Toward recursive spherical harmonics issued bi-filters: Part II: an associated spherical harmonics entropy for optimal modeling

  • Malika Jallouli
  • Wafa Bel Hadj Khélifa
  • Anouar Ben MabroukEmail author
  • Mohamed Ali Mahjoub
Methodologies and Application


Image processing continues to be a challenging topic in many scientific fields such as medicine, computational physics and informatics especially with the discovery and development of 3D cases. Therefore, development of suitable tools that guarantee a best treatment is a necessity. Spherical shapes are a big class of 3D images whom processing necessitates adoptable tools. This encourages researchers to develop special mathematical bases suitable for 3D spherical shapes. The present work lies in this whole topic with the application of special spherical harmonics bases. In Jallouli et al. (Soft Comput 2018., theoretical framework of spherical harmonics filters adapted to image processing has been developed. In the present paper, new approach based on Jallouli et al. (Soft Comput 2018. is proposed for the reconstruction of images provided with spherical harmonics Shannon-type entropy to evaluate the order/disorder of the reconstructed image. Efficiency and accuracy of the approach are demonstrated by a simulation study on several spherical models.


Spherical harmonics Filters Bi-filters Recursive methods Image processing Entropy 

List of symbols

1D, 2D, 3D

One-dimensional, two-dimensional, three-dimensional


Spherical harmonics












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


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

\(\nabla ^2=\Delta \)

Laplace’s operator or Laplacian


Legendre polynomial of degree l and order m


The spherical harmonics of degree l and order m


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


Shannon’s entropy

\(\hbox {SHsE}(l,m)\)

Spherical harmonics energy at the level l and the order m

\(\hbox {SHsE}(l)\)

Spherical harmonics energy of the level l

\(\hbox {SHsE}\)

Total spherical harmonics energy

\(\hbox {SHsE}_L\)

The L-level approximation of the spherical harmonics energy

\(\hbox {SHsP}(l)\)

Spherical harmonics probability of the level l

\(\hbox {SHsEnt}\)

Spherical harmonics entropy

\(\hbox {SHsEnt}_L\)

The L-level approximation of the spherical harmonics entropy


Magnetic resonance imaging


Compliance with ethical standards

Conflict of interest

The authors declare no conflict of interest for the present work.

Human and animal rights

The authors declare that no animals were involved in the study and that this article does not contain any studies with human participants performed by any of the authors.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Malika Jallouli
    • 1
  • Wafa Bel Hadj Khélifa
    • 1
  • Anouar Ben Mabrouk
    • 2
    • 3
    • 4
    Email author
  • Mohamed Ali Mahjoub
    • 1
  1. 1.LATIS - Laboratory of Advanced Technology and Intelligent Systems, Ecole Nationale d’Ingénieurs de SousseUniversité de SousseSousseTunisia
  2. 2.Laboratory of Algebra, Number Theory and Nonlinear Analysis UR11ES50, Department of Mathematics, Faculty of SciencesUniversity of MonastirMonastirTunisia
  3. 3.Department of Mathematics, Higher Institute of Applied Mathematics and Informatics, Street of Assad Ibn AlfouratUniversity of KairouanKairouanTunisia
  4. 4.Department of Mathematics, College of SciencesUniversity of TabukTabukKingdom of Saudi Arabia

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