Skip Navigation Links
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA


Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania


An Improved Approach for Image Segmentation and Three-Dimensional Reconstruction

Discontinuity, Nonlinearity, and Complexity 9(2) (2020) 199--215 | DOI:10.5890/DNC.2020.06.003

K. Bellaj$^{1}$, S. Boujena$^{1}$, E. EL Guarmah$^{2}$

$^{1}$ MACS laboratory, Mathematics and Computing Department, Ain Chock Sciences Faculty, Hassan II University of Casablanca. Km 8 Route El Jadida POB 5366 Maarif, Casablanca, Morocco

$^{2}$ LIRIMA-LERMA laboratories, Royal Air School, Mathematics and Informatics Department, DFST, BEFRA, POB 40002, Menara, Marrakech, Morocco

Download Full Text PDF



Themain contribution of this paper consists of introducing a novelmodel of three-dimensional reconstruction from multiple two-dimensional images. Actually, our proposedmodel presents the followingmain advantages; first, we improve the original region fitting energy in the general region-based level set method by an anisotropic diffusion to evolve the contour. Second, we use the Dijkstra algorithm to improve and allow simple and free initializations. Third, we adopt the domain decomposition method to reduce the computational cost for high-resolution images. Finally, in order to prove the efficiency and accuracy of our proposed method, experiments were performed on synthetic and real images.


  1. [1]  Cao, M., Jia, W., Li, S., Li, Y., Zheng, L., and Liu, X. (2018), GPU-accelerated feature tracking for 3D reconstruction, Optics and Laser Technology, 107, 617-676.
  2. [2]  Cao, M., Jia, W., Li, Y., Lv, Z., Li, L., Zheng, L., and Liu, X. (2018), Fast and robust local feature extraction for 3D reconstruction, Computers and Electrical Engineering, 17, 657-666.
  3. [3]  Stahlberg, H., Biyani, N., and Engel, A. (2015), 3D reconstruction of two-dimensional crystals, Archives of biochemistry and biophysics, 581, 68-77.
  4. [4]  Yan, H., Wang, Y.F., Zhou, Y.G., and Sun, Y.H. (2014), 3D ECT reconstruction by an improved Landweber iteration algorithm, Flow Measurement and Instrumentation, 37, 92-98.
  5. [5]  Li, C., Xu, C., Gui, C., and Fox, M.D. (2010), Distance regularized level set evolution and its application to image segmentation, IEEE transactions on image processing, 19, 3243.
  6. [6]  Bettahar, S., Lambert, P., and Stambouli, A.B. (2017), PDE-based efficient method for colour image restoration, Computers and Mathematics with Applications, 74, 577-590.
  7. [7]  Wen, W. (2014), Adaptive active contours based on local and global intensity information for image segmentation, Optik-International Journal for Light and Electron Optics, 125, 6995-7001.
  8. [8]  Dong, F., Chen, Z., and Wang, J. (2013), A new level set method for inhomogeneous image segmentation, Image and Vision Computing, 31, 809-822.
  9. [9]  Wu, Y.D., Zhu, Q.X., Sun, S.X., and Zhang, H.Y. (2006), Image restoration using variational PDE-based neural network, Neurocomputing, 69, 2364-2368.
  10. [10]  Jidesh, P. and Holla, S. (2018), Non-local total variation regularization models for image restoration, Computers and Electrical Engineering, 67, 114-133.
  11. [11]  Liu, J., Huang, T.Z., Selesnick, I.W., Lv, X.G., and Chen, P.Y. (2015), Image restoration using total variation with overlapping group sparsity, Information Sciences, 295, 232-246.
  12. [12]  Zhao, X., Huang, K.,Wang, X., Shi, M., Zhu, X., Gao, Q., and Yu, Z. (2018), Reaction diffusion equation based image restoration, Applied Mathematics and Computation, 338, 588-606.
  13. [13]  Boujena, S., El Guarmah, E., Gouasnouane, O., and Pousin, J. (2017), An Improved Nonlinear Model for Image Restoration, Pure and Applied Functional Analysis, 2, 599-623.
  14. [14]  Rao, A.S. and Nagendra, N. (2015),Thermal radiation effects on Oldroyd-B nano fluid from a stretching sheet in a non-Darcy porous medium, Global Journal of Pure and Applied Mathematics (GJPAM), 11, 2015.
  15. [15]  Rudin, L.I., Osher, S., and Fatemi, E. (1992), Nonlinear total variation based noise removal algorithms, Physica D: nonlinear phenomena, 60, 259-268.
  16. [16]  Getreuer, P. (2012), Rudin-Osher-Fatemi total variation denoising using split Bregman, Image Processing On Line, 2, 74-95.
  17. [17]  Perona, P. and Malik, J. (1990), Scale-space and edge detection using anisotropic diffusion, IEEE Transactions on pattern analysis and machine intelligence, 12, 629-639.
  18. [18]  Weickert, J. (1998), Anisotropic diffusion in image processing, Stuttgart: Teubner, Vol. 1, 59-60.
  19. [19]  Kichenassamy, S. (2007), Theorie des semi-groupes pour l’equation de PeronaMalik, Comptes RendusMathematique, 344, 225-229.
  20. [20]  Aboulaich, R., Boujena, S., and El Guarmah, E. (2007), Sur un modele non-lineaire pour le debruitage de l’image, Comptes Rendus Mathematique, 345, 425-429.
  21. [21]  Aboulaich, R., Boujena, S., and El Guarmah, E. (2008), A nonlinear parabolic model in processing of medical image, Mathematical Modelling of Natural Phenomena, 3, 131-145.
  22. [22]  Boujena, S., Bellaj, K., El Guarmah, E., and Gouasnouane, O. (2015), An Improved Nonlinear Model for Image Inpainting, Applied Mathematical Sciences, 9, 6189-6205.
  23. [23]  Catte, F., Lions, P.L., Morel, J.M., and Coll, T. (1992), Image selective smoothing and edge detection by nonlinear diffusion, SIAM Journal on Numerical Analysis, 29, 182-193.
  24. [24]  Li, S. and Li, P. (2018), Image segmentation and selective smoothing based on p-harmonicMumford Shah functional, Optik, 168, 13-26.
  25. [25]  Saadatmand-Tarzjan, M. (2015), Self-affine snake for medical image segmentation, Pattern Recognition Letters, 59, 1-10.
  26. [26]  Nithila, E.E. and Kumar, S.S. (2019), Segmentation of lung from CT using various active contour models, Biomedical Signal Processing and Control, 47, 57-62.
  27. [27]  Han, B. and Wu, Y. (2017), Active contours driven by median global image fitting energy for SAR river image segmentation, Digital Signal Processing, 71, 46-60.
  28. [28]  Rudin, L.I., Osher, S., and Fatemi, E. (1992), Nonlinear total variation based noise removal algorithms, Physica D: nonlinear phenomena, 60, 259-268.
  29. [29]  Sun, Q., Hou, Y., and Tan, Q. (2016), A subpixel edge detection method based on an arctangent edge model, Optik- International Journal for Light and Electron Optics, 127, 5702-5710.
  30. [30]  Ciecholewski, M. (2016), An edge-based active contour model using an inflation deflation force with a damping coefficient, Expert Systems with Applications, 44, 22-36.
  31. [31]  Keatmanee, C., Chaumrattanakul, U., Kotani, K., and Makhanov, S.S. (2017), Initialization of active contours for segmentation of breast cancer via fusion of ultrasound, Doppler, and elasticity images, Ultrasonics, 30, 603-624.
  32. [32]  Talu, M.F. (2013), ORACM: Online region-based active contour model, Expert Systems with Applications, 40, 6233- 6240.
  33. [33]  Chan, T.F. and Vese, L.A. (2001),Active contourswithout edges, IEEE Transactions on image processing, 10, 266-277.
  34. [34]  Vese, L.A. and Chan, T.F. (2002), A multiphase level set framework for image segmentation using the Mumford and Shah model, International journal of computer vision, 50, 271-293.
  35. [35]  Li, C., Xu, C., Gui, C., and Fox, M.D. (2005), Level set evolution without re-initialization: a new variational formulation, Computer Vision and Pattern Recognition, 2005. CVPR 2005. IEEE Computer Society Conference, Vol. 1, 430-436.
  36. [36]  Li, L., Bai, P.R., Liu, Q.Y., Teng, S.H., Li, J., and Cao, M.Y. (2013), A novel region-based level set method initialized with mean shift clustering for automated medical image segmentation, Computers in biology and medicine, 43, 1827- 1832.
  37. [37]  Bellaj, K., Boujena, S., and EL Guarmah, E. (2018), Adaptive level set evolution for image segmentation, JAMS Journal, 11, 265-273.
  38. [38]  Abe, H., Tani, T., Mashiko, H., Kitamura, N., Miyakawa, N., Mimura, K., andWatakabe, A. (2017), 3D reconstruction of brain section images for creating axonal projection maps in marmosets, Journal of neuroscience methods, 286, 102-113.
  39. [39]  Zankel, A., Wagner, J., and Poelt, P. (2014), Serial sectioning methods for 3D investigations in materials science, Micron, 62, 66-78.
  40. [40]  Schwarz, H.A. (1972), Gesammelte mathematische abhandlungen, American Mathematical Soc, Vol. 260.
  41. [41]  Bernard, P.E., Gorris, T., and Stainier, L. (2015), A study of transverse cracking in laminates by the Thick Level Set approach, Mechanics of Materials, 90, 118-130.
  42. [42]  Lankton, S. and Tannenbaum, A. (2014), Localizing region-based active contours, IEEE transactions on image processing, 17, 2029-2039.