-38.  doi: 10.1007/978-0-8176-8316-0_1.  Google Scholar

[14]

G. Kutyniok and P. Petersen, Classification of edges using compactly supported shearlets, Appl. Comput. Harmon. Anal., 42 (2017), 245-293.  doi: 10.1016/j.acha.2015.08.006.  Google Scholar

[15]

D. Labate and K. Guo, Optimally sparse multidimensional representation using shearlets, SIAM J. Math. Anal., 39 (2007), 298-318.  doi: 10.1137/060649781.  Google Scholar

[16]

D. Langemann and J. Prestin, Multivariate periodic wavelet analysis, Appl. Comput. Harmon. Anal., 28 (2010), 46-66.  doi: 10.1016/j.acha.2009.07.001.  Google Scholar

[17]

I. E. Maksimenko and M. A. Skopina, Multidimensional periodic wavelets, St. Petersbg. Math. J., 15 (2004), 165-190.  doi: 10.1090/S1061-0022-04-00808-8.  Google Scholar

[18] S. Mallat, A Wavelet Tour of Signal Processing, Academic Press, Inc., San Diego, CA, 1998.   Google Scholar
[19]

S. Mallat and W. L. Hwang, Singularity detection and processing with wavelets, IEEE Trans. Inf. Theory, 38 (1992), 617-643.  doi: 10.1109/18.119727.  Google Scholar

[20]

F. G. Meyer and R. R. Coifman, Brushlets: A tool for directional image analysis and image compression, Appl. Comput. Harmon. Anal., 4 (1997), 147-187.  doi: 10.1006/acha.1997.0208.  Google Scholar

[21] I. R. Porteous, Geometric Differentiation: For the Intelligence of Curves and Surfaces, Cambridge University Press, 2001.   Google Scholar
[22]

K. SchoberJ. Prestin and S. A. Stasyuk, Edge detection with trigonometric polynomial shearlets, Adv. Comput. Math., 47 (2021), 147-187.  doi: 10.1007/s10444-020-09838-3.  Google Scholar

show all references

References:
[1]

R. Bergmann and J. Prestin, Multivariate periodic wavelets of de la Vallée Poussin-type, J. Fourier Anal. Appl., 21 (2015), 342-369.  doi: 10.1007/s00041-014-9372-z.  Google Scholar

[2]

E. J. Candès and D. L. Donoho, New tight frames of curvelets and optimal representations of objects with piecewise $ C^2$ singularities, Commun. Pure Appl. Math., 57 (2004), 219-266.  doi: 10.1002/cpa.10116.  Google Scholar

[3]

E. J. Candès and D. L. Donoho, Ridgelets: A key to higher–dimensional intermittency?, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 357 (1999), 2495-2509.  doi: 10.1098/rsta.1999.0444.  Google Scholar

[4]

F. J. Canny, A computational approach to edge detection, Readings in Computer Vision, (1987), 184-203.  doi: 10.1016/B978-0-08-051581-6.50024-6.  Google Scholar

[5]

D. L. Donoho, Wedgelets: Nearly minimax estimation of edges, Ann. Statist., 27 (1999), 859-897.  doi: 10.1214/aos/1018031261.  Google Scholar

[6]

D. L. Donoho, Sparse components of images and optimal atomic decompositions, Constr. Approx., 17 (2001), 353-382.  doi: 10.1007/s003650010032.  Google Scholar

[7]

P. Grohs and G. Kutyniok, Parabolic molecules, Found. Comput. Math., 14 (2014), 299-337.  doi: 10.1007/s10208-013-9170-z.  Google Scholar

[8]

P. GrohsS. KeiperG. Kutyniok and M. Schäfer, $ \alpha$-molecules, Appl. Comput. Harmon. Anal., 41 (2016), 297-336.  doi: 10.1016/j.acha.2015.10.009.  Google Scholar

[9]

P. Grohs and Z. Kereta, Analysis of edge and corner points using parabolic dictionaries, Appl. Comput. Harmon. Anal., 48 (2020), 655-681.  doi: 10.1016/j.acha.2018.08.005.  Google Scholar

[10]

K. Guo and D. Labate, Characterization and analysis of edges using the continuous shearlet transform, SIAM J. Imaging Sci., 2 (2009), 959-986.  doi: 10.1137/080741537.  Google Scholar

[11]

K. Guo and D. Labate, Detection of singularities by discrete multiscale directional representations, J. Geom. Anal., 28 (2018), 2102-2128.  doi: 10.1007/s12220-017-9897-x.  Google Scholar

[12]

K. Guo and D. Labate, Characterization and analysis of edges in piecewise smooth functions, Appl. Comput. Harmon. Anal., 41 (2016), 139-163.  doi: 10.1016/j.acha.2015.10.007.  Google Scholar

[13]

G. Kutyniok and D. Labate, Introduction to shearlets, Appl. Numer. Harmon. Anal. Birkhäuser, (2012), 1-38.  doi: 10.1007/978-0-8176-8316-0_1.  Google Scholar

[14]

G. Kutyniok and P. Petersen, Classification of edges using compactly supported shearlets, Appl. Comput. Harmon. Anal., 42 (2017), 245-293.  doi: 10.1016/j.acha.2015.08.006.  Google Scholar

[15]

D. Labate and K. Guo, Optimally sparse multidimensional representation using shearlets, SIAM J. Math. Anal., 39 (2007), 298-318.  doi: 10.1137/060649781.  Google Scholar

[16]

D. Langemann and J. Prestin, Multivariate periodic wavelet analysis, Appl. Comput. Harmon. Anal., 28 (2010), 46-66.  doi: 10.1016/j.acha.2009.07.001.  Google Scholar

[17]

I. E. Maksimenko and M. A. Skopina, Multidimensional periodic wavelets, St. Petersbg. Math. J., 15 (2004), 165-190.  doi: 10.1090/S1061-0022-04-00808-8.  Google Scholar

[18] S. Mallat, A Wavelet Tour of Signal Processing, Academic Press, Inc., San Diego, CA, 1998.   Google Scholar
[19]

S. Mallat and W. L. Hwang, Singularity detection and processing with wavelets, IEEE Trans. Inf. Theory, 38 (1992), 617-643.  doi: 10.1109/18.119727.  Google Scholar

[20]

F. G. Meyer and R. R. Coifman, Brushlets: A tool for directional image analysis and image compression, Appl. Comput. Harmon. Anal., 4 (1997), 147-187.  doi: 10.1006/acha.1997.0208.  Google Scholar

[21] I. R. Porteous, Geometric Differentiation: For the Intelligence of Curves and Surfaces, Cambridge University Press, 2001.   Google Scholar
[22]

K. SchoberJ. Prestin and S. A. Stasyuk, Edge detection with trigonometric polynomial shearlets, Adv. Comput. Math., 47 (2021), 147-187.  doi: 10.1007/s10444-020-09838-3.  Google Scholar

Figure 1.  Left: Cartoon-like function with jump discontinuities in the zeroth (red), first (blue) and second (black) order directional derivative on a circle with radius $ 2 $. Right: Zoom into the green window in the left picture
Figure 1 with colored boundary lines where the function has directional jump discontinuities of different orders. Right: Magnitudes of $ \mathcal{L}^{(\mathfrak{i}), \mathrm{max}}_{\ell} $ and $ \mathcal{L}^{(\mathfrak{i}), \mathrm{min}}_{\ell} $ from (17) as functions of the orientation angles $ \theta_{10, \ell}^{(\mathfrak{i})} $">Figure 2.  Left: Schematic visualization of the function from Figure 1 with colored boundary lines where the function has directional jump discontinuities of different orders. Right: Magnitudes of $ \mathcal{L}^{(\mathfrak{i}), \mathrm{max}}_{\ell} $ and $ \mathcal{L}^{(\mathfrak{i}), \mathrm{min}}_{\ell} $ from (17) as functions of the orientation angles $ \theta_{10, \ell}^{(\mathfrak{i})} $
Figure 3.  Left: Star-like set $ T\in\mathrm{STAR}^2 $ (red). Right: Zoom into the small window of the left picture to see the neighborhood $ B_\varepsilon({\bf{y}}) $ around $ {\bf{y}}\in \mathcal{P}({\bf{N}}_{j, \ell}^{(\mathfrak{i})}) $ and $ U_\varepsilon({\bf{y}}) $ on the boundary $ \partial T $ with the interval $ [a_{k^*}, a_{k^*+1}) $
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