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.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

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

    Article  Google Scholar 

  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

    Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

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

    Book  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  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

    Article  Google Scholar 

  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

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  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

    Article  Google Scholar 

  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

    Article  Google Scholar 

  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

    MathSciNet  MATH  Google Scholar 

  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

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  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

    MathSciNet  MATH  Google Scholar 

  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

    Article  Google Scholar 

  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

    Article  MATH  Google Scholar 

  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

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  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

    Article  Google Scholar 

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

    Article  Google Scholar 

  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

    MathSciNet  Article  MATH  Google Scholar 

  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

    MathSciNet  Article  Google Scholar 

  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

    Article  Google Scholar 

  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

    Article  Google Scholar 

  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

    Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  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

    Article  Google Scholar 

  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

    Article  Google Scholar 

  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

    Article  Google Scholar 

  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

    Google Scholar 

  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

    Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  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

    Google Scholar 

  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)

    MathSciNet  Article  MATH  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Anouar Ben Mabrouk.

Ethics declarations

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.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Communicated by V. Loia.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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). https://doi.org/10.1007/s00500-019-04274-y

Download citation

Keywords

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