Relative Convex Hull Determination from Convex Hulls in the Plane

A new algorithm for the determination of the relative convex hull in the plane of a simple polygon A with respect to another simple polygon B which contains A, is proposed. The relative convex hull is also known as geodesic convex hull, and the problem of its determination in the plane is equivalent to find the shortest curve among all Jordan curves lying in the difference set of B and A and encircling A. Algorithms solving this problem known from Computational Geometry are based on the triangulation or similar decomposition of that difference set. The algorithm presented here does not use such decomposition, but it supposes that A and B are given as ordered sequences of vertices. The algorithm is based on convex hull calculations of A and B and of smaller polygons and polylines, it produces the output list of vertices of the relative convex hull from the sequence of vertices of the convex hull of A.Comment: 15 pages, 4 figures, Conference paper published. We corrected two typing errors in Definition 2: $I_S$ has to be defined based on $O_S$, and $I_E$ has to be defined based on $O_E$ (not just using $O$). These errors appeared in the text of the original conference paper, which also contained the pseudocode of an algorithm where $I_S$ and $I_E$ appeared as correctly define

Marker-free human motion capture in dynamic cluttered environments from a single view-point

Human Motion Capture is a widely used technique to obtain motion data for animation of virtual characters. Commercial optical motion capture systems are marker-based. This thesis is about marker-free motion capture. The pose and motion estimation of an observed person is carried out in an optimization framework for articulated objects. The motion function is formulated with kinematic chains consisting of rotations around arbitrary axes in 3D space. This formulation leads to a Nonlinear Least Squares problem, which is solved with gradient-based methods. With the formulation in this thesis the necessary derivatives can be derived analytically. This speeds up processing and increases accuracy. Different gradient based methods are compared to solve the Nonlinear Least Squares problem, which allows the integration of second order motion derivatives as well. The pose estimation requires correspondences between known model of the person and observed data. To obtain this model, a new method is developed, which fits a template model to a specific person from 6 posture images taken by a single camera. Various types of correspondences are integrated in the optimization simultaneously without making approximations to the motion or optimization function, namely 3D-3D correspondences from stereo algorithms and 3D-2D correspondences from image silhouettes and 2D point tracking. Of major importance for the developed methods is the processing time and robustness to cluttered and dynamic background. Experiments show, that complex motion with 24 degrees of freedom is track-able from a single stereo view until body parts get totally occluded. Further methods are developed to estimate pose from a single camera view with cluttered dynamic background. Similar to other work on 2D-3D pose estimation, correspondences between model and image silhouette of the person are established by analyzing the gray value gradient near the predicted model silhouette. To increase the accuracy of silhouette correspondences, color histograms for each body part are combined with image gradient search. The combination of 3D depth data and 2D image data is tested with depth data from a PMD camera (Photonic Mixer Device), which measures the depth to scene points by the time of flight of ligh

Learning Non-rigid, 3D Shape Variations using Statistical, Physical and Geometric Models

3D shape modelling is a fundamental component in computer vision and computer graphics. Applications include shape interpolation and extrapolation, shape reconstruction, motion capture and mesh editing, etc. By “modelling” we mean the process of learning a parameter-driven model. This thesis focused on the scope of statistical modelling for 3D non-rigid shapes, such as human faces and bodies. The problem is challenging due to highly non-linear deformations, high dimensionality, and data sparsity. Several new algorithms are proposed for 3D shape modelling, 3D shape matching (computing dense correspondence) and applications. First, we propose a variant of Principal Component Analysis called “Shell PCA” which provides a physically-inspired statistical shape model. This is our first attempt to use a physically plausible metric (specifically, the discrete shell model) for statistical shape modelling. Second, we further develop this line of work into a fully Riemannian approach called “Shell PGA”. We demonstrate how to perform Principal Geodesic Analysis in the space of discrete shells. To achieve this, we present an alternate formulation of PGA which avoids working in the tangent space and deals with shapes lying on the manifold directly. Unlike displacement-based methods, Shell PGA is invariant to rigid body motion, and therefore alignment preprocessing such as Procrustes analysis is not needed. Third, we propose a groupwise shape matching method using functional map representation. Targeting at near-isometric deformations, we consider groupwise optimisation of consistent functional maps over a product of Stiefel manifolds, and optimise over a minimal subset of the transformations for efficiency. Last, we show that our proposed shape model achieves state-of-the-art performance in two very challenging applications: handle-based mesh editing, and model fitting using motion capture data. We also contribute a new algorithm for human body shape estimation using clothed scan sequence, along with a new dataset “BUFF” for evaluation

Geometric uncertainty models for correspondence problems in digital image processing

Many recent advances in technology rely heavily on the correct interpretation of an enormous amount of visual information. All available sources of visual data (e.g. cameras in surveillance networks, smartphones, game consoles) must be adequately processed to retrieve the most interesting user information. Therefore, computer vision and image processing techniques gain significant interest at the moment, and will do so in the near future. Most commonly applied image processing algorithms require a reliable solution for correspondence problems. The solution involves, first, the localization of corresponding points -visualizing the same 3D point in the observed scene- in the different images of distinct sources, and second, the computation of consistent geometric transformations relating correspondences on scene objects. This PhD presents a theoretical framework for solving correspondence problems with geometric features (such as points and straight lines) representing rigid objects in image sequences of complex scenes with static and dynamic cameras. The research focuses on localization uncertainty due to errors in feature detection and measurement, and its effect on each step in the solution of a correspondence problem. Whereas most other recent methods apply statistical-based models for spatial localization uncertainty, this work considers a novel geometric approach. Localization uncertainty is modeled as a convex polygonal region in the image space. This model can be efficiently propagated throughout the correspondence finding procedure. It allows for an easy extension toward transformation uncertainty models, and to infer confidence measures to verify the reliability of the outcome in the correspondence framework. Our procedure aims at finding reliable consistent transformations in sets of few and ill-localized features, possibly containing a large fraction of false candidate correspondences. The evaluation of the proposed procedure in practical correspondence problems shows that correct consistent correspondence sets are returned in over 95% of the experiments for small sets of 10-40 features contaminated with up to 400% of false positives and 40% of false negatives. The presented techniques prove to be beneficial in typical image processing applications, such as image registration and rigid object tracking

Methods for 3D Geometry Processing in the Cultural Heritage Domain

This thesis presents methods for 3D geometry processing under the aspects of cultural heritage applications. After a short overview over the relevant basics in 3D geometry processing, the present thesis investigates the digital acquisition of 3D models. A particular challenge in this context are on the one hand difficult surface or material properties of the model to be captured. On the other hand, the fully automatic reconstruction of models even with suitable surface properties that can be captured with Laser range scanners is not yet completely solved. This thesis presents two approaches to tackle these challenges. One exploits a thorough capture of the object’s appearance and a coarse reconstruction for a concise and realistic object representation even for objects with problematic surface properties like reflectivity and transparency. The other method concentrates on digitisation via Laser-range scanners and exploits 2D colour images that are typically recorded with the range images for a fully automatic registration technique. After reconstruction, the captured models are often still incomplete, exhibit holes and/or regions of insufficient sampling. In addition to that, holes are often deliberately introduced into a registered model to remove some undesired or defective surface part. In order to produce a visually appealing model, for instance for visualisation purposes, for prototype or replica production, these holes have to be detected and filled. Although completion is a well-established research field in 2D image processing and many approaches do exist for image completion, surface completion in 3D is a fairly new field of research. This thesis presents a hierarchical completion approach that employs and extends successful exemplar-based 2D image processing approaches to 3D and fills in detail-equipped surface patches into missing surface regions. In order to identify and construct suitable surface patches, selfsimilarity and coherence properties of the surface context of the hole are exploited. In addition to the reconstruction and repair, the present thesis also investigates methods for a modification of captured models via interactive modelling. In this context, modelling is regarded as a creative process, for instance for animation purposes. On the other hand, it is also demonstrated how this creative process can be used to introduce human expertise into the otherwise automatic completion process. This way, reconstructions are feasible even of objects where already the data source, the object itself, is incomplete due to corrosion, demolition, or decay.Methoden zur 3D-Geometrieverarbeitung im Kulturerbesektor In dieser Arbeit werden Methoden zur Bearbeitung von digitaler 3D-Geometrie unter besonderer Berücksichtigung des Anwendungsbereichs im Kulturerbesektor vorgestellt. Nach einem kurzen Überblick über die relevanten Grundlagen der dreidimensionalen Geometriebehandlung wird zunächst die digitale Akquise von dreidimensionalen Objekten untersucht. Eine besondere Herausforderung stellen bei der Erfassung einerseits ungünstige Oberflächen- oder Materialeigenschaften der Objekte dar (wie z.B. Reflexivität oder Transparenz), andererseits ist auch die vollautomatische Rekonstruktion von solchen Modellen, die sich verhältnismäßig problemlos mit Laser-Range Scannern erfassen lassen, immer noch nicht vollständig gelöst. Daher bilden zwei neuartige Verfahren, die diesen Herausforderungen begegnen, den Anfang. Auch nach der Registrierung sind die erfassten Datensätze in vielen Fällen unvollständig, weisen Löcher oder nicht ausreichend abgetastete Regionen auf. Darüber hinaus werden in vielen Anwendungen auch, z.B. durch Entfernen unerwünschter Oberflächenregionen, Löcher gewollt hinzugefügt. Für eine optisch ansprechende Rekonstruktion, vor allem zu Visualisierungszwecken, im Bildungs- oder Unterhaltungssektor oder zur Prototyp- und Replik-Erzeugung müssen diese Löcher zunächst automatisch detektiert und anschließend geschlossen werden. Obwohl dies im zweidimensionalen Fall der Bildbearbeitung bereits ein gut untersuchtes Forschungsfeld darstellt und vielfältige Ansätze zur automatischen Bildvervollständigung existieren, ist die Lage im dreidimensionalen Fall anders, und die Übertragung von zweidimensionalen Ansätzen in den 3D stellt vielfach eine große Herausforderung dar, die bislang keine zufriedenstellenden Lösungen erlaubt hat. Nichtsdestoweniger wird in dieser Arbeit ein hierarchisches Verfahren vorgestellt, das beispielbasierte Konzepte aus dem 2D aufgreift und Löcher in Oberflächen im 3D unter Ausnutzung von Selbstähnlichkeiten und Kohärenzeigenschaften des Oberflächenkontextes schließt. Um plausible Oberflächen zu erzeugen werden die Löcher dabei nicht nur glatt gefüllt, sondern auch feinere Details aus dem Kontext rekonstruiert. Abschließend untersucht die vorliegende Arbeit noch die Modifikation der vervollständigten Objekte durch Freiformmodellierung. Dies wird dabei zum einen als kreativer Prozess z.B. zu Animationszwecken betrachtet. Zum anderen wird aber auch untersucht, wie dieser kreative Prozess benutzt werden kann, um etwaig vorhandenes Expertenwissen in die ansonsten automatische Vervollständigung mit einfließen zu lassen. Auf diese Weise werden auch Rekonstruktionen ermöglicht von Objekten, bei denen schon die Datenquelle, also das Objekt selbst z.B. durch Korrosion oder mutwillige Zerstörung unvollständig ist

Inferring Human Pose and Motion from Images

As optical gesture recognition technology advances, touchless human computer interfaces of the future will soon become a reality. One particular technology, markerless motion capture, has gained a large amount of attention, with widespread application in diverse disciplines, including medical science, sports analysis, advanced user interfaces, and virtual arts. However, the complexity of human anatomy makes markerless motion capture a non-trivial problem: I) parameterised pose configuration exhibits high dimensionality, and II) there is considerable ambiguity in surjective inverse mapping from observation to pose configuration spaces with a limited number of camera views. These factors together lead to multimodality in high dimensional space, making markerless motion capture an ill-posed problem. This study challenges these difficulties by introducing a new framework. It begins with automatically modelling specific subject template models and calibrating posture at the initial stage. Subsequent tracking is accomplished by embedding naturally-inspired global optimisation into the sequential Bayesian filtering framework. Tracking is enhanced by several robust evaluation improvements. Sparsity of images is managed by compressive evaluation, further accelerating computational efficiency in high dimensional space

Advanced methods for relightable scene representations in image space

The realistic reproduction of visual appearance of real-world objects requires accurate computer graphics models that describe the optical interaction of a scene with its surroundings. Data-driven approaches that model the scene globally as a reflectance field function in eight parameters deliver high quality and work for most material combinations, but are costly to acquire and store. Image-space relighting, which constrains the application to create photos with a virtual, fix camera in freely chosen illumination, requires only a 4D data structure to provide full fidelity. This thesis contributes to image-space relighting on four accounts: (1) We investigate the acquisition of 4D reflectance fields in the context of sampling and propose a practical setup for pre-filtering of reflectance data during recording, and apply it in an adaptive sampling scheme. (2) We introduce a feature-driven image synthesis algorithm for the interpolation of coarsely sampled reflectance data in software to achieve highly realistic images. (3) We propose an implicit reflectance data representation, which uses a Bayesian approach to relight complex scenes from the example of much simpler reference objects. (4) Finally, we construct novel, passive devices out of optical components that render reflectance field data in real-time, shaping the incident illumination into the desired imageDie realistische Wiedergabe der visuellen Erscheinung einer realen Szene setzt genaue Modelle aus der Computergraphik für die Interaktion der Szene mit ihrer Umgebung voraus. Globale Ansätze, die das Verhalten der Szene insgesamt als Reflektanzfeldfunktion in acht Parametern modellieren, liefern hohe Qualität für viele Materialtypen, sind aber teuer aufzuzeichnen und zu speichern. Verfahren zur Neubeleuchtung im Bildraum schränken die Anwendbarkeit auf fest gewählte Kameras ein, ermöglichen aber die freie Wahl der Beleuchtung, und erfordern dadurch lediglich eine 4D - Datenstruktur für volle Wiedergabetreue. Diese Arbeit enthält vier Beiträge zu diesem Thema: (1) wir untersuchen die Aufzeichnung von 4D Reflektanzfeldern im Kontext der Abtasttheorie und schlagen einen praktischen Aufbau vor, der Reflektanzdaten bereits während der Messung vorfiltert. Wir verwenden ihn in einem adaptiven Abtastschema. (2) Wir führen einen merkmalgesteuerten Bildsynthesealgorithmus für die Interpolation von grob abgetasteten Reflektanzdaten ein. (3) Wir schlagen eine implizite Beschreibung von Reflektanzdaten vor, die mit einem Bayesschen Ansatz komplexe Szenen anhand des Beispiels eines viel einfacheren Referenzobjektes neu beleuchtet. (4) Unter der Verwendung optischer Komponenten schaffen wir passive Aufbauten zur Darstellung von Reflektanzfeldern in Echtzeit, indem wir einfallende Beleuchtung direkt in das gewünschte Bild umwandeln

Collection of abstracts of the 24th European Workshop on Computational Geometry

International audienceThe 24th European Workshop on Computational Geomety (EuroCG'08) was held at INRIA Nancy - Grand Est & LORIA on March 18-20, 2008. The present collection of abstracts contains the 63 scientific contributions as well as three invited talks presented at the workshop

Spatial Database Support for Virtual Engineering

The development, design, manufacturing and maintenance of modern engineering products is a very expensive and complex task. Shorter product cycles and a greater diversity of models are becoming decisive competitive factors in the hard-fought automobile and plane market. In order to support engineers to create complex products when being pressed for time, systems are required which answer collision and similarity queries effectively and efficiently. In order to achieve industrial strength, the required specialized functionality has to be integrated into fully-fledged database systems, so that fundamental services of these systems can be fully reused, including transactions, concurrency control and recovery. This thesis aims at the development of theoretical sound and practical realizable algorithms which effectively and efficiently detect colliding and similar complex spatial objects. After a short introductory Part I, we look in Part II at different spatial index structures and discuss their integrability into object-relational database systems. Based on this discussion, we present two generic approaches for accelerating collision queries. The first approach exploits available statistical information in order to accelerate the query process. The second approach is based on a cost-based decompositioning of complex spatial objects. In a broad experimental evaluation based on real-world test data sets, we demonstrate the usefulness of the presented techniques which allow interactive query response times even for large data sets of complex objects. In Part III of the thesis, we discuss several similarity models for spatial objects. We show by means of a new evaluation method that data-partitioning similarity models yield more meaningful results than space-partitioning similarity models. We introduce a very effective similarity model which is based on a new paradigm in similarity search, namely the use of vector set represented objects. In order to guarantee efficient query processing, suitable filters are introduced for accelerating similarity queries on complex spatial objects. Based on clustering and the introduced similarity models we present an industrial prototype which helps the user to navigate through massive data sets.Ein schneller und reibungsloser Entwicklungsprozess neuer Produkte ist ein wichtiger Faktor für den wirtschaftlichen Erfolg vieler Unternehmen insbesondere aus der Luft- und Raumfahrttechnik und der Automobilindustrie. Damit Ingenieure in immer kürzerer Zeit immer anspruchsvollere Produkte entwickeln können, werden effektive und effiziente Kollisions- und Ähnlichkeitsanfragen auf komplexen räumlichen Objekten benötigt. Um den hohen Anforderungen eines produktiven Einsatzes zu genügen, müssen entsprechend spezialisierte Zugriffsmethoden in vollwertige Datenbanksysteme integriert werden, so dass zentrale Datenbankdienste wie Trans-aktionen, kontrollierte Nebenläufigkeit und Wiederanlauf sichergestellt sind. Ziel dieser Doktorarbeit ist es deshalb, effektive und effiziente Algorithmen für Kollisions- und Ähnlichkeitsanfragen auf komplexen räumlichen Objekten zu ent-wickeln und diese in kommerzielle Objekt-Relationale Datenbanksysteme zu integrieren. Im ersten Teil der Arbeit werden verschiedene räumliche Indexstrukturen zur effizienten Bearbeitung von Kollisionsanfragen diskutiert und auf ihre Integrationsfähigkeit in Objekt-Relationale Datenbanksysteme hin untersucht. Daran an-knüpfend werden zwei generische Verfahren zur Beschleunigung von Kollisionsanfragen vorgestellt. Das erste Verfahren benutzt statistische Informationen räumlicher Indexstrukturen, um eine gegebene Anfrage zu beschleunigen. Das zweite Verfahren beruht auf einer kostenbasierten Zerlegung komplexer räumlicher Datenbank- Objekte. Diese beiden Verfahren ergänzen sich gegenseitig und können unabhängig voneinander oder zusammen eingesetzt werden. In einer ausführlichen experimentellen Evaluation wird gezeigt, dass die beiden vorgestellten Verfahren interaktive Kollisionsanfragen auf umfangreichen Datenmengen und komplexen Objekten ermöglichen. Im zweiten Teil der Arbeit werden verschiedene Ähnlichkeitsmodelle für räum-liche Objekte vorgestellt. Es wird experimentell aufgezeigt, dass datenpartitionierende Modelle effektiver sind als raumpartitionierende Verfahren. Weiterhin werden geeignete Filtertechniken zur Beschleunigung des Anfrageprozesses entwickelt und experimentell untersucht. Basierend auf Clustering und den entwickelten Ähnlichkeitsmodellen wird ein industrietauglicher Prototyp vorgestellt, der Benutzern hilft, durch große Datenmengen zu navigieren