Properties of a Variational Model for Video Inpainting

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.