Further results on snakes in powers of complete graphs

AbstractBy a snake in a finite graph G is meant a cycle without chords. Denoted by S(G) the length of a longest snake in G. In this paper we obtain a new lower bound for S(G) in the case where G is the product of d copies of the complete graph on n vertices

The Brownian map is the scaling limit of uniform random plane quadrangulations

76 pages, 7 figures, improved versionWe prove that uniform random quadrangulations of the sphere with n faces, endowed with the usual graph distance and renormalized by n −1/4 , converge as n → ∞ in distribution for the Gromov-Hausdorff topology to a limiting metric space. We validate a conjecture by Le Gall, by showing that the limit is (up to a scale constant) the so-called Brownian map, which was introduced by Marckert & Mokkadem and Le Gall as the most natural candidate for the scaling limit of many models of random plane maps. The proof relies strongly on the concept of geodesic stars in the map, which are configurations made of several geodesics that only share a common endpoint and do not meet elsewhere

An extensive English language bibliography on graph theory and its applications, supplement 1

Graph theory and its applications - bibliography, supplement

Mathematical Formula Recognition and Automatic Detection and Translation of Algorithmic Components into Stochastic Petri Nets in Scientific Documents

A great percentage of documents in scientific and engineering disciplines include mathematical formulas and/or algorithms. Exploring the mathematical formulas in the technical documents, we focused on the mathematical operations associations, their syntactical correctness, and the association of these components into attributed graphs and Stochastic Petri Nets (SPN). We also introduce a formal language to generate mathematical formulas and evaluate their syntactical correctness. The main contribution of this work focuses on the automatic segmentation of mathematical documents for the parsing and analysis of detected algorithmic components. To achieve this, we present a synergy of methods, such as string parsing according to mathematical rules, Formal Language Modeling, optical analysis of technical documents in forms of images, structural analysis of text in images, and graph and Stochastic Petri Net mapping. Finally, for the recognition of the algorithms, we enriched our rule based model with machine learning techniques to acquire better results

Combinatorial Solutions for Shape Optimization in Computer Vision

This thesis aims at solving so-called shape optimization problems, i.e. problems where the shape of some real-world entity is sought, by applying combinatorial algorithms. I present several advances in this field, all of them based on energy minimization. The addressed problems will become more intricate in the course of the thesis, starting from problems that are solved globally, then turning to problems where so far no global solutions are known. The first two chapters treat segmentation problems where the considered grouping criterion is directly derived from the image data. That is, the respective data terms do not involve any parameters to estimate. These problems will be solved globally. The first of these chapters treats the problem of unsupervised image segmentation where apart from the image there is no other user input. Here I will focus on a contour-based method and show how to integrate curvature regularity into a ratio-based optimization framework. The arising optimization problem is reduced to optimizing over the cycles in a product graph. This problem can be solved globally in polynomial, effectively linear time. As a consequence, the method does not depend on initialization and translational invariance is achieved. This is joint work with Daniel Cremers and Simon Masnou. I will then proceed to the integration of shape knowledge into the framework, while keeping translational invariance. This problem is again reduced to cycle-finding in a product graph. Being based on the alignment of shape points, the method actually uses a more sophisticated shape measure than most local approaches and still provides global optima. It readily extends to tracking problems and allows to solve some of them in real-time. I will present an extension to highly deformable shape models which can be included in the global optimization framework. This method simultaneously allows to decompose a shape into a set of deformable parts, based only on the input images. This is joint work with Daniel Cremers. In the second part segmentation is combined with so-called correspondence problems, i.e. the underlying grouping criterion is now based on correspondences that have to be inferred simultaneously. That is, in addition to inferring the shapes of objects, one now also tries to put into correspondence the points in several images. The arising problems become more intricate and are no longer optimized globally. This part is divided into two chapters. The first chapter treats the topic of real-time motion segmentation where objects are identified based on the observations that the respective points in the video will move coherently. Rather than pre-estimating motion, a single energy functional is minimized via alternating optimization. The main novelty lies in the real-time capability, which is achieved by exploiting a fast combinatorial segmentation algorithm. The results are furthermore improved by employing a probabilistic data term. This is joint work with Daniel Cremers. The final chapter presents a method for high resolution motion layer decomposition and was developed in combination with Daniel Cremers and Thomas Pock. Layer decomposition methods support the notion of a scene model, which allows to model occlusion and enforce temporal consistency. The contributions are twofold: from a practical point of view the proposed method allows to recover fine-detailed layer images by minimizing a single energy. This is achieved by integrating a super-resolution method into the layer decomposition framework. From a theoretical viewpoint the proposed method introduces layer-based regularity terms as well as a graph cut-based scheme to solve for the layer domains. The latter is combined with powerful continuous convex optimization techniques into an alternating minimization scheme. Lastly I want to mention that a significant part of this thesis is devoted to the recent trend of exploiting parallel architectures, in particular graphics cards: many combinatorial algorithms are easily parallelized. In Chapter 3 we will see a case where the standard algorithm is hard to parallelize, but easy for the respective problem instances

Methods for Learning Structured Prediction in Semantic Segmentation of Natural Images

Automatic segmentation and recognition of semantic classes in natural images is an important open problem in computer vision. In this work, we investigate three different approaches to recognition: without supervision, with supervision on level of images, and with supervision on the level of pixels. The thesis comprises three parts. The first part introduces a clustering algorithm that optimizes a novel information-theoretic objective function. We show that the proposed algorithm has clear advantages over standard algorithms from the literature on a wide array of datasets. Clustering algorithms are an important building block for higher-level computer vision applications, in particular for semantic segmentation. The second part of this work proposes an algorithm for automatic segmentation and recognition of object classes in natural images, that learns a segmentation model solely from annotation in the form of presence and absence of object classes in images. The third and main part of this work investigates one of the most popular approaches to the task of object class segmentation and semantic segmentation, based on conditional random fields and structured prediction. We investigate several learning algorithms, in particular in combination with approximate inference procedures. We show how structured models for image segmentation can be learned exactly in practical settings, even in the presence of many loops in the underlying neighborhood graphs. The introduced methods provide results advancing the state-of-the-art on two complex benchmark datasets for semantic segmentation, the MSRC-21 Dataset of RGB images and the NYU V2 Dataset or RGB-D images of indoor scenes. Finally, we introduce a software library that al- lows us to perform extensive empirical comparisons of state-of-the-art structured learning approaches. This allows us to characterize their practical properties in a range of applications, in particular for semantic segmentation and object class segmentation.Methoden zum Lernen von Strukturierter Vorhersage in Semantischer Segmentierung von Natürlichen Bildern Automatische Segmentierung und Erkennung von semantischen Klassen in natür- lichen Bildern ist ein wichtiges offenes Problem des maschinellen Sehens. In dieser Arbeit untersuchen wir drei möglichen Ansätze der Erkennung: ohne Überwachung, mit Überwachung auf Ebene von Bildern und mit Überwachung auf Ebene von Pixeln. Diese Arbeit setzt sich aus drei Teilen zusammen. Im ersten Teil der Arbeit schlagen wir einen Clustering-Algorithmus vor, der eine neuartige, informationstheoretische Zielfunktion optimiert. Wir zeigen, dass der vorgestellte Algorithmus üblichen Standardverfahren aus der Literatur gegenüber klare Vorteile auf vielen verschiedenen Datensätzen hat. Clustering ist ein wichtiger Baustein in vielen Applikationen des machinellen Sehens, insbesondere in der automatischen Segmentierung. Der zweite Teil dieser Arbeit stellt ein Verfahren zur automatischen Segmentierung und Erkennung von Objektklassen in natürlichen Bildern vor, das mit Hilfe von Supervision in Form von Klassen-Vorkommen auf Bildern in der Lage ist ein Segmentierungsmodell zu lernen. Der dritte Teil der Arbeit untersucht einen der am weitesten verbreiteten Ansätze zur semantischen Segmentierung und Objektklassensegmentierung, Conditional Random Fields, verbunden mit Verfahren der strukturierten Vorhersage. Wir untersuchen verschiedene Lernalgorithmen des strukturierten Lernens, insbesondere im Zusammenhang mit approximativer Vorhersage. Wir zeigen, dass es möglich ist trotz des Vorhandenseins von Kreisen in den betrachteten Nachbarschaftsgraphen exakte strukturierte Modelle zur Bildsegmentierung zu lernen. Mit den vorgestellten Methoden bringen wir den Stand der Kunst auf zwei komplexen Datensätzen zur semantischen Segmentierung voran, dem MSRC-21 Datensatz von RGB-Bildern und dem NYU V2 Datensatz von RGB-D Bildern von Innenraum-Szenen. Wir stellen außerdem eine Software-Bibliothek vor, die es erlaubt einen weitreichenden Vergleich der besten Lernverfahren für strukturiertes Lernen durchzuführen. Unsere Studie erlaubt uns eine Charakterisierung der betrachteten Algorithmen in einer Reihe von Anwendungen, insbesondere der semantischen Segmentierung und Objektklassensegmentierung