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doi: 10.3934/ipi.2021072
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## Euler equations and trace properties of minimizers of a functional for motion compensated inpainting

 1 Istituto per le Applicazioni del Calcolo, CNR, Via dei Taurini 19, 00185 Roma, Italy 2 Dipartimento di Matematica e Informatica - Università della Calabria, Via Pietro Bucci, 31B, 87036 Rende (CS), Italy

*Corresponding author: Giuseppe Riey

Received  October 2020 Revised  September 2021 Early access December 2021

Fund Project: G.R. has been supported by the Italian PRIN Research Project 2017 "Qualitative and quantitative aspects of nonlinear PDE"

We compute the Euler equations of a functional useful for simultaneous video inpainting and motion estimation, which was obtained in [17] as the relaxation of a modified version of the functional proposed in [16]. The functional is defined on vectorial functions of bounded variations, therefore we also get the Euler equations holding on the singular sets of minimizers, highlighting in particular the conditions on the jump sets. Such conditions are expressed by means of traces of geometrically meaningful vector fields and characterized as pointwise limits of averages on cylinders with axes parallel to the unit normals to the jump sets.

Citation: Riccardo March, Giuseppe Riey. Euler equations and trace properties of minimizers of a functional for motion compensated inpainting. Inverse Problems & Imaging, doi: 10.3934/ipi.2021072
##### References:
 [1] G. Alberti, Rank one property for derivatives of functions with bounded variation, Proc. Roy. Soc. Edinburgh Sect. A, 123 (1993), 239-274.  doi: 10.1017/S030821050002566X.  Google Scholar [2] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Clarendon Press - Oxford, 2000.  Google Scholar [3] G. Anzellotti, Pairings between measures and bounded functions and compensated compactness, Ann. Mat. Pura Appl., 135 (1983), 293-318.  doi: 10.1007/BF01781073.  Google Scholar [4] G. Anzellotti, The Euler equation for functionals with linear growth, Trans. Amer. Math. Soc., 290 (1985), 483-501.  doi: 10.1090/S0002-9947-1985-0792808-4.  Google Scholar [5] G. Anzellotti, On the minima of functionals with linear growth, Rend. Sem. Mat. Univ. Padova, 75 (1986), 91-110.   Google Scholar [6] G. Anzellotti, Traces of bounded vector-fields and the Divergence Theorem, Dipartimento di Matematica dell'Università di Trento, U.T.M., 131 (1983). Google Scholar [7] H. Attouch, G. Buttazzo and G. Michaille, Variational Analysis in Sobolev and BV Spaces, SIAM, 2006.  Google Scholar [8] G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations, 2$^nd$ edition, Springer, New York, 2006.  Google Scholar [9] T. Brox, A. Bruhn, N. Papenberg and J. Weickert, High accuracy optical flow estimation based on a theory for warping, Proceedings of the 8th European Conference on Computer Vision, (eds. T. Pajdla and J. Matas), 3024 (2004), 25–36. doi: 10.1007/978-3-540-24673-2_3.  Google Scholar [10] G. Buttazzo, Semicontinuity, Relaxation and Integral Representation in the Calculus of Variations, Pitman Research Notes in Mathematics Series, Vol. 207, Longman Scientific & Technical, UK, 1989.  Google Scholar [11] T. F. Chan, G. H. Golub and P. Mulet, A nonlinear primal-dual method for total variation-based image restoration, SIAM J. Sci. Comput., 20 (1999), 1964-1977.  doi: 10.1137/S1064827596299767.  Google Scholar [12] C. Goffman and J. Serrin, Sublinear functions of measures and variationals integrals, Duke Math. J., 31 (1964), 159-178.   Google Scholar [13] B. K. P. Horn and B. G. Schunck, Determining optical flow, Artificial Intelligence, 17 (1981), 185-203.  doi: 10.1016/0004-3702(81)90024-2.  Google Scholar [14] S. H. Keller, F. Lauze and M. Nielsen, Deintarlacing using variational methods, IEEE Trans. Image Proc., 17 (2008), 2015-2028.  doi: 10.1109/TIP.2008.2003394.  Google Scholar [15] S. H. Keller, F. Lauze and M. Nielsen, Video super-resolution using simultaneous motion and intensity calculations, IEEE Trans. Image Proc., 20 (2011), 1870-1884.  doi: 10.1109/TIP.2011.2106793.  Google Scholar [16] F. Lauze and M. Nielsen, A Variational algorithm for motion compensated inpainting, British Machine Vision Conference, (eds. S. Barman, A. Hoppe, T. Ellis), 2 (2004), 777–787. doi: 10.5244/C.18.80.  Google Scholar [17] R. March and G. Riey, Analysis of a variational model for motion compensated inpainting, Inverse Probl. Imaging, 11 (2017), 997-1025.  doi: 10.3934/ipi.2017046.  Google Scholar [18] R. March and G. Riey, Properties of a variational model for video inpainting, Networks and Spatial Economics, 2019 doi: 10.1007/s11067-019-09459-4.  Google Scholar [19] S. Uras, F. Girosi, A. Verri and V. Torre, A computational approach to motion perception, Biol. Cybern., 60 (1988), 79-87.  doi: 10.1007/BF00202895.  Google Scholar

show all references

##### References:
 [1] G. Alberti, Rank one property for derivatives of functions with bounded variation, Proc. Roy. Soc. Edinburgh Sect. A, 123 (1993), 239-274.  doi: 10.1017/S030821050002566X.  Google Scholar [2] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Clarendon Press - Oxford, 2000.  Google Scholar [3] G. Anzellotti, Pairings between measures and bounded functions and compensated compactness, Ann. Mat. Pura Appl., 135 (1983), 293-318.  doi: 10.1007/BF01781073.  Google Scholar [4] G. Anzellotti, The Euler equation for functionals with linear growth, Trans. Amer. Math. Soc., 290 (1985), 483-501.  doi: 10.1090/S0002-9947-1985-0792808-4.  Google Scholar [5] G. Anzellotti, On the minima of functionals with linear growth, Rend. Sem. Mat. Univ. Padova, 75 (1986), 91-110.   Google Scholar [6] G. Anzellotti, Traces of bounded vector-fields and the Divergence Theorem, Dipartimento di Matematica dell'Università di Trento, U.T.M., 131 (1983). Google Scholar [7] H. Attouch, G. Buttazzo and G. Michaille, Variational Analysis in Sobolev and BV Spaces, SIAM, 2006.  Google Scholar [8] G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations, 2$^nd$ edition, Springer, New York, 2006.  Google Scholar [9] T. Brox, A. Bruhn, N. Papenberg and J. Weickert, High accuracy optical flow estimation based on a theory for warping, Proceedings of the 8th European Conference on Computer Vision, (eds. T. Pajdla and J. Matas), 3024 (2004), 25–36. doi: 10.1007/978-3-540-24673-2_3.  Google Scholar [10] G. Buttazzo, Semicontinuity, Relaxation and Integral Representation in the Calculus of Variations, Pitman Research Notes in Mathematics Series, Vol. 207, Longman Scientific & Technical, UK, 1989.  Google Scholar [11] T. F. Chan, G. H. Golub and P. Mulet, A nonlinear primal-dual method for total variation-based image restoration, SIAM J. Sci. Comput., 20 (1999), 1964-1977.  doi: 10.1137/S1064827596299767.  Google Scholar [12] C. Goffman and J. Serrin, Sublinear functions of measures and variationals integrals, Duke Math. J., 31 (1964), 159-178.   Google Scholar [13] B. K. P. Horn and B. G. Schunck, Determining optical flow, Artificial Intelligence, 17 (1981), 185-203.  doi: 10.1016/0004-3702(81)90024-2.  Google Scholar [14] S. H. Keller, F. Lauze and M. Nielsen, Deintarlacing using variational methods, IEEE Trans. Image Proc., 17 (2008), 2015-2028.  doi: 10.1109/TIP.2008.2003394.  Google Scholar [15] S. H. Keller, F. Lauze and M. Nielsen, Video super-resolution using simultaneous motion and intensity calculations, IEEE Trans. Image Proc., 20 (2011), 1870-1884.  doi: 10.1109/TIP.2011.2106793.  Google Scholar [16] F. Lauze and M. Nielsen, A Variational algorithm for motion compensated inpainting, British Machine Vision Conference, (eds. S. Barman, A. Hoppe, T. Ellis), 2 (2004), 777–787. doi: 10.5244/C.18.80.  Google Scholar [17] R. March and G. Riey, Analysis of a variational model for motion compensated inpainting, Inverse Probl. Imaging, 11 (2017), 997-1025.  doi: 10.3934/ipi.2017046.  Google Scholar [18] R. March and G. Riey, Properties of a variational model for video inpainting, Networks and Spatial Economics, 2019 doi: 10.1007/s11067-019-09459-4.  Google Scholar [19] S. Uras, F. Girosi, A. Verri and V. Torre, A computational approach to motion perception, Biol. Cybern., 60 (1988), 79-87.  doi: 10.1007/BF00202895.  Google Scholar
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