Dr. Marcel Campen|
Now at New York University,
but this information is still up to date.
|SIGGRAPH Asia 2016||Course||Directional Field Synthesis, Design, and Processing|
|SIGGRAPH 2016||Paper||Bijective Maps from Simplicial Foliations|
|SGP 2016||Course||Quad Mesh Generation|
|SGP 2016||Paper||Scale-Invariant Directional Alignment of Surface Parametrizations|
|Eurographics 2016||Paper||Directional Field Synthesis, Design, and Processing|
|SIGGRAPH Asia 2015||Paper||Quantized Global Parametrization|
|Eurographics 2015||Paper||Quad Layout Embedding via Aligned Parameterization|
|LGG, EPFL||Invited Talk||From Quad Meshes to Quad Layouts|
|SIGGRAPH Asia 2014||Paper||Dual Strip Weaving: Interactive Design of Quad Layouts using Elastica Strips|
|Titane Seminar, INRIA Sophia Antipolis||Invited Talk||Quad Layouts on Surfaces: Generation and Optimization|
|SGP 2013||Paper||Practical Anisotropic Geodesy|
|SGP 2013||Course||Fixing Mesh Defects - Algorithms and Techniques|
|SIGGRAPH 2012||Paper||Dual Loops Meshing: Quality Quad Layouts on Manifolds|
|Eurographics 2012||Tutorial||Polygon Mesh Repairing|
|Eurographics 2011||Paper||Hybrid Booleans|
|Eurographics 2011||Paper||Walking On Broken Mesh: Defect-Tolerant Geodesic Distances and Parameterizations|
|SGP 2010||Paper||Polygonal Boundary Evaluation of Minkowski Sums and Swept Volumes|
|Eurographics 2010||Paper||Exact and Robust (Self-)Intersections for Polygonal Meshes|
|New Trends in Applied Geometry 2010||Talk||Hybrid Geometry Processing: Booleans, Offsets, Repairing|
|GI Informatik-Tage 2009||Poster||A Framework for Geometry Processing based on Hybrid Surface Representations|
This paper presents a method for bijective parametrization of 2D and 3D objects over canonical domains. While a range of solutions for the two-dimensional case are well-known, our method guarantees bijectivity of mappings also for a large, combinatorially-defined class of tetrahedral meshes (shellable meshes). The key concept in our method is the piecewise-linear (PL) foliation, decomposing the mesh into one-dimensional submanifolds and reducing the mapping problem to parametrization of a lower-dimensional manifold (a foliation section). The maps resulting from these foliations are proved to be bijective and continuous, and shown to have provably bijective PL approximations. We describe exact, numerically robust evaluation methods and demonstrate our implementation's capabilities on a large variety of meshes.
Parametrization based methods have recently become very popular for the generation of high quality quad meshes. In contrast to previous approaches, they allow for intuitive user control in order to accommodate all kinds of application driven constraints and design intentions. A major obstacle in practice, however, are the relatively long computations that lead to response times of several minutes already for input models of moderate complexity. In this paper we introduce a novel strategy to handle highly complex input meshes with up to several millions of triangles such that quad meshes can still be created and edited within an interactive workflow. Our method is based on representing the input model on different levels of resolution with a mechanism to propagate parametrizations from coarser to finer levels. The major challenge is to guarantee consistent parametrizations even in the presence of charts, transition functions, and singularities. Moreover, the remaining degrees of freedom on coarser levels of resolution have to be chosen carefully in order to still achieve low distortion parametrizations. We demonstrate a prototypic system where the user can interactively edit quad meshes with powerful high-level operations such as guiding constraints, singularity repositioning, and singularity connections.
Direction fields and vector fields play an increasingly important role in computer graphics and geometry processing. The synthesis of directional fields on surfaces, or other spatial domains, is a fundamental step in numerous applications, such as mesh generation, deformation, texture mapping, and many more. The wide range of applications resulted in definitions for many types of directional fields: from vector and tensor fields, over line and cross fields, to frame and vector-set fields. Depending on the application at hand, researchers have used various notions of objectives and constraints to synthesize such fields. These notions are defined in terms of fairness, feature alignment, symmetry, or field topology, to mention just a few. To facilitate these objectives, various representations, discretizations, and optimization strategies have been developed. These choices come with varying strengths and weaknesses. This report provides a systematic overview of directional field synthesis for graphics applications, the challenges it poses, and the methods developed in recent years to address these challenges.
Various applications of global surface parametrization benefit from the alignment of parametrization isolines with principal curvature directions. This is particularly true for recent parametrization-based meshing approaches, where this directly translates into a shape-aware edge flow, better approximation quality, and reduced meshing artifacts. Existing methods to influence a parametrization based on principal curvature directions suffer from scale-dependence, which implies the necessity of parameter variation, or try to capture complex directional shape features using simple 1D curves. Especially for non-sharp features, such as chamfers, fillets, blends, and even more for organic variants thereof, these abstractions can be unfit. We present a novel approach which respects and exploits the 2D nature of such directional feature regions, detects them based on coherence and homogeneity properties, and controls the parametrization process accordingly. This approach enables us to provide an intuitive, scale-invariant control parameter to the user. It also allows us to consider non-local aspects like the topology of a feature, enabling further improvements. We demonstrate that, compared to previous approaches, global parametrizations of higher quality can be generated without user intervention.
Global surface parametrization often requires the use of cuts or charts due to non-trivial topology. In recent years a focus has been on so-called seamless parametrizations, where the transition functions across the cuts are rigid transformations with a rotation about some multiple of 90 degrees. Of particular interest, e.g. for quadrilateral meshing, paneling, or texturing, are those instances where in addition the translational part of these transitions is integral (or more generally: quantized). We show that finding not even the optimal, but just an arbitrary valid quantization (one that does not imply parametric degeneracies), is a complex combinatorial problem. We present a novel method that allows us to solve it, i.e. to find valid as well as good quality quantizations. It is based on an original approach to quickly construct solutions to linear Diophantine equation systems, exploiting the specific geometric nature of the parametrization problem. We thereby largely outperform the state-of-the-art, sometimes by several orders of magnitude.
The most effective and popular tools for obtaining feature aligned quad meshes from triangular input meshes are based on cross field guided parametrization. These methods are incarnations of a conceptual three-step pipeline: (1) cross field computation, (2) field-guided surface parametrization, (3) quad mesh extraction. While in most meshing scenarios the user prescribes a desired target quad size or edge length, this information is typically taken into account from step 2 onwards only, but not in the cross field computation step. This turns into a problem in the presence of small scale geometric or topological features or noise in the input mesh: closely placed singularities are induced in the cross field, which are not properly reproducible by vertices in a quad mesh with the prescribed edge length, causing severe distortions or even failure of the meshing algorithm. We reformulate the construction of cross fields as well as field-guided parametrizations in a scale-aware manner which effectively suppresses densely spaced features and noise of geometric as well as topological kind. Dominant large-scale features are adequately preserved in the output by relying on the unaltered input mesh as the computational domain.
We introduce Dual Strip Weaving, a novel concept for the interactive design of quad layouts, i.e. partitionings of freeform surfaces into quadrilateral patch networks. In contrast to established tools for the design of quad layouts or subdivision base meshes, which are often based on creating individual vertices, edges, and quads, our method takes a more global perspective, operating on a higher level of abstraction: the atomic operation of our method is the creation of an entire cyclic strip, delineating a large number of quad patches at once. The global consistency-preserving nature of this approach reduces demands on the user’s expertise by requiring less advance planning. Efficiency is achieved using a novel method at the heart of our system, which automatically proposes geometrically and topologically suitable strips to the user. Based on this we provide interaction tools to influence the design process to any desired degree and visual guides to support the user in this task.
Quad layouting, i.e. the partitioning of a surface into a coarse network of quadrilateral patches, is a fundamental step in application scenarios ranging from animation and simulation to reverse engineering and meshing. This process involves determining the layout's combinatorial structure as well as its geometric embedding in the surface. We present a novel quad layout algorithm that focuses on the embedding optimization, thereby complementing recent methods focusing on the structure optimization aspect. It takes as input a description of the target layout structure and computes a complete embedding in form of a parameterization globally optimized for isometry and, in particular, principal direction alignment. Besides being suited for fully automatic workflows, our method can also incorporate user constraints and support the tedious but common procedure of manual layouting.
The efficient, computer aided or automatic generation of quad layouts, i.e. the partitioning of an object’s surface into simple networks of conforming quadrilateral patches, is a task that – despite its importance and utility in Computer Graphics and Geometric Modeling – received relatively low attention in the past. As a consequence, this task is most often performed manually by well-trained experts in practice, where quad layouts are of particular interest for surface representation and parameterization tasks. Deeper analysis reveals the inherent complexity of this problem, which might be one of the underlying reasons for this situation. In this thesis we investigate the structure of the problem and the commonly relevant quality criteria. Based on this we develop novel efficient solution strategies and algorithms for the generation of high quality quad layouts. In particular, we present a fully automatic as well as an interactive pipeline for this task. Both are based on splitting the hard problem into sub-problems with a simpler structure each. For each sub-problem we design efficient, custom-tailored optimization algorithms motivated by the geometric nature of these problems. In this process we pay attention to compatibility, such that these algorithms can be applied in sequence, forming the stages of efficient quad layouting pipelines.
Quadrilateral remeshing approaches based on global parametrization enable many desirable mesh properties. Two of the most important ones are (1) high regularity due to explicit control over irregular vertices and (2) smooth distribution of distortion achieved by convex variational formulations. Apart from these strengths, state-of-the-art techniques suffer from limited reliability on real-world input data, i.e. the determined map might have degeneracies like (local) non-injectivities and consequently often cannot be used directly to generate a quadrilateral mesh. In this paper we propose a novel convex Mixed-Integer Quadratic Programming (MIQP) formulation which ensures by construction that the resulting map is within the class of so called Integer-Grid Maps that are guaranteed to imply a quad mesh. In order to overcome the NP-hardness of MIQP and to be able to remesh typical input geometries in acceptable time we propose two additional problem specific optimizations: a complexity reduction algorithm and singularity separating conditions. While the former decouples the dimension of the MIQP search space from the input complexity of the triangle mesh and thus is able to dramatically speed up the computation without inducing inaccuracies, the latter improves the continuous relaxation, which is crucial for the success of modern MIQP optimizers. Our experiments show that the reliability of the resulting algorithm does not only annihilate the main drawback of parametrization based quad-remeshing but moreover enables the global search for high-quality coarse quad layouts – a difficult task solely tackled by greedy methodologies before.
The most popular and actively researched class of quad remeshing techniques is
the family of parametrization based quad meshing methods. They all strive
to generate an integer-grid map, i.e. a parametrization of the input surface
into R2 such that the canonical grid of integer iso-lines forms a
quad mesh when mapped back onto the surface in R3. An essential,
albeit broadly neglected aspect of these methods is the quad extraction
step, i.e. the materialization of an actual quad mesh from the mere “quad
texture”. Quad (mesh) extraction is often believed to be a trivial matter but
quite the opposite is true: Numerous special cases, ambiguities induced by
numerical inaccuracies and limited solver precision, as well as imperfections
in the maps produced by most methods (unless costly countermeasures are taken)
pose significant challenges to the quad extractor. We present a method to
sanitize a provided parametrization such that it becomes numerically
consistent even in a limited precision floating point representation. Based
on this we are able to provide a comprehensive and sound description of how to
perform quad extraction robustly and without the need for any complex
tolerance thresholds or disambiguation rules. On top of that we develop a
novel strategy to cope with common local fold-overs in the parametrization.
This allows our method, dubbed QEx, to generate all-quadrilateral meshes
where otherwise holes, non-quad polygons or no output at all would have been
produced. We thus enable the practical use of an entire class of maps that was
previously considered defective. Since state of the art quad meshing methods
spend a significant share of their run time solely to prevent local
fold-overs, using our method it is now possible to obtain quad meshes
significantly quicker than before. We also provide
libQEx, an open source
C++ reference implementation of our method and thus significantly lower the
bar to enter the field of quad meshing.
In the context of shape segmentation and retrieval object-wide distributions of measures are needed to accurately evaluate and compare local regions of shapes. Lien et al. proposed two point-wise concavity measures in the context of Approximate Convex Decompositions of polygons measuring the distance from a point to the polygon’s convex hull: an accurate Shortest Path-Concavity (SPC) measure and a Straight Line-Concavity (SLC) approximation of the same. While both are practicable on 2D shapes, the exponential costs of SPC in 3D makes it inhibitively expensive for a generalization to meshes. In this paper we propose an efficient and straight forward approximation of the Shortest Path-Concavity measure to 3D meshes. Our approximation is based on discretizing the space between mesh and convex hull, thereby reducing the continuous Shortest Path search to an efficiently solvable graph problem. Our approach works out-of-the-box on complex mesh topologies and requires no complicated handling of genus. Besides presenting a rigorous evaluation of our method on a variety of input meshes, we also define an SPC-based Shape Descriptor and show its superior retrieval and runtime performance compared with the recently presented results on the Convexity Distribution by Lian et al.
Nowadays, digital 3D models are in widespread and ubiquitous use, and each specific application dealing with 3D geometry has its own quality requirements that restrict the class of acceptable and supported models. This article analyzes typical defects that make a 3D model unsuitable for key application contexts, and surveys existing algorithms that process, repair, and improve its structure, geometry, and topology to make it appropriate to case-by-case requirements. The analysis is focused on polygon meshes, which constitute by far the most common 3D object representation. In particular, this article provides a structured overview of mesh repairing techniques from the point of view of the application context. Different types of mesh defects are classified according to the upstream application that produced the mesh, whereas mesh quality requirements are grouped by representative sets of downstream applications where the mesh is to be used. The numerous mesh repair methods that have been proposed during the last two decades are analyzed and classified in terms of their capabilities, properties, and guarantees. Based on these classifications, guidelines can be derived to support the identification of repairing algorithms best-suited to bridge the compatibility gap between the quality provided by the upstream process and the quality required by the downstream applications in a given geometry processing scenario.
The computation of intrinsic, geodesic distances and geodesic paths on surfaces is a fundamental low-level building block in countless Computer Graphics and Geometry Processing applications. This demand led to the development of numerous algorithms – some for the exact, others for the approximative computation, some focussing on speed, others providing strict guarantees. Most of these methods are designed for computing distances according to the standard Riemannian metric induced by the surface’s embedding in Euclidean space. Generalization to other, especially anisotropic, metrics – which more recently gained interest in several application areas – is not rarely hampered by fundamental problems. We explore and discuss possibilities for the generalization and extension of well-known methods to the anisotropic case, evaluate their relative performance in terms of accuracy and speed, and propose a novel algorithm, the Short-Term Vector Dijkstra. This algorithm is strikingly simple to implement and proves to provide practical accuracy at a higher speed than generalized previous methods.
We present a theoretical framework and practical method for the automatic construction of simple, all-quadrilateral patch layouts on manifold surfaces. The resulting layouts are coarse, surface-embedded cell complexes well adapted to the geometric structure, hence they are ideally suited as domains and base complexes for surface parameterization, spline fitting, or subdivision surfaces and can be used to generate quad meshes with a high-level patch structure that are advantageous in many application scenarios. Our approach is based on the careful construction of the layout graph's combinatorial dual. In contrast to the primal this dual perspective provides direct control over the globally interdependent structural constraints inherent to quad layouts. The dual layout is built from curvature-guided, crossing loops on the surface. A novel method to construct these efficiently in a geometry- and structure-aware manner constitutes the core of our approach.
In mechanical engineering and architecture, structural elements with low material consumption and high load-bearing capabilities are essential for light-weight and even self-supporting constructions. This paper deals with so called point-folding elements - non-planar, pyramidal panels, usually formed from thin metal sheets, which exploit the increased structural capabilities emerging from folds or creases. Given a triangulated free-form surface, a corresponding point-folding structure is a collection of pyramidal elements basing on the triangles. User-specified or material-induced geometric constraints often imply that each individual folding element has a different shape, leading to immense fabrication costs. We present a rationalization method for such structures which respects the prescribed aesthetic and production constraints and ?nds a minimal set of molds for the production process, leading to drastically reduced costs. For each base triangle we compute and parametrize the range of feasible folding elements that satisfy the given constraints within the allowed tolerances. Then we pose the rationalization task as a geometric intersection problem, which we solve so as to maximize the re-use of mold dies. Major challenges arise from the high precision requirements and the non-trivial parametrization of the search space. We evaluate our method on a number of practical examples where we achieve rationalization gains of more than 90%.
Digital 3D models are key components in many industrial and scientific sectors. In numerous domains polygon meshes have become a de facto standard for model representation. In practice meshes often have a number of defects and flaws that make them incompatible with quality requirements of specific applications. Hence, repairing such defects in order to achieve compatibility is a highly important task – in academic as well as industrial applications. In this tutorial we first systematically analyze typical application contexts together with their requirements and issues, as well as the various types of defects that typically play a role. Subsequently, we consider existing techniques to process, repair, and improve the structure, geometry, and topology of imperfect meshes, aiming at making them appropriate to case-by-case requirements. We present seminal works and key algorithms, discuss extensions and improvements, and analyze the respective advantages and disadvantages depending on the application context. Furthermore, we outline directions where further research is particularly important or promising.
Several theoretical and practical geometry applications are based on polygon meshes with planar faces. The planar panelization of freeform surfaces is a prominent example from the field of architectural geometry. One approach to obtain a certain kind of such meshes is by intersection of suitably distributed tangent planes. Unfortunately, this simple tangent plane intersection (TPI) idea is limited to the generation of hex-dominant meshes: as vertices are in general defined by three intersecting planes, the resulting meshes are basically duals of triangle meshes.
The explicit computation of intersection points furthermore requires dedicated handling of special cases and degenerate constellations to achieve robustness on freeform surfaces. Another limitation is the small number of degrees of freedom for incorporating design parameters.
Using a variational re-formulation, we equip the concept of TPI with additional degrees of freedom and present a robust, unified approach for creating polygonal structures with planar faces that is readily able to integrate various objectives and constraints needed in different applications scenarios. We exemplarily demonstrate the abilities of our approach on three common problems in geometry processing.
Efficient methods to compute intrinsic distances and geodesic paths have been presented for various types of surface representations, most importantly polygon meshes. These meshes are usually assumed to be well-structured and manifold. In practice, however, they often contain defects like holes, gaps, degeneracies, non-manifold configurations – or they might even be just a soup of polygons. The task of repairing these defects is computationally complex and in many cases exhibits various ambiguities demanding tedious manual efforts. We present a computational framework that enables the computation of meaningful approximate intrinsic distances and geodesic paths on raw meshes in a way which is tolerant to such defects. Holes and gaps are bridged up to a user-specified tolerance threshold such that distances can be computed plausibly even across multiple connected components of inconsistent meshes. Further, we show ways to locally parameterize a surface based on geodesic distance fields, easily facilitating the application of textures and decals on raw meshes. We do all this without explicitly repairing the input, thereby avoiding the costly additional efforts. In order to enable broad applicability we provide details on two implementation variants, one optimized for performance, the other optimized for memory efficiency. Using the presented framework many applications can readily be extended to deal with imperfect meshes. Since we abstract from the input applicability is not even limited to meshes, other representations can be handled as well.
We present a novel technique for the efficient boundary evaluation of sweep operations applied to objects in polygonal boundary representation. These sweep operations include Minkowski addition, offsetting, and sweeping along a discrete rigid motion trajectory. Many previous methods focus on the construction of a polygonal superset (containing self-intersections and spurious internal geometry) of the boundary of the volumes which are swept. Only few are able to determine a clean representation of the actual boundary, most of them in a discrete volumetric setting. We unify such superset constructions into a succinct common formulation and present a technique for the robust extraction of a polygonal mesh representing the outer boundary, i.e. it makes no general position assumptions and always yields a manifold, watertight mesh. It is exact for Minkowski sums and approximates swept volumes polygonally. By using plane-based geometry in conjunction with hierarchical arrangement computations we avoid the necessity of arbitrary precision arithmetics and extensive special case handling. By restricting operations to regions containing pieces of the boundary, we significantly enhance the performance of the algorithm.
A WebService employing this method is available.
We present a new technique to implement operators that modify the topology of polygonal meshes at intersectionsand self-intersections. Depending on the modification strategy, this effectively results in operators for Boolean combinations or for the construction of outer hulls that are suited for mesh repair tasks and accurate meshbased front tracking of deformable materials that split and merge. By combining an adaptive octree with nested binary space partitions (BSP), we can guarantee exactness (= correctness) and robustness (= completeness) of the algorithm while still achieving higher performance and less memory consumption than previous approaches. The efficiency and scalability in terms of runtime and memory is obtained by an operation localization scheme. We restrict the essential computations to those cells in the adaptive octree where intersections actually occur. Within those critical cells, we convert the input geometry into a plane-based BSP-representation which allows us to perform all computations exactly even with fixed precision arithmetics. We carefully analyze the precision requirements of the involved geometric data and predicates in order to guarantee correctness and show how minimal input mesh quantization can be used to safely rely on computations with standard floating point numbers. We properly evaluate our method with respect to precision, robustness, and efficiency.
A WebService employing this method is available.
Talk at Eurographics 2011
In this paper we present a novel method to compute Boolean operation polygonal meshes. Given a Boolean expression over an arbitrary number input meshes we reliably and efficiently compute an output mesh which faithfully preserves the existing sharp features and precisely reconstructs the new features appearing along the intersections of the input meshes. The term "hybrid" applies to our method in two ways: First, our algorithm operates on a hybrid data structure which stores the original input polygons (surface data) in an adaptively refined octree (volume data). By this we combine the robustness of volumetric techniques with the accuracy of surface-oriented techniques. Second, we generate a new triangulation only in a close vicinity around the intersections of the input meshes and thus preserve as much of the original mesh structure as possible (hybrid mesh). Since the actual processing of the Boolean operation is confined to a very small region around the intersections of the input meshes, we can achieve very high adaptive refinement resolutions and hence very high precision. We demonstrate our method on a number of challenging examples.
We present a framework that allows for the composition of custom-tailored data structures for hybrid representation of geometry and supports the development of associated geometry processing methods. Besides others, a novel hybrid approach for the evaluation of Boolean expressions on polygon meshes is elaborated in this context. By relying on the hybrid geometry information it is – in contrast to previous methods – able to perform such operations robustly as well as accurately.
We present a framework that allows for the composition of custom-tailored data structures for hybrid representation of geometry and supports the development of associated geometry processing methods. Besides others, a novel hybrid approach for the evaluation of Boolean expressions on polygon meshes is elaborated in this context.