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Poisson Equation Solution and Its Gradient Vector Field to Geometric Features Detection

  • Mengzhe Chen
  • Nikolay Metodiev Sirakov
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11324)

Abstract

In this paper we solve the Poisson partial differential equation (PDE) with a free right side, which is a function of the image and its gradient. We call such a PDE Poisson Image (PI) equation. Further, we define the function \(\phi ={u+\left\| \nabla u\right\| }^2\), where u is the PI’s solution. Then, we generate the Poisson gradient vector fields (PGVFs) \(\nabla u\) and \(\nabla \phi \) and study the patterns of their trajectories in the vicinity of the singular points (SPs). Next, we use the critical points (CPs) of u and \(\phi \), the SPs of \(\nabla u\) and \( \nabla \phi \), and their relations, to determine the image objects’ concavities and convexities, and use them for automatic objects partitioning. We validated the theoretical concepts with experiments on above 80 synthetic and real-life images, and show some of them in the paper. At the end we compare the new method with contemporary methods in the field and list its contributions, advantages and bottlenecks.

Keywords

Critical and singular points Trajectory patterns Objects partitioning 

Notes

Acknowledgments

We thank to the anonymous reviewers for the useful notes, to A. Bowden for providing the active contour code, and Dr. M. Celik and Dr. T. Wang for the useful discussions on CPs and SPs.

References

  1. 1.
    Caselles, V., Kimmel, R., Sapiro, G.: Geodesic active contours. Int. J. Comput. Vis. 22(1), 61–79 (1997)CrossRefGoogle Scholar
  2. 2.
    Sirakov, N.M.: A new active convex hull model for image regions. J. Math. Imaging Vis. 26, 309–325 (2006)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bowden, A., Todorov, D., Sirakov, N.M.: Implementation of the Euler-Lagrange and Poisson equations to extract one connected region. In: Proceedings of AMiTANs, vol. 1629, pp. 400–407. American Institute of Physics (2014).  https://doi.org/10.1063/1.4902301
  4. 4.
    Bowden, A., Sirakov, M.N.: Applications of the Euler-Lagrange Poisson active contour in vector fields, overcoming noise, and line integrals. Dyn. Contin., Discret. Impuls. Syst. Ser. B Appl. Algorithms 23, 59–73 (2016)Google Scholar
  5. 5.
    Li, B., Acton, B.: Active contour external force using vector field convolution for image segmentation. IEEE TIP 16(8), 2096–2106 (2007)MathSciNetGoogle Scholar
  6. 6.
    Aubert, G., Barlaud, M., Faugeras, O., Jehan-Besson, S.: Image segmentation using active contours: calculus of variations or shape gradients? SIAM J. Appl. Math. 63(6), 2128–2154 (2003)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Gorelick, L., Galun, M., Sharon, E., Basri, R., Brandt, A.: Shape representation and classification using the Poisson equation. IEEE Trans. PAMI 28, 1991–2005 (2007)CrossRefGoogle Scholar
  8. 8.
    Tari, S., Genctav, M.: From a non-local ambrosio-tortorelli phase field to a randomized part hierarchy tree. JMIV 49(1), 69–86 (2014)CrossRefGoogle Scholar
  9. 9.
    Sosinsky, Vector fields on the plane (2015). http://ium.mccme.ru/postscript/s16/topology1-Lec7.pdf
  10. 10.
    Ferreira, N., Klosowski, J.T., Scheidegger, C.E., Silva, C.T.: Vector field k-means: clustering trajectories by fitting multiple vector fields. Comput. Graph. Forum 32(3), 201–210 (2013)CrossRefGoogle Scholar
  11. 11.
    Zhang, E., Konstantin, M., Greg, T.: Vector fields design on surfaces. ACM Trans. Graph. 25(4), 1294–1326 (2006)CrossRefGoogle Scholar
  12. 12.
    Wei, L., Eraldo, R.: Detecting singular patterns in 2-D vector fields using weighted Laurent polynomial. Pattern Recogn. 45(11), 3912–3925 (2012)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of MathematicsTexas A&M University CommerceCommerceUSA

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