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


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.

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1D, 2D, 3D:

One-dimensional, two-dimensional, three-dimensional


Spherical harmonics











\(S^2\) :

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

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

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

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

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 \(0\le k\le l\)


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


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Correspondence to Anouar Ben Mabrouk.

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Jallouli, M., Bel Hadj Khélifa, W., Ben Mabrouk, A. et al. Toward recursive spherical harmonics issued bi-filters: Part II: an associated spherical harmonics entropy for optimal modeling. Soft Comput 24, 5231–5243 (2020).

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  • Spherical harmonics
  • Filters
  • Bi-filters
  • Recursive methods
  • Image processing
  • Entropy