Calculus of Variations and Geometric Measure Theory

R. March - G. Riey

Euler equations and trace properties of minimizers of a functional for motion compensated inpainting

created by march on 16 Nov 2021
modified on 05 Oct 2022


Published Paper

Inserted: 16 nov 2021
Last Updated: 5 oct 2022

Journal: Inverse Problems and Imaging
Volume: 16
Number: 4
Pages: 703-737
Year: 2022
Doi: doi:10.3934/ipi.2021072


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 variation, 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.