Long term motion analysis for object level grouping and nonsmooth optimization methods = Langzeitanalyse von Bewegungen zur objektorientierten Gruppierung und nichglatte Optimierungsmethoden

This work deals with theoretical and practical aspects of convex and nonconvex optimization algorithms for several classes of problems. They are applied to several (low-level) computer vision tasks. The optical flow motion estimation problem, which is among them, is a potential source for improving motion based segmentation methods. The second, more practical part of this work focuses on such an object-level motion segmentation method for videos. The part about convex optimization extends the analysis of the so-called “Heavy-ball method” for smooth, strongly convex objective functions to nonsmooth, strongly convex problems. The algorithm that is developed is motivated by the extension of the gradient descent method to forward–backward splitting methods. Its optimality in the sense of rates of convergence for the considered class of problems is proved. We termed the algorithm “iPiasco”. It extends the optimal rate of convergence that is known from the Heavy-ball method to the generalized setting. The iteration structure of iPiasco, i.e., the way the next iterate is computed, even generalizes to a certain class of nonconvex optimization problems. This opportunity is used to introduce “iPiano”, which is an algorithm that is applicable to the sum of a nonsmooth, nonconvex function and a smooth, nonconvex function. Since convergence rates in the nonconvex setting are hard to find, the focus of this research is to proof convergence. Key for the convergence analysis is the so-called Kurdyka–Łojasiewicz (KL) property. Convergence is obtained by extending a preexisting convergence result for abstract descent methods to fit the setting of iPiano. As the KL property is a rather weak assumption—many functions emerging in computer vision are shown to have this property—iPiano is a promising algorithm for a wealth of problems. Although iPiano is an efficient algorithm that is proved to converge, it is limited to composite functions where one part is differentiable. In order to deal with hard, nondifferentiable objective functions the “IRconvex” algorithm is introduced. It belongs to the class of majorization minimization (MM) algorithms. For several instances of problems explicit convex majorizers are constructed and minimized. This leads to well-known algorithms like the iteratively reweighted (IR) least squares (IRLS) or the IR `1 (IRL1) algorithm, but also allows for new algorithm like the IR tight convex (IRTight) or the IR Huber algorithm (IRHuber). In the unified, much more general framework of MM algorithms, convergence is proved under some mild assumption, among them the KL property. An extensive numerical analysis of IRconvex algorithms shows that it is particularly efficient for nonsmooth optimization problems, whereas iPiano should (in most cases) be preferred when a part of the function is smooth. The part of the thesis about optimization attaches importance to a thorough introduction of basic results from convex and nonconvex analysis and a recapitulation of the most prominent convex optimization algorithms (for computer vision applications). IRconvex is applied to the optical flow problem in computer vision. It shows some favorable properties, however further investigation is required to benefit in practice. There is potential to improve optical flow and therefore also many motion based algorithms. The second part of the thesis is devoted to motion