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Toward recursive spherical harmonics-issued bi-filters: Part I: theoretical framework

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Abstract

This work lies in the whole topic of signal/image processing based on frequency decompositions. These are indeed applied in a variety of subjects such as estimation, reconstruction, shape recognition, filtering \(\dots \). Among these decompositions, spherical harmonics are widely used. Mathematically, spherical harmonics are special functions obtained as particular solutions of the Laplace equation generated by Legendre polynomials. Using the three-level recurrence relation of these polynomials, spherical harmonics recursive bases are revisited allowing the decompositions of signals in eigenmodes similar to Fourier ones. Special filters have been constructed and proved to be more efficient and accurate than existing ones as they lead to faster and more accurate algorithms.

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Abbreviations

1D, 2D, 3D:

One-dimensional, two-dimensional, three-dimensional

SHs:

Spherical harmonics

N :

North

C :

Center

S :

South

E :

East

W :

West

\(S^2\) :

The unit sphere in the Euclidian space \(\mathbb {R}^3\)

\(L^2(S^2)\) :

The set of functions (images, signals) supported on \(S^2\) with finite energy (variance)

\(\nabla ^2=\varDelta \) :

Laplace’s operator or Laplacian

\(P_{l,m}\) :

Legendre polynomial of degree l and order m

\(Y_{l,m}\) :

The spherical harmonics of degree l and order m

\(C_l^k\) :

\(=\displaystyle \frac{l!}{k!(l-k)!}\), for \(l,k\in \mathbb {N}\) such that \(0\le k\le l\)

MRI:

Magnetic resonance imaging

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Authors and Affiliations

  1. Laboratory of Advanced Technology and Intelligent Systems LATIS, National Engineering School of Sousse, University of Sousse, Sousse, Tunisia

    Malika Jallouli, Makerem Zemni & Mohamed Ali Mahjoub

  2. Laboratory of Algebra, Number Theory and Nonlinear Analysis LR18ES15, Department of Mathematics, Faculty of Sciences, University of Monastir, 5000, Monastir, Tunisia

    Anouar Ben Mabrouk

  3. Department of Mathematics, Higher Institute of Applied Mathematics and Informatics, Street of Assad Ibn Alfourat, University of Kairouan, 3100, Kairouan, Tunisia

    Anouar Ben Mabrouk

  4. Department of Mathematics, College of Sciences, University of Tabuk, Tabuk, Kingdom of Saudi Arabia

    Anouar Ben Mabrouk

Authors
  1. Malika Jallouli
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  2. Makerem Zemni
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  3. Anouar Ben Mabrouk
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  4. Mohamed Ali Mahjoub
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Correspondence to Anouar Ben Mabrouk.

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Jallouli, M., Zemni, M., Ben Mabrouk, A. et al. Toward recursive spherical harmonics-issued bi-filters: Part I: theoretical framework. Soft Comput 23, 10415–10428 (2019). https://doi.org/10.1007/s00500-018-3596-9

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