Analysis of optimal control problems for the optical flow equation under mild regularity assumptions
Philipp Paul Jarde · mediaTUM – the media and publications repository of the Technical University Munich (Technical University Munich) · 2018
In many fields of image and video processing like image stabilization and video compression information about movements of image values are required.These are frequently given as an optical flow.The optical flow of an image sequence is identified as the velocity field of apparent points of movements of objects projected to the image plane.The determination of the optical flow is normally carried out by solving an optimization problem.In this thesis, we investigate an optimal control problem for a given image sequence with the transport equation as a side constraint which yields a time-continuous optical flow field of the image sequence as the optimal control.The corresponding optimal state then represents a time-continuous image interpolation of the sequence.For the transport equation we use results about well-posedness of Ambrosio for BV -regular vector fields.Furthermore, we improve existing stability results of solutions in the setting of spatial BV -regularity of the vector fields.With the aid of these results we show the existence of minima of the objective function under various regularization terms.In the second part of the thesis, we attend to differentiability of the problem.In a first step we show Fréchet differentiability of the control-to-state operator with BV -regular initial values of the transport equation.By smoothing this operator, Fréchet differentiability of the composition of the control-to-state operator with the tracking term of the objective function can be immediately proven.In a further proof, we show directly Fréchet differentiability of this composition under the requirement that the image sequence satisfies a certain condition.In the case that the condition is not fulfilled we are able to prove that the composition still possesses a one-sided directional derivative.At the end, we show two duality relations which are based on the adjoint equation of the transport equation on the one hand and on the backward transport equation on the other hand.With the aid of these results we find two different representations of the gradient of the composition.The thesis finally ends with necessary optimality conditions of first order.