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Item Quad Layouts via Constrained T-Mesh Quantization(The Eurographics Association and John Wiley & Sons Ltd., 2021) Lyon, Max; Campen, Marcel; Kobbelt, Leif; Mitra, Niloy and Viola, IvanWe present a robust and fast method for the creation of conforming quad layouts on surfaces. Our algorithm is based on the quantization of a T-mesh, i.e. an assignment of integer lengths to the sides of a non-conforming rectangular partition of the surface. This representation has the benefit of being able to encode an infinite number of layout connectivity options in a finite manner, which guarantees that a valid layout can always be found. We carefully construct the T-mesh from a given seamless parametrization such that the algorithm can provide guarantees on the results' quality. In particular, the user can specify a bound on the angular deviation of layout edges from prescribed directions. We solve an integer linear program (ILP) to find a coarse quad layout adhering to that maximal deviation. Our algorithm is guaranteed to yield a conforming quad layout free of T-junctions together with bounded angle distortion. Our results show that the presented method is fast, reliable, and achieves high quality layouts.Item Procedural Interpolation of Historical City Maps(The Eurographics Association and John Wiley and Sons Ltd., 2012) Krecklau, Lars; Manthei, Christopher; Kobbelt, Leif; P. Cignoni and T. ErtlWe propose a novel approach for the temporal interpolation of city maps. The input to our algorithm is a sparse set of historical city maps plus optional additional knowledge about construction or destruction events. The output is a fast forward animation of the city map development where roads and buildings are constructed and destroyed over time in order to match the sparse historical facts and to look plausible where no precise facts are available. A smooth transition between any real-world data could be interesting for educational purposes, because our system conveys an intuition of the city development. The insertion of data, like when and where a certain building or road existed, is efficiently performed by an intuitive graphical user interface. Our system collects all this information into a global dependency graph of events. By propagating time intervals through the dependency graph we can automatically derive the earliest and latest possible date for each event which are guaranteeing temporal as well as geographical consistency (e.g. buildings can only appear along roads that have been constructed before). During the simulation of the city development, events are scheduled according to a score function that rates the plausibility of the development (e.g. cities grow along major roads). Finally, the events are properly distributed over time to control the dynamics of the city development. Based on the city map animation we create a procedural city model in order to render a 3D animation of the city development over decades.Item An Intuitive Interface for Interactive High Quality Image-Based Modeling(The Eurographics Association and Blackwell Publishing Ltd, 2009) Habbecke, Martin; Kobbelt, LeifWe present the design of an interactive image-based modeling tool that enables a user to quickly generate detailed 3D models with texture from a set of calibrated input images. Our main contribution is an intuitive user interface that is entirely based on simple 2D painting operations and does not require any technical expertise by the user or difficult pre-processing of the input images. One central component of our tool is a GPU-based multi-view stereo reconstruction scheme, which is implemented by an incremental algorithm, that runs in the background during user interaction so that the user does not notice any significant response delay.Item Efficient Enforcement of Hard Articulation Constraints in the Presence of Closed Loops and Contacts(The Eurographics Association and John Wiley and Sons Ltd., 2014) Tomcin, Robin; Sibbing, Dominik; Kobbelt, Leif; B. Levy and J. KautzIn rigid body simulation, one must distinguish between contacts (so-called unilateral constraints) and articulations (bilateral constraints). For contacts and friction, iterative solution methods have proven most useful for interactive applications, often in combination with Shock-Propagation in cases with strong interactions between contacts (such as stacks), prioritizing performance and plausibility over accuracy. For articulation constraints, direct solution methods are preferred, because one can rely on a factorization with linear time complexity for tree-like systems, even in ill-conditioned cases caused by large mass-ratios or high complexity. Despite recent advances, combining the advantages of direct and iterative solution methods wrt. performance has proven difficult and the intricacy of articulations in interactive applications is often limited by the convergence speed of the iterative solution method in the presence of closed kinematic loops (i.e. auxiliary constraints) and contacts. We identify common performance bottlenecks in the dynamic simulation of unilateral and bilateral constraints and are able to present a simulation method, that scales well in the number of constraints even in ill-conditioned cases with frictional contacts, collisions and closed loops in the kinematic graph. For cases where many joints are connected to a single body, we propose a technique to increase the sparsity of the positive definite linear system. A solution to these bottlenecks is presented in this paper to make the simulation of a wider range of mechanisms possible in real-time without extensive parameter tuning.Item Surface Maps via Adaptive Triangulations(The Eurographics Association and John Wiley & Sons Ltd., 2023) Schmidt, Patrick; Pieper, Dörte; Kobbelt, Leif; Myszkowski, Karol; Niessner, MatthiasWe present a new method to compute continuous and bijective maps (surface homeomorphisms) between two or more genus-0 triangle meshes. In contrast to previous approaches, we decouple the resolution at which a map is represented from the resolution of the input meshes. We discretize maps via common triangulations that approximate the input meshes while remaining in bijective correspondence to them. Both the geometry and the connectivity of these triangulations are optimized with respect to a single objective function that simultaneously controls mapping distortion, triangulation quality, and approximation error. A discrete-continuous optimization algorithm performs both energy-based remeshing as well as global second-order optimization of vertex positions, parametrized via the sphere. With this, we combine the disciplines of compatible remeshing and surface map optimization in a unified formulation and make a contribution in both fields. While existing compatible remeshing algorithms often operate on a fixed pre-computed surface map, we can now globally update this correspondence during remeshing. On the other hand, bijective surface-to-surface map optimization previously required computing costly overlay meshes that are inherently tied to the input mesh resolution. We achieve significant complexity reduction by instead assessing distortion between the approximating triangulations. This new map representation is inherently more robust than previous overlay-based approaches, is less intricate to implement, and naturally supports mapping between more than two surfaces. Moreover, it enables adaptive multi-resolution schemes that, e.g., first align corresponding surface regions at coarse resolutions before refining the map where needed. We demonstrate significant speedups and increased flexibility over state-of-the art mapping algorithms at similar map quality, and also provide a reference implementation of the method.Item Interactive Curve Constrained Functional Maps(The Eurographics Association and John Wiley & Sons Ltd., 2018) Gehre, Anne; Bronstein, Michael M.; Kobbelt, Leif; Solomon, Justin; Ju, Tao and Vaxman, AmirFunctional maps have gained popularity as a versatile framework for representing intrinsic correspondence between 3D shapes using algebraic machinery. A key ingredient for this framework is the ability to find pairs of corresponding functions (typically, feature descriptors) across the shapes. This is a challenging problem on its own, and when the shapes are strongly non-isometric, nearly impossible to solve automatically. In this paper, we use feature curve correspondences to provide flexible abstractions of semantically similar parts of non-isometric shapes. We design a user interface implementing an interactive process for constructing shape correspondence, allowing the user to update the functional map at interactive rates by introducing feature curve correspondences. We add feature curve preservation constraints to the functional map framework and propose an efficient numerical method to optimize the map with immediate feedback. Experimental results show that our approach establishes correspondences between geometrically diverse shapes with just a few clicks.Item Cost Minimizing Local Anisotropic Quad Mesh Refinement(The Eurographics Association and John Wiley & Sons Ltd., 2020) Lyon, Max; Bommes, David; Kobbelt, Leif; Jacobson, Alec and Huang, QixingQuad meshes as a surface representation have many conceptual advantages over triangle meshes. Their edges can naturally be aligned to principal curvatures of the underlying surface and they have the flexibility to create strongly anisotropic cells without causing excessively small inner angles. While in recent years a lot of progress has been made towards generating high quality uniform quad meshes for arbitrary shapes, their adaptive and anisotropic refinement remains difficult since a single edge split might propagate across the entire surface in order to maintain consistency. In this paper we present a novel refinement technique which finds the optimal trade-off between number of resulting elements and inserted singularities according to a user prescribed weighting. Our algorithm takes as input a quad mesh with those edges tagged that are prescribed to be refined. It then formulates a binary optimization problem that minimizes the number of additional edges which need to be split in order to maintain consistency. Valence 3 and 5 singularities have to be introduced in the transition region between refined and unrefined regions of the mesh. The optimization hence computes the optimal trade-off and places singularities strategically in order to minimize the number of consistency splits —- or avoids singularities where this causes only a small number of additional splits. When applying the refinement scheme iteratively, we extend our binary optimization formulation such that previous splits can be undone if this prevents degenerate cells with small inner angles that otherwise might occur in anisotropic regions or in the vicinity of singularities. We demonstrate on a number of challenging examples that the algorithm performs well in practice.Item Feature Curve Co-Completion in Noisy Data(The Eurographics Association and John Wiley & Sons Ltd., 2018) Gehre, Anne; Lim, Isaak; Kobbelt, Leif; Gutierrez, Diego and Sheffer, AllaFeature curves on 3D shapes provide important hints about significant parts of the geometry and reveal their underlying structure. However, when we process real world data, automatically detected feature curves are affected by measurement uncertainty, missing data, and sampling resolution, leading to noisy, fragmented, and incomplete feature curve networks. These artifacts make further processing unreliable. In this paper we analyze the global co-occurrence information in noisy feature curve networks to fill in missing data and suppress weakly supported feature curves. For this we propose an unsupervised approach to find meaningful structure within the incomplete data by detecting multiple occurrences of feature curve configurations (cooccurrence analysis). We cluster and merge these into feature curve templates, which we leverage to identify strongly supported feature curve segments as well as to complete missing data in the feature curve network. In the presence of significant noise, previous approaches had to resort to user input, while our method performs fully automatic feature curve co-completion. Finding feature reoccurrences however, is challenging since naïve feature curve comparison fails in this setting due to fragmentation and partial overlaps of curve segments. To tackle this problem we propose a robust method for partial curve matching. This provides us with the means to apply symmetry detection methods to identify co-occurring configurations. Finally, Bayesian model selection enables us to detect and group re-occurrences that describe the data well and with low redundancy.Item Procedural Modeling of Interconnected Structures(The Eurographics Association and Blackwell Publishing Ltd., 2011) Krecklau, Lars; Kobbelt, Leif; M. Chen and O. DeussenThe complexity and detail of geometric scenes that are used in today's computer animated films and interactive games have reached a level where the manual creation by traditional 3D modeling tools has become infeasible. This is why procedural modeling concepts have been developed which generate highly complex 3D models by automatically executing a set of formal construction rules. Well-known examples are variants of L-systems which describe the bottom-up growth process of plants and shape grammars which define architectural buildings by decomposing blocks in a top-down fashion. However, none of these approaches allows for the easy generation of interconnected structures such as bridges or roller coasters where a functional interaction between rigid and deformable parts of an object is needed. Our approach mainly relies on the top-down decomposition principle of shape grammars to create an arbitrarily complex but well structured layout. During this process, potential attaching points are collected in containers which represent the set of candidates to establish interconnections. Our grammar then uses either abstract connection patterns or geometric queries to determine elements in those containers that are to be connected. The two different types of connections that our system supports are rigid object chains and deformable beams. The former type is constructed by inverse kinematics, the latter by spline interpolation. We demonstrate the descriptive power of our grammar by example models of bridges, roller coasters, and wall-mounted catenaries.Item Linear Analysis of Nonlinear Constraints for Interactive Geometric Modeling(The Eurographics Association and John Wiley and Sons Ltd., 2012) Habbecke, Martin; Kobbelt, Leif; P. Cignoni and T. ErtlThanks to its flexibility and power to handle even complex geometric relations, 3D geometric modeling with nonlinear constraints is an attractive extension of traditional shape editing approaches. However, existing approaches to analyze and solve constraint systems usually fail to meet the two main challenges of an interactive 3D modeling system: For each atomic editing operation, it is crucial to adjust as few auxiliary vertices as possible in order to not destroy the user's earlier editing effort. Furthermore, the whole constraint resolution pipeline is required to run in real-time to enable a fluent, interactive workflow. To address both issues, we propose a novel constraint analysis and solution scheme based on a key observation: While the computation of actual vertex positions requires nonlinear techniques, under few simplifying assumptions the determination of the minimal set of to-be-updated vertices can be performed on a linearization of the constraint functions. Posing the constraint analysis phase as the solution of an under-determined linear system with as few non-zero elements as possible enables us to exploit an efficient strategy for the Cardinality Minimization problem known from the field of Compressed Sensing, resulting in an algorithm capable of handling hundreds of vertices and constraints in real-time. We demonstrate at the example of an image-based modeling system for architectural models that this approach performs very well in practical applications.