Image labeling and grouping by minimizing linear functionals over cones

We consider energy minimization problems related to image labeling, partitioning, and grouping, which typically show up at mid-level stages of computer vision systems. A common feature of these problems is their intrinsic combinatorial complexity from an optimization pointof-view. Rather than trying to compute the global minimum - a goal we consider as elusive in these cases - we wish to design optimization approaches which exhibit two relevant properties: First, in each application a solution with guaranteed degree of suboptimality can be computed. Secondly, the computations are based on clearly defined algorithms which do not comprise any (hidden) tuning parameters. In this paper, we focus on the second property and introduce a novel and general optimization technique to the field of computer vision which amounts to compute a sub optimal solution by just solving a convex optimization problem. As representative examples, we consider two binary quadratic energy functionals related to image labeling and perceptual grouping. Both problems can be considered as instances of a general quadratic functional in binary variables, which is embedded into a higher-dimensional space such that sub optimal solutions can be computed as minima of linear functionals over cones in that space (semidefinite programs). Extensive numerical results reveal that, on the average, sub optimal solutions can be computed which yield a gap below 5% with respect to the global optimum in case where this is known

Image Partitioning based on Semidefinite Programming

Many tasks in computer vision lead to combinatorial optimization problems. Automatic image partitioning is one of the most important examples in this context: whether based on some prior knowledge or completely unsupervised, we wish to find coherent parts of the image. However, the inherent combinatorial complexity of such problems often prevents to find the global optimum in polynomial time. For this reason, various approaches have been proposed to find good approximative solutions for image partitioning problems. As an important example, we will first consider different spectral relaxation techniques: based on straightforward eigenvector calculations, these methods compute suboptimal solutions in short time. However, the main contribution of this thesis is to introduce a novel optimization technique for discrete image partitioning problems which is based on a semidefinite programming relaxation. In contrast to approximation methods employing annealing algorithms, this approach involves solving a convex optimization problem, which does not suffer from possible local minima. Using interior point techniques, the solution of the relaxation can be found in polynomial time, and without elaborate parameter tuning. High quality solutions to the original combinatorial problem are then obtained with a randomized rounding technique. The only potential drawback of the semidefinite relaxation approach is that the number of variables of the optimization problem is squared. Nevertheless, it can still be applied to problems with up to a few thousand variables, as is demonstrated for various computer vision tasks including unsupervised segmentation, perceptual grouping and image restoration. Concerning problems of higher dimensionality, we study two different approaches to effectively reduce the number of variables. The first one is based on probabilistic sampling: by considering only a small random fraction of the pixels in the image, our semidefinite relaxation method can be applied in an efficient way while maintaining a reliable quality of the resulting segmentations. The second approach reduces the problem size by computing an over-segmentation of the image in a preprocessing step. After that, the image is partitioned based on the resulting "superpixels" instead of the original pixels. Since the real world does not consist of pixels, it can even be argued that this is the more natural image representation. Initially, our semidefinite relaxation method is defined only for binary partitioning problems. To derive image segmentations into multiple parts, one possibility is to apply the binary approach in a hierarchical way. Besides this natural extension, we also discuss how multiclass partitioning problems can be solved in a direct way based on semidefinite relaxation techniques

Automatische bildbasierte Segmentierung organischer Objekte einer gleichartigen Gruppe: Abgeleitet vom Problem der Stammschnittflächensegmentierung

Diese Arbeit adressiert die automatische bildbasierte Segmentierung von organo-Gruppen, Gruppen gleichartiger organischer Objekte. Die Segmentierung einer organo-Gruppe ermöglicht Anwendungen zur automatischen Vermessung, Inspektion oder Sortierung. In dieser Arbeit werden, ausgehend vom Problem der Stammschnittflächen, drei Segmentierungskonzepte entwickelt und quantitativ evaluiert. Ausgehend von den Konzepten wird eine allgemeinere Lösung für organo-Gruppen entwickelt und am Beispiel von Plattfischen, Kartoffeln und Äpfeln evaluiert, wobei gute bis sehr gute Ergebnisse erzielt werden