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

## Abstract

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. https://doi.org/10.1007/s00500-018-3596-9), 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. https://doi.org/10.1007/s00500-018-3596-9) 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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## 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 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 $$0\le k\le l$$

ShE:

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

MRI:

Magnetic resonance imaging

## References

1. Antoine JP, Carrette P, Murenzi R, Piette R (1993) Image analysis with two-dimensional continuous wavelet transform. Signal Process. 31:241–272

2. Antoine JP, Murenzi R, Vandergheynst P (1996) Two-dimensional directional wavelets in image processing. Int. J. of Imaging Systems and Technology, 7(3):152–165

3. Antoine JP, Demanet L, Jacques L, Vandergheynst P (2002) Wavelets on the sphere: implementation and approximation. Appl Comput Harmonic Anal 13:177–200

4. Arfaoui S, Rezgui I, Ben Mabrouk A (2017) Wavelet analysis on the sphere, spheroidal wavelets, Degryuter, Berlin

5. Attard PP (2008) The Second Entropy: A Variational Principle for Time-dependent Systems. Entropy 10:380–390

6. Aytekin C, Cricri F, Aksu E (2019) Compressibility loss for neural network weights. arXiv:1905.01044v1 [cs.LG], 7 pp

7. Bessa RJ, Miranda V, Gama J (2009) Entropy and Correntropy Against Minimum Square Error in Offline and Online Three-Day Ahead Wind Power Forecasting. IEEE Transactions on Power Systems 24(4):10

8. Borwein JM, Lewis AS, Limber MN, Noll D (1995) Maximum entropy reconstruction using derivative information part 2: computational results. Numer. Math. 69:243–256

9. Broadbridge Ph (2008) Entropy Diagnostics for Fourth Order Partial Differential Equations in Conservation Form. Entropy 10:365–379

10. Bulow Th, Daniilidis K (2001) Surface representations using spherical harmonics and gabor wavelets on the sphere. Technical Reports (CIS) No. MS-CIS-01-37. Department of Computer and Information Science. University of Pennsylvania. January 2001, 21 pages

11. Burns B, Wilson NE, Furuyama JK, Thomas MA (2014) Non-uniformly under-sampled multi-dimensional spectroscopic imaging in vivo: maximum entropy versus compressed sensing reconstruction. NMR Biomed. 27:191–201

12. Chambodut A, Panet I, Mandea M, Diament M, Holschneider M, Jamet O (2005) Wavelet frames: an alternative to spherical harmonic representation of potential fields. Geophysical Journal International 163:875–899

13. Dine M (2010) Special functions: Legendre functions, spherical harmonics, and Bessel functions. Presentation, Physics 212 2010, Electricity and Magnetism. Department of Physics, University of California, Santa Cruz

14. Dudık M, Phillips SJ, Schapire RE (2007) Maximum Entropy Density Estimation with Generalized Regularization and an Application to Species Distribution Modeling. Journal of Machine Learning Research 8:1217–1260

15. Fan J, Hu T, Wu Q, Zhou DX (2016) Consistency Analysis of an Empirical Minimum Error Entropy Algorithm. Appl. Comput. Harmon. Anal. 41:164–189

16. Fradkov A (2000) Speed-gradient Entropy Principle for Nonstationary Processes. Entropy 10:757–764

17. Gomes-Gonçalves E, Gzyl H, Mayoral S (2014) Density Reconstructions with Errors in the Data. Entropy 16:3257–3272

18. Gonzalez D, Cueto E, Doblare M (2007) A higher-order method based on local maximum entropy approximation. Int. J. Numer. Meth. Engineering 83(6):741–764

19. Guan X (2011) Spherical image processing for immersive visualisation and view generation. PhD, University of Central Lancashire

20. Hielscher R, Schaeben H, Chateigner D (2007) On the entropy to texture index relationship in quantitative texture analysis. J. Appl. Cryst. 40:371–375

21. Jallouli M, Zemni Z, Ben Mabrouk A, Mahjoub MA (2018) Toward Recursive Spherical Harmonics Issued Bi-Filters: Part I: Theoretical Framework. Soft Computing. https://doi.org/10.1007/s00500-018-3596-9

22. Kahali S, Sing JK, Saha PK (2018) A new entropy-based approach for fuzzy c-means clustering and its application to brain MR image segmentation. Soft Computing. https://doi.org/10.1007/s00500-018-3594-y

23. Labat D (2005) Recent advances in wavelet analyses: Part 1. A review of Concepts. J Hydrol 314:275–288

24. Lombardi D, Pant S (2016) Nonparametric $$k$$-nearest-neighbor entropy estimator. Physical Review E 93(013310):12

25. Low FE (2004) Classical field theory, Electromagnetism and Gravitation. Willey-VCH Verlag GmbH & Co,

26. Lyubushin A (2013) How Soon would the next Mega-Earthquake Occur in Japan Nat. Sci. 5:1–7

27. Muller I (2008) Entropy and energy: a universal competition. Entropy 10:462–476

28. Muller I (2008) Extended thermodynamics: a theory of symmetric hyperbolic field equations. Entropy 10:477–492

29. Ng AY (2004) Feature selection, $$L_1$$ vs. $$L_2$$ regularization, and rotational invariance. In: Proceeding ICML’04 Proceedings of the twenty-first international conference on machine learning, pp 78–85

30. Nicholson T, Sambridge M, Gudmundsson O (2000) On Entropy and Clustering in Earthquake Hypocentre Distributions. Geophys. J. Int. 142:37–51

31. Nicolis O, Mateu J (2015) 2D Anisotropic Wavelet Entropy with an Application to Earthquakes in Chile. Entropy 17:4155–4172

32. Nose-Filho K, Fantinato DG, Attux R, Neves A, Romano JMT (2013) A novel entropy-based equalization performance measure and relations to $$L_p$$-norm deconvolution. XXXI Simposio Brasileiro de Telecomunicacoes-SBrT2013, 1–4 De Setembro De 2013, Fortaleza, CE, 5 pp

33. Prestin J, Wülker Ch (2016) Fast fourier transforms for spherical Gauss–Laguerre basis functions. arXiv:1604.05140v3 [math.NA], 27 pp

34. Ranocha H (2017) Comparison of some entropy conservative numerical fluxes for the Euler equations. J. Sci Comput. https://doi.org/10.1007/s10915-017-0618-1

35. Riezler S, Vasserman A (2004) Incremental feature selection and $$l_1$$ regularization for relaxed maximum-entropy modeling. In: Proceedings of the 2004 conference on empirical methods in natural language processing, EMNLP 2004, A meeting of SIGDAT, a Special Interest Group of the ACL, held in conjunction with ACL 2004, 25-26 July 2004, Barcelona, Spain. 8 pp

36. Robinson DW (2008) Entropy and Uncertainty. Entropy 10:493–506

37. Rosso O, Blanco S, Yordanowa J, Kolev V, Figliola A, Schürmann M, Bacsar E (2001) Wavelet Entropy: A New Tool for Analysis of Short Duration Brain Electrical Signals. J. Neurosci. Methods 105:65–75

38. Ruggeri T (2008) The Entropy Principle from Continuum Mechanics to Hyperbolic Systems of Balance Laws: The Modern Theory of Extended Thermodynamics. Entropy 10:319–333

39. Sakai Y, Iwata KI (2016) Extremal relations between shannon entropy and $$l_\alpha$$-norm. arXiv:1601.07678v1 [cs.IT], 35 pp

40. Sello S (2003) Wavelet Entropy and the Multi-peaked Structure of Solar Cycle Maximum. New Astron. 8:105–117

41. Shannon C (1949) A Mathematical Theory of Communication. Bell Syst. Tech. J. 27:379–423

42. Silva LEV, Duque JJ, Felipe JC, Murta LO Jr, Humeau-Heurtier A (2018) Two-Dimensional Multiscale Entropy Analysis: Applications to Image Texture Evaluation. Signal Processing 147:224–232. https://doi.org/10.1016/j.sigpro.2018.02.004

43. Telesca L, Lapenna V, Lovallo M (2004) Information Entropy Analysis of Seismicity of Umbria-Marche Region (Central Italy). Nat. Hazards Earth Syst. Sci. 4:691–695

44. Telesca L, Lovallo M, Molist JM, Moreno CL, Mendelez RA (2014) Using the Fisher-Shannon Method to Characterize Continuous Seismic Signal during Volcanic Eruptions: Application to 2011–2012 El Hierro (Canary Islands) Eruption. Terra Nova 26:425–429

45. Tomczak JM, Gonczarek A (2015) Sparse hidden units activation in Restricted Boltzmann Machine. In: Selvaraj H et al (eds) Progress in systems engineering: proceedings of the twenty-third international conference on systems engineering, Advances in Intelligent Systems and Computing 1089. Springer, Berlin, pp 181–185. https://doi.org/10.1007/978-3-319-08422-0_27

46. Vazquez PP, Feixas M, Sbert M, Heidrich W (2001) Viewpoint selection using viewpoint entropy, Stuttgart, Germany

47. Wang S, Zhao Z, Zhang X, Zhang J, Wang S, Ma S, Gao W (2016) Improved entropy of primitive for visual information estimation. In: 2016 Visual communications and image processing (VCIP), 4 pp. https://doi.org/10.1109/VCIP.2016.7805589

48. Wang B, Chen LL, Cheng J (2018) New result on maximum entropy threshold image segmentation based on P system. Optik 163:81–85

49. Williams DM (2019) An analysis of discontinuous Galerkinmethods for the compressible Euler equations: entropy and $$L_2$$ stability. Numer Math 141:1079–1120

50. Xu L, Lee TY, Shen HW (2011) An Information-Theoretic Framework for Flow Visualization. IEEE Transactions on Visualization and Computer Graphics 16(6):1216–1224

51. Zemni M, Jallouli M, Ben Mabrouk A, Mahjoub MA (2019) Explicit Haar-Schauder multiwavelet filters and algorithms. Part II: Relative entropy-based estimation for optimal modeling of biomedical signals. International Journal 1 of Wavelets, Multiresolution and Information Processing 17(4):1950038. https://doi.org/10.1142/S0219691319500383 (25 pages)

## Author information

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Correspondence to Anouar Ben Mabrouk.

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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.