We compute the Euler equations of a functional useful for simultaneous video inpainting and motion estimation, which was obtained in [
Citation: |
[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.![]() ![]() ![]() |
[2] |
L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Clarendon Press - Oxford, 2000.
![]() ![]() |
[3] |
G. Anzellotti, Pairings between measures and bounded functions and compensated compactness, Ann. Mat. Pura Appl., 135 (1983), 293-318.
doi: 10.1007/BF01781073.![]() ![]() ![]() |
[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.![]() ![]() ![]() |
[5] |
G. Anzellotti, On the minima of functionals with linear growth, Rend. Sem. Mat. Univ. Padova, 75 (1986), 91-110.
![]() ![]() |
[6] |
G. Anzellotti, Traces of bounded vector-fields and the Divergence Theorem, Dipartimento di Matematica dell'Università di Trento, U.T.M., 131 (1983).
![]() |
[7] |
H. Attouch, G. Buttazzo and G. Michaille, Variational Analysis in Sobolev and BV Spaces, SIAM, 2006.
![]() ![]() |
[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.
![]() ![]() |
[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.![]() ![]() |
[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.
![]() ![]() |
[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.![]() ![]() ![]() |
[12] |
C. Goffman and J. Serrin, Sublinear functions of measures and variationals integrals, Duke Math. J., 31 (1964), 159-178.
![]() ![]() |
[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.![]() ![]() |
[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.![]() ![]() ![]() |
[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.![]() ![]() ![]() |
[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.![]() ![]() |
[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.![]() ![]() ![]() |
[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.![]() ![]() |
[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.![]() ![]() |