Explicit calculation of singular integrals of tensorial polyadic kernels
Authors:
M. Perrin and F. Gruy
Journal:
Quart. Appl. Math. 81 (2023), 65-86
MSC (2020):
Primary 46F12, 42A38, 42B20; Secondary 44-04, 65R10
DOI:
https://doi.org/10.1090/qam/1629
Published electronically:
September 26, 2022
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Additional Information
Abstract: The Riesz transform of $u$ : $\mathcal {S}(\mathbb {R}^n) \rightarrow \mathcal {S’}(\mathbb {R}^n)$ is defined as a convolution by a singular kernel, and can be conveniently expressed using the Fourier transform and a simple multiplier. We extend this analysis to higher order Riesz transforms, i.e. some type of singular integrals that contain tensorial polyadic kernels and define an integral transform for functions $\mathcal {S}(\mathbb {R}^n) \rightarrow \mathcal {S’}(\mathbb {R}^{ n \times n \times \dots n})$. We show that the transformed kernel is also a polyadic tensor, and propose a general method to compute explicitely the Fourier mutliplier. Analytical results are given, as well as a recursive algorithm, to compute the coefficients of the transformed kernel. We compare the result to direct numerical evaluation, and discuss the case $n=2$, with application to image analysis.
References
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References
- L. Zhang and H. Li, Image and Vision Computing 30 (2012), no. 12, 1043–1051.
- K. Langley and S. J. Anderson, Vision Research 50 (2010), no. 17, 748–1765.
- A. Abbasi, C. S. Woo, and S. Shamshirband, Measurement 74 (2015), 116–129.
- Tadeusz Iwaniec and Gaven Martin, Riesz transforms and related singular integrals, J. Reine Angew. Math. 473 (1996), 25–57. MR 1390681
- Loukas Grafakos, Classical Fourier analysis, 3rd ed., Graduate Texts in Mathematics, vol. 249, Springer, New York, 2014. MR 3243734, DOI 10.1007/978-1-4939-1194-3
- Mikael Sørensen, Lieven De Lathauwer, Pierre Comon, Sylvie Icart, and Luc Deneire, Canonical polyadic decomposition with a columnwise orthonormal factor matrix, SIAM J. Matrix Anal. Appl. 33 (2012), no. 4, 1190–1213. MR 3023470, DOI 10.1137/110830034
- F. Gruy, M. Perrin, and V. Rabiet, https://hal.archives-ouvertes.fr/hal-03043716/.
- Matlab programs have been made available under the D.O.I. : 10.5281/zenodo.6452062
- http://extremelearning.com.au
- M. E. Muller, Commun. Ass. Comput. Math. 2 (1959), 19–20.
- https://fr.mathworks.com/help/stats/generating-quasi-random-numbers.html\#br5k9hi-9
- R. Marchant and P. Jackway, Local feature analysis using a sinusoidal signal model derived from higher-order Riesz transforms, 2013 IEEE International Conference on Image Processing, 2013, pp. 3489–3493, doi: 10.1109/ICIP.2013.6738720.
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- C. Harris and M. Stephens, A combined corner and edge detector, Proceedings of the 4th Alvey Vision Conference, 1988, pp. 147–151.
- G. S. Adkins, Indian J. of Math., Dharma Prakash Gupta memorial volume, Vol. 60, No.1, 2018, 65–84.
- Z. Fu, L. Grafakos, Y. Lin, Y. Wu, and S. Yang, arXiv:2111.04027v1, 2021.
- J. Carroll and J. Chang, Psychometrika 9 (1970), 267–283.
- Tamara G. Kolda and Brett W. Bader, Tensor decompositions and applications, SIAM Rev. 51 (2009), no. 3, 455–500. MR 2535056, DOI 10.1137/07070111X
- Milton Abramowitz and Irene A. Stegun (eds.), Handbook of mathematical functions, with formulas, graphs, and mathematical tables, Dover Publications, Inc., New York, 1966. MR 0208797
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Additional Information
M. Perrin
Affiliation:
Université de Bordeaux, CNRS, LOMA, UMR 5798, F-33405 Talence, France
Address at time of publication:
LOMA - UMR 5798, 351 Cours de la libération, 33400 Talence, France
ORCID:
0000-0003-0735-6598
Email:
mathias.perrin@u-bordeaux.fr
F. Gruy
Affiliation:
Mines Saint-Etienne, Univ. Lyon, CNRS, UMR 5307 LGF, Centre SPIN 42023 Saint-Etienne, France
MR Author ID:
835354
Email:
fgruy@emse.fr
Received by editor(s):
April 20, 2022
Received by editor(s) in revised form:
July 5, 2022
Published electronically:
September 26, 2022
Article copyright:
© Copyright 2022
Brown University