Advertisement

Advances in Homotopy Applied to Object Deformation

  • Jose Alejandro Salazar-Castro
  • Ana Cristina Umaquinga-Criollo
  • Lilian Dayana Cruz-Cruz
  • Luis Omar Alpala-Alpala
  • Catalina González-Castaño
  • Miguel A. Becerra-Botero
  • Diego Hernán Peluffo-Ordóñez
  • Cesar Germán Castellanos-Domínguez
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10814)

Abstract

This work explores novel alternatives to conventional linear homotopy to enhance the quality of resulting transitions from object deformation applications. Studied/introduced approaches extend the linear mapping to other representations that provides smooth transitions when deforming objects while homotopy conditions are fulfilled. Such homotopy approaches are based on transcendental functions (TFH) in both simple and parametric versions. As well, we propose a variant of an existing quality indicator based on the ratio between the coefficients curve of resultant homotopy and that of a less-realistic, reference homotopy. Experimental results depict the effect of proposed TFH approaches regarding its usability and benefit for interpolating images formed by homotopic objects with smooth changes.

Keywords

Homotopy Object deformation Transcendental functions Smooth transitions 

Notes

Acknowledgments

This work is supported by the “Smart Data Analysis Systems - SDAS” group (http://sdas-group.com).

References

  1. 1.
    Alpala, L.O., Peluffo-Ordóñez, D.H., González-Castaño, C., Guasmayan, F.A.: Deforming objects via exponential homotopy: a first approach. In: 2015 20th Symposium on Signal Processing, Images and Computer Vision (STSIVA), pp. 1–6. IEEE (2015)Google Scholar
  2. 2.
    Riehl, E.: Categorical Homotopy Theory, vol. 24. Cambridge University Press, Cambridge (2014)CrossRefzbMATHGoogle Scholar
  3. 3.
    He, J.H.: Homotopy perturbation method: a new nonlinear analytical technique. Appl. Math. Comput. 135(1), 73–79 (2003)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Babolian, E., Azizi, A., Saeidian, J.: Some notes on using the homotopy perturbation method for solving time-dependent differential equations. Math. Comput. Modell. 50(1), 213–224 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Molina-Abril, H., Real, P.: A homological-based description of subdivided nD objects. In: Real, P., Diaz-Pernil, D., Molina-Abril, H., Berciano, A., Kropatsch, W. (eds.) CAIP 2011. LNCS, vol. 6854, pp. 42–50. Springer, Heidelberg (2011).  https://doi.org/10.1007/978-3-642-23672-3_6CrossRefGoogle Scholar
  6. 6.
    Peluffo-Ordóñez, D.H., Alvarado-Pérez, J.C., Lee, J.A., Verleysen, M., et al.: Geometrical homotopy for data visualization. In: European Symposium on Artificial Neural Networks, Computational Intelligence and Machine Learning (ESANN 2015) (2015)Google Scholar
  7. 7.
    Salazar-Castro, J., Rosas-Narváez, Y., Pantoja, A., Alvarado-Pérez, J.C., Peluffo-Ordóñez, D.H.: Interactive interface for efficient data visualization via a geometric approach. In: 2015 20th Symposium on Signal Processing, Images and Computer Vision (STSIVA), pp. 1–6. IEEE (2015)Google Scholar
  8. 8.
    Feng, X.: Geometric and topological modeling techniques for large and complex shapes. Michigan State University (2014)Google Scholar
  9. 9.
    Georghiades, A., Belhumeur, P., Kriegman, D.: Yale face database, vol. 2. Center for computational Vision and Control at Yale University (1997). http://cvc.yale.edu/projects/yalefaces/yalefa

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Jose Alejandro Salazar-Castro
    • 1
    • 2
  • Ana Cristina Umaquinga-Criollo
    • 3
    • 4
  • Lilian Dayana Cruz-Cruz
    • 2
  • Luis Omar Alpala-Alpala
    • 5
  • Catalina González-Castaño
    • 6
  • Miguel A. Becerra-Botero
    • 7
  • Diego Hernán Peluffo-Ordóñez
    • 8
  • Cesar Germán Castellanos-Domínguez
    • 1
  1. 1.Universidad Nacional de Colombia, sede ManizalesManizalesColombia
  2. 2.Corporación Universitaria Autónoma de NariñoPastoColombia
  3. 3.Universidad Técnica del NorteIbarraEcuador
  4. 4.Universidad de SalamancaSalamancaSpain
  5. 5.Universidad Cooperativa de ColombiaPastoColombia
  6. 6.Universitat Rovira i VirgiliTarragonaSpain
  7. 7.Institución Universitaria Pascual BravoMedellínColombia
  8. 8.Universidad Yachay TechUrcuquíEcuador

Personalised recommendations