Submodular relaxation for inference in Markov random fields

In this paper we address the problem of finding the most probable state of a discrete Markov random field (MRF), also known as the MRF energy minimization problem. The task is known to be NP-hard in general and its practical importance motivates numerous approximate algorithms. We propose a submodular relaxation approach (SMR) based on a Lagrangian relaxation of the initial problem. Unlike the dual decomposition approach of Komodakis et al., 2011 SMR does not decompose the graph structure of the initial problem but constructs a submodular energy that is minimized within the Lagrangian relaxation. Our approach is applicable to both pairwise and high-order MRFs and allows to take into account global potentials of certain types. We study theoretical properties of the proposed approach and evaluate it experimentally.Comment: This paper is accepted for publication in IEEE Transactions on Pattern Analysis and Machine Intelligenc

Blending Learning and Inference in Structured Prediction

In this paper we derive an efficient algorithm to learn the parameters of structured predictors in general graphical models. This algorithm blends the learning and inference tasks, which results in a significant speedup over traditional approaches, such as conditional random fields and structured support vector machines. For this purpose we utilize the structures of the predictors to describe a low dimensional structured prediction task which encourages local consistencies within the different structures while learning the parameters of the model. Convexity of the learning task provides the means to enforce the consistencies between the different parts. The inference-learning blending algorithm that we propose is guaranteed to converge to the optimum of the low dimensional primal and dual programs. Unlike many of the existing approaches, the inference-learning blending allows us to learn efficiently high-order graphical models, over regions of any size, and very large number of parameters. We demonstrate the effectiveness of our approach, while presenting state-of-the-art results in stereo estimation, semantic segmentation, shape reconstruction, and indoor scene understanding

Playing with Duality: An Overview of Recent Primal-Dual Approaches for Solving Large-Scale Optimization Problems

Optimization methods are at the core of many problems in signal/image processing, computer vision, and machine learning. For a long time, it has been recognized that looking at the dual of an optimization problem may drastically simplify its solution. Deriving efficient strategies which jointly brings into play the primal and the dual problems is however a more recent idea which has generated many important new contributions in the last years. These novel developments are grounded on recent advances in convex analysis, discrete optimization, parallel processing, and non-smooth optimization with emphasis on sparsity issues. In this paper, we aim at presenting the principles of primal-dual approaches, while giving an overview of numerical methods which have been proposed in different contexts. We show the benefits which can be drawn from primal-dual algorithms both for solving large-scale convex optimization problems and discrete ones, and we provide various application examples to illustrate their usefulness

Worst-case Optimal Submodular Extensions for Marginal Estimation

Submodular extensions of an energy function can be used to efficiently compute approximate marginals via variational inference. The accuracy of the marginals depends crucially on the quality of the submodular extension. To identify the best possible extension, we show an equivalence between the submodular extensions of the energy and the objective functions of linear programming (LP) relaxations for the corresponding MAP estimation problem. This allows us to (i) establish the worst-case optimality of the submodular extension for Potts model used in the literature; (ii) identify the worst-case optimal submodular extension for the more general class of metric labeling; and (iii) efficiently compute the marginals for the widely used dense CRF model with the help of a recently proposed Gaussian filtering method. Using synthetic and real data, we show that our approach provides comparable upper bounds on the log-partition function to those obtained using tree-reweighted message passing (TRW) in cases where the latter is computationally feasible. Importantly, unlike TRW, our approach provides the first practical algorithm to compute an upper bound on the dense CRF model.Comment: Accepted to AISTATS 201

Combining Features and Semantics for Low-level Computer Vision

Visual perception of depth and motion plays a significant role in understanding and navigating the environment. Reconstructing outdoor scenes in 3D and estimating the motion from video cameras are of utmost importance for applications like autonomous driving. The corresponding problems in computer vision have witnessed tremendous progress over the last decades, yet some aspects still remain challenging today. Striking examples are reflecting and textureless surfaces or large motions which cannot be easily recovered using traditional local methods. Further challenges include occlusions, large distortions and difficult lighting conditions. In this thesis, we propose to overcome these challenges by modeling non-local interactions leveraging semantics and contextual information. Firstly, for binocular stereo estimation, we propose to regularize over larger areas on the image using object-category specific disparity proposals which we sample using inverse graphics techniques based on a sparse disparity estimate and a semantic segmentation of the image. The disparity proposals encode the fact that objects of certain categories are not arbitrarily shaped but typically exhibit regular structures. We integrate them as non-local regularizer for the challenging object class 'car' into a superpixel-based graphical model and demonstrate its benefits especially in reflective regions. Secondly, for 3D reconstruction, we leverage the fact that the larger the reconstructed area, the more likely objects of similar type and shape will occur in the scene. This is particularly true for outdoor scenes where buildings and vehicles often suffer from missing texture or reflections, but share similarity in 3D shape. We take advantage of this shape similarity by localizing objects using detectors and jointly reconstructing them while learning a volumetric model of their shape. This allows to reduce noise while completing missing surfaces as objects of similar shape benefit from all observations for the respective category. Evaluations with respect to LIDAR ground-truth on a novel challenging suburban dataset show the advantages of modeling structural dependencies between objects. Finally, motivated by the success of deep learning techniques in matching problems, we present a method for learning context-aware features for solving optical flow using discrete optimization. Towards this goal, we present an efficient way of training a context network with a large receptive field size on top of a local network using dilated convolutions on patches. We perform feature matching by comparing each pixel in the reference image to every pixel in the target image, utilizing fast GPU matrix multiplication. The matching cost volume from the network's output forms the data term for discrete MAP inference in a pairwise Markov random field. Extensive evaluations reveal the importance of context for feature matching.Die visuelle Wahrnehmung von Tiefe und Bewegung spielt eine wichtige Rolle bei dem Verständnis und der Navigation in unserer Umwelt. Die 3D Rekonstruktion von Szenen im Freien und die Schätzung der Bewegung von Videokameras sind von größter Bedeutung für Anwendungen, wie das autonome Fahren. Die Erforschung der entsprechenden Probleme des maschinellen Sehens hat in den letzten Jahrzehnten enorme Fortschritte gemacht, jedoch bleiben einige Aspekte heute noch ungelöst. Beispiele hierfür sind reflektierende und texturlose Oberflächen oder große Bewegungen, bei denen herkömmliche lokale Methoden häufig scheitern. Weitere Herausforderungen sind niedrige Bildraten, Verdeckungen, große Verzerrungen und schwierige Lichtverhältnisse. In dieser Arbeit schlagen wir vor nicht-lokale Interaktionen zu modellieren, die semantische und kontextbezogene Informationen nutzen, um diese Herausforderungen zu meistern. Für die binokulare Stereo Schätzung schlagen wir zuallererst vor zusammenhängende Bereiche mit objektklassen-spezifischen Disparitäts Vorschlägen zu regularisieren, die wir mit inversen Grafik Techniken auf der Grundlage einer spärlichen Disparitätsschätzung und semantischen Segmentierung des Bildes erhalten. Die Disparitäts Vorschläge kodieren die Tatsache, dass die Gegenstände bestimmter Kategorien nicht willkürlich geformt sind, sondern typischerweise regelmäßige Strukturen aufweisen. Wir integrieren sie für die komplexe Objektklasse 'Auto' in Form eines nicht-lokalen Regularisierungsterm in ein Superpixel-basiertes grafisches Modell und zeigen die Vorteile vor allem in reflektierenden Bereichen. Zweitens nutzen wir für die 3D-Rekonstruktion die Tatsache, dass mit der Größe der rekonstruierten Fläche auch die Wahrscheinlichkeit steigt, Objekte von ähnlicher Art und Form in der Szene zu enthalten. Dies gilt besonders für Szenen im Freien, in denen Gebäude und Fahrzeuge oft vorkommen, die unter fehlender Textur oder Reflexionen leiden aber ähnlichkeit in der Form aufweisen. Wir nutzen diese ähnlichkeiten zur Lokalisierung von Objekten mit Detektoren und zur gemeinsamen Rekonstruktion indem ein volumetrisches Modell ihrer Form erlernt wird. Dies ermöglicht auftretendes Rauschen zu reduzieren, während fehlende Flächen vervollständigt werden, da Objekte ähnlicher Form von allen Beobachtungen der jeweiligen Kategorie profitieren. Die Evaluierung auf einem neuen, herausfordernden vorstädtischen Datensatz in Anbetracht von LIDAR-Entfernungsdaten zeigt die Vorteile der Modellierung von strukturellen Abhängigkeiten zwischen Objekten. Zuletzt, motiviert durch den Erfolg von Deep Learning Techniken bei der Mustererkennung, präsentieren wir eine Methode zum Erlernen von kontextbezogenen Merkmalen zur Lösung des optischen Flusses mittels diskreter Optimierung. Dazu stellen wir eine effiziente Methode vor um zusätzlich zu einem Lokalen Netzwerk ein Kontext-Netzwerk zu erlernen, das mit Hilfe von erweiterter Faltung auf Patches ein großes rezeptives Feld besitzt. Für das Feature Matching vergleichen wir mit schnellen GPU-Matrixmultiplikation jedes Pixel im Referenzbild mit jedem Pixel im Zielbild. Das aus dem Netzwerk resultierende Matching Kostenvolumen bildet den Datenterm für eine diskrete MAP Inferenz in einem paarweisen Markov Random Field. Eine umfangreiche Evaluierung zeigt die Relevanz des Kontextes für das Feature Matching