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.

[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.

[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.

[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.

[5]

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

[6]

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

[7]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[15]

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

[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.

[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.

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

[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.

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

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.

[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.

[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.

[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.

[5]

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

[6]

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

[7]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[15]

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

[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.

[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.

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

[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.

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

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 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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