Advertisement

Properties of a Variational Model for Video Inpainting

  • Riccardo March
  • Giuseppe RieyEmail author
Article
  • 18 Downloads

Abstract

We consider a variational model analyzed in March and Riey (Inverse Probl Imag 11(6): 997–1025, 2017) for simultaneous video inpainting and motion estimation. The model has applications in the field of recovery of missing data in archive film materials. A gray-value video content is reconstructed in a spatiotemporal region where the video data is lost. A variational method for motion compensated video inpainting is used, which is based on the simultaneous estimation of apparent motion in the video data. Apparent motion is mathematically described by a vector field of velocity, denoted optical flow, which is estimated through gray-value variations of the video data. The functional to be minimized is defined on a space of vector valued functions of bounded variation and the relaxation method of the Calculus of Variations is used. We introduce in the functional analyzed in March and Riey(Inverse Probl Imag 11(6): 997–1025, 2017) a suitable positive weight, and we show that diagonal minimizing sequences of the functional converge, up to subsequences as the weight tends to infinity, to minimizers of an appropriate limit functional. Such a limit functional is the relaxed version of a functional, modified with suitable improvements, proposed by Lauze and Nielsen (2004) and which permits an accurate joint reconstruction both of the optical flow and of the video content.

Keywords

Calculus of variations Functional relaxation Video inpainting Optical flow estimation 

Notes

References

  1. Amar M, Chiricotto M, Giacomelli L, Riey G (2013) Mass-constrained minimization of a one-homogeneous functional arising in strain-gradient plasticity. J Math Anal Appl 397(1):381–401CrossRefGoogle Scholar
  2. Ambrosio L (1989) Variational problems in SBV and image segmentation. Acta Appl Math 17:1–40CrossRefGoogle Scholar
  3. Ambrosio L, Fusco N, Pallara D (2000) Functions of bounded variation and free discontinuity problems. Clarendon Press, OxfordGoogle Scholar
  4. Anzellotti G (1985) The Euler equation for functionals with linear growth. Trans American Mat Soc 290(2):483–501CrossRefGoogle Scholar
  5. Anzellotti G (1986) On the minima of functionals with linear growth. Rendiconti del Seminario Matematico dell’Università, di Padova 75:91–110Google Scholar
  6. Aubert G, Deriche R, Kornprobst P (1999) Computing optimal flow via variational techniques. SIAM J App Math 60(1):156–182CrossRefGoogle Scholar
  7. Aubert G, Kornprobst P (1999) A mathematical study of the relaxed optical flow problem in the space B V (Ω). SIAM J Math Anal 30(6):1282–1308CrossRefGoogle Scholar
  8. Aubert G, Kornprobst P (2006) Mathematical problems in image processing: partial differential equations and the calculus of variations, 2nd Edition. Springer, New YorkGoogle Scholar
  9. Bertalmio M, Bertozzi A, Sapiro G (2001) Navier-Stokes, fluid-dynamics and image and video inpainting. In: Proceedings of computer vision and pattern recognition, vol 2001, pp 355–362Google Scholar
  10. Brox T, Bruhn A, Papenberg N, Weickert J (2004) High accuracy optical flow estimation based on a theory for warping. In: Pajdla T, Matas J (eds) Proceedings of the 8th European conference on computer vision, vol 4. Springer, pp 25–36Google Scholar
  11. Buttazzo G (1989) Semicontinuity, relaxation and integral representation in the calculus of variations pitman research notes in mathematics series, vol 207. Longman Scientific &Technical, UKGoogle Scholar
  12. Carlsson JG, Carlsson E, Devulapalli R (2016) Shadow prices in territory division. Networks and Spatial Economics 16:893–931CrossRefGoogle Scholar
  13. Cocquerez JP, Chanas L, Blanc-Talon J (2003) Simultaneous inpainting and motion estimation of highly degraded video-sequences. In: Scandinavian conference on image analysis LNCS 2749, pp 523-530. SpringerGoogle Scholar
  14. Horn BKP, Schunck BG (1981) Determining optical flow. Artif Intell 17:185–203CrossRefGoogle Scholar
  15. Keller SH, Lauze F, Nielsen M (2008) Deintarlacing using variational methods. IEEE Trans Image Proc 17:2015–2028CrossRefGoogle Scholar
  16. Keller SH, Lauze F, Nielsen M (2011) Video super-resolution using simultaneous motion and intensity calculations. IEEE Trans Image Proc 20:1870–1884CrossRefGoogle Scholar
  17. Lauze F, Nielsen M (2004) A Variational algorithm for motion compensated inpainting. In: Barman S, Hoppe A, Ellis T (eds) British machine vision conference, vol 2. BMVA, pp 777–787Google Scholar
  18. Low SH, Srikant R (2004) A mathematical framework for designing a low-loss, low-delay Internet. Networks and Spatial Economics 4:75101CrossRefGoogle Scholar
  19. March R, Riey G (2017) Analysis of a variational model for motion compensated inpainting. Inverse Probl Imag 11(6):997–1025CrossRefGoogle Scholar
  20. Peeta S, Ziliaskopoulos AK (2001) Foundations of dynamic traffic assignment: the past, the present and the future. Networks and Spatial Economics 1:233–265CrossRefGoogle Scholar
  21. Uras S, Girosi F, Verri A, Torre V (1988) Computational approach to motion perception. A Biol Cybern 60:79–87CrossRefGoogle Scholar
  22. Ziemer WP (1989) Weakly differentiable functions Springer - New YorkGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Istituto per le Applicazioni del CalcoloConsiglio Nazionale delle RicercheRomaItaly
  2. 2.Dipartimento di Matematica e InformaticaUniversità della CalabriaCosenzaItaly

Personalised recommendations