Detection of Intensity and Motion Edges within Optical Flow via Multidimensional Control

Abstract

In this paper, we propose a new optimization approach for the simultaneous computation of optical flow and edge detection therein. Instead of using an Ambrosio-Tortorelli type energy functional, we reformulate the optical flow problem as a multidimensional control problem. The optimal control problem is solved by discretization methods and large-scale optimization techniques. The edge detector can be immediately built from the control variables. We provide three series of numerical examples. The first shows that the mere presence of a gradient restriction has a regularizing effect, while the second demonstrates how to balance the regularizing effects of a term within the objective and the control restriction. The third series of numerical results is concerned with the direct evaluation of a TV-regularization term by introduction of control variables with sign restrictions.

A Guide to the TV Zoo

“…e.g. [43,76,195,196], inpainting (cf. [52,64,65,84]), and image decomposition into a cartoon and texture part (cf.…”

Quasiconvex relaxation of multidimensional control problems with integrandsf(t,ξ,v)

“…Then in analogy to [8,19], the simultaneous detection of the "discontinuities" of x (i.e. regions with large gradients ∇x 1 , ∇x 2 ) will be made possible where the indicator corresponds to the distance Dist Jx(t), ∂K .…”

On the lower semicontinuous quasiconvex envelope for unbounded integrands (I)

“…As yet, relaxation theorems for multidimensional control problems of this type have been obtained in the case of one variable only (n = 1) 5 . In contrast to this situation, the comprehension of control restrictions within problems of image processing with possibly nonconvex integrands 6 , as for example the smoothing of color image data or the determination of the optical flow 7 , requires the extension of the known results to the case n 2. Then in analogy to the multidimensional Calculus of Variations, one has to look for the quasiconvex relaxation of the problem instead of a convex one.…”

“…Following the remark ibid., p. 160, we may add norm balls to these bodies, thus arriving at the assertion. Proof of Theorem 5.7 (6). The restriction of the lower semicontinuous function f c to the face Φ is lower semicontinuous as well since Φ consists of accumulation points only (see [7], p. 136, Thm.…”