A Mathematical Study of the Relaxed Optical Flow Problem in the Space $BV (\Omega)$

Abstract

This paper describes a variational approach for estimating a discontinuous optical ow from a sequence of images. De ned as the apparent motion of the image brightness pattern, the optical ow is very important in the computer vision community where its accurate estimation is strongly needed. After a fast overview of existing methods, we present a new variational method that we study in the space of Bounded Variations. We rst present an integral representation of the optical ow problem which appears to be not lower semicontinuous. The relaxed functional is then calculated. We conclude by challenging questions about the possible numerical analysis of the abstract results.

Pixelwise-adaptive blind optical flow assuming nonstationary statistics

“…The first way is by using a robust approach, i.e., minimizing an adaptively truncated quadratic cost function. 2 And the second way is implicit, by the very fact that we do not use an area-based regression. More specifically, we have a pointwise regularization, where the level of regularization is controlled at any pixel entirely by the data.…”

“…If within the local neighborhood (aperture), where the information is combined, the image intensity variations are of lower dimensional nature, then the information in that local neighborhood cannot be used to disambiguate correctly the motion components. One systematic solution to this problem is the multiscale approach [1], [2], [26], [57], [69].…”

A Phase Field Method for Joint Denoising, Edge Detection, and Motion Estimation in Image Sequence Processing

“…A rigorous validation of this limit behavior in terms of Γ-convergence is still open. For results on Γ-convergence for the optical flow problem in the context of TV type models we refer to [4,22].…”

“…First advances in this direction were investigated in [33,27,28,9,23,30]. In particular, Kornprobst et al [22,3,4] have considered piecewise smooth motion patterns on image sequences characterized by piecewise smooth objects. Their results are phrased rigorously on the space of functions of bounded variation (BV), and they propose suitable approximations for the numerical implementation.…”

“…Already, in [22] a joint approach for the segmentation of moving objects in front of a still background and the computation of motion velocities has been proposed. For a given intensity function on an image sequence, a total variation (TV) type functional for the motion field-which allows for jumps in the optical flow velocity-is analyzed in [4,3]. Recently, Papenberg et al [29] considered another TV regularization of the motion field and optical flow constraints involving higher order gradients.…”