# Publications

## 3D Shape Generation with Grid-based Implicit Functions

Previous approaches to generate shapes in a 3D setting train a GAN on the latent space of an autoencoder (AE). Even though this produces convincing results, it has two major shortcomings. As the GAN is limited to reproduce the dataset the AE was trained on, we cannot reuse a trained AE for novel data. Furthermore, it is difficult to add spatial supervision into the generation process, as the AE only gives us a global representation. To remedy these issues, we propose to train the GAN on grids (i.e. each cell covers a part of a shape). In this representation each cell is equipped with a latent vector provided by an AE. This localized representation enables more expressiveness (since the cell-based latent vectors can be combined in novel ways) as well as spatial control of the generation process (e.g. via bounding boxes). Our method outperforms the current state of the art on all established evaluation measures, proposed for quantitatively evaluating the generative capabilities of GANs. We show limitations of these measures and propose the adaptation of a robust criterion from statistical analysis as an alternative.

@inproceedings {ibing20213Dshape,

title = {3D Shape Generation with Grid-based Implicit Functions},

author = {Ibing, Moritz and Lim, Isaak and Kobbelt, Leif},

booktitle = {IEEE Computer Vision and Pattern Recognition (CVPR)},

pages = {},

year = {2021}

}

## Learning Direction Fields for Quad Mesh Generation

State of the art quadrangulation methods are able to reliably and robustly convert triangle meshes into quad meshes. Most of these methods rely on a dense direction field that is used to align a parametrization from which a quad mesh can be extracted. In this context, the aforementioned direction field is of particular importance, as it plays a key role in determining the structure of the generated quad mesh. If there are no user-provided directions available, the direction field is usually interpolated from a subset of principal curvature directions. To this end, a number of heuristics that aim to identify significant surface regions have been proposed. Unfortunately, the resulting fields often fail to capture the structure found in meshes created by human experts. This is due to the fact that experienced designers can leverage their domain knowledge in order to optimize a mesh for a specific application. In the context of physics simulation, for example, a designer might prefer an alignment and local refinement that facilitates a more accurate numerical simulation. Similarly, a character artist may prefer an alignment that makes the resulting mesh easier to animate. Crucially, this higher level domain knowledge cannot be easily extracted from local curvature information alone. Motivated by this issue, we propose a data-driven approach to the computation of direction fields that allows us to mimic the structure found in existing meshes, which could originate from human experts or other sources. More specifically, we make use of a neural network that aggregates global and local shape information in order to compute a direction field that can be used to guide a parametrization-based quad meshing method. Our approach is a first step towards addressing this challenging problem with a fully automatic learning-based method. We show that compared to classical techniques our data-driven approach combined with a robust model-driven method, is able to produce results that more closely exhibit the ground truth structure of a synthetic dataset (i.e. a manually designed quad mesh template fitted to a variety of human body types in a set of different poses).

@article{dielen2021learning_direction_fields,

title={Learning Direction Fields for Quad Mesh Generation},

author={Dielen, Alexander and Lim, Isaak and Lyon, Max and Kobbelt, Leif},

year={2021},

journal={Computer Graphics Forum},

volume={40},

number={5},

}

## Simpler Quad Layouts using Relaxed Singularities

A common approach to automatic quad layout generation on surfaces is to, in a first stage, decide on the positioning of irregular layout vertices, followed by finding sensible layout edges connecting these vertices and partitioning the surface into quadrilateral patches in a second stage. While this two-step approach reduces the problem's complexity, this separation also limits the result quality. In the worst case, the set of layout vertices fixed in the first stage without consideration of the second may not even permit a valid quad layout. We propose an algorithm for the creation of quad layouts in which the initial layout vertices can be adjusted in the second stage. Whenever beneficial for layout quality or even validity, these vertices may be moved within a prescribed radius or even be removed. Our algorithm is based on a robust quantization strategy, turning a continuous T-mesh structure into a discrete layout. We show the effectiveness of our algorithm on a variety of inputs.

» Show BibTeX

@article{lyon2021simplerlayouts,

title={Simpler Quad Layouts using Relaxed Singularities},

author={Lyon, Max and Campen, Marcel and Kobbelt, Leif},

year={2021},

journal={Computer Graphics Forum},

volume={40},

number={5},

}

## Surface Map Homology Inference

A homeomorphism between two surfaces not only defines a (continuous and bijective) geometric correspondence of points but also (by implication) an identification of topological features, i.e. handles and tunnels, and how the map twists around them. However, in practice, surface maps are often encoded via sparse correspondences or fuzzy representations that merely approximate a homeomorphism and are therefore inherently ambiguous about map topology. In this work, we show a way to infer topological information from an imperfect input map between two shapes. In particular, we compute a homology map, a linear map that transports homology classes of cycles from one surface to the other, subject to a global consistency constraint. Our inference robustly handles imperfect (e.g., partial, sparse, fuzzy, noisy, outlier-ridden, non-injective) input maps and is guaranteed to produce homology maps that are compatible with true homeomorphisms between the input shapes. Homology maps inferred by our method can be directly used to transfer homological information between shapes, or serve as foundation for the construction of a proper homeomorphism guided by the input map, e.g., via compatible surface decomposition.

This work has received the best paper award at SGP 2021.

» Show BibTeX

@article{born2021surface,

title={Surface Map Homology Inference},

author={Born, Janis and Schmidt, Patrick and Campen, Marcel and Kobbelt, Leif},

year={2021},

journal={Computer Graphics Forum},

volume={40},

number={5},

}

## Geodesic Distance Computation via Virtual Source Propagation

We present a highly practical, efficient, and versatile approach for computing approximate geodesic distances. The method is designed to operate on triangle meshes and a set of point sources on the surface. We also show extensions for all kinds of geometric input including inconsistent triangle soups and point clouds, as well as other source types, such as lines. The algorithm is based on the propagation of virtual sources and hence easy to implement. We extensively evaluate our method on about 10000 meshes taken from the Thingi10k and the Tet Meshing in the Wild data sets. Our approach clearly outperforms previous approximate methods in terms of runtime efficiency and accuracy. Through careful implementation and cache optimization, we achieve runtimes comparable to other elementary mesh operations (e.g. smoothing, curvature estimation) such that geodesic distances become a "first-class citizen" in the toolbox of geometric operations. Our method can be parallelized and we observe up to 6× speed-up on the CPU and 20× on the GPU. We present a number of mesh processing tasks easily implemented on the basis of fast geodesic distances. The source code of our method will be provided as a C++ library under the MIT license.

Note: we are currently in the process of cleaning up and documenting the source code. A basic implementation can already be found in the supplemental material.

## Sampling from Quadric-Based CSG Surfaces

We present an efficient method to create samples directly on surfaces defined by constructive solid geometry (CSG) trees or graphs. The generated samples can be used for visualization or as an approximation to the actual surface with strong guarantees. We chose to use quadric surfaces as CSG primitives as they can model classical primitives such as planes, cubes, spheres, cylinders, and ellipsoids, but also certain saddle surfaces. More importantly, they are closed under affine transformations, a desirable property for a modeling system. We also propose a rendering method that performs local quadric ray-tracing and clipping to achieve pixel-perfect accuracy and hole-free rendering.

## Layout Embedding via Combinatorial Optimization

We consider the problem of injectively embedding a given graph connectivity (a layout) into a target surface. Starting from prescribed positions of layout vertices, the task is to embed all layout edges as intersection-free paths on the surface. Besides merely geometric choices (the shape of paths) this problem is especially challenging due to its topological degrees of freedom (how to route paths around layout vertices). The problem is typically addressed through a sequence of shortest path insertions, ordered by a greedy heuristic. Such insertion sequences are not guaranteed to be optimal: Early path insertions can potentially force later paths into unexpected homotopy classes. We show how common greedy methods can easily produce embeddings of dramatically bad quality, rendering such methods unsuitable for automatic processing pipelines. Instead, we strive to find the optimal order of insertions, i.e. the one that minimizes the total path length of the embedding. We demonstrate that, despite the vast combinatorial solution space, this problem can be effectively solved on simply-connected domains via a custom-tailored branch-and-bound strategy. This enables directly using the resulting embeddings in downstream applications which cannot recover from initializations in a wrong homotopy class. We demonstrate the robustness of our method on a shape dataset by embedding a common template layout per category, and show applications in quad meshing and inter-surface mapping.

This work has received honorable mentions for the Günter Enderle Best Paper Award and best full paper talk at Eurographics 2021.

» Show BibTeX

@article{born2021layout,

title={Layout Embedding via Combinatorial Optimization},

author={Born, Janis and Schmidt, Patrick and Kobbelt, Leif},

year={2021},

journal={Computer Graphics Forum},

volume={40},

number={2},

}

## Compression and Rendering of Textured Point Clouds via Sparse Coding

Splat-based rendering techniques produce highly realistic renderings from 3D scan data without prior mesh generation. Mapping high-resolution photographs to the splat primitives enables detailed reproduction of surface appearance. However, in many cases these massive datasets do not fit into GPU memory. In this paper, we present a compression and rendering method that is designed for large textured point cloud datasets. Our goal is to achieve compression ratios that outperform generic texture compression algorithms, while still retaining the ability to efficiently render without prior decompression. To achieve this, we resample the input textures by projecting them onto the splats and create a fixed-size representation that can be approximated by a sparse dictionary coding scheme. Each splat has a variable number of codeword indices and associated weights, which define the final texture as a linear combination during rendering. For further reduction of the memory footprint, we compress geometric attributes by careful clustering and quantization of local neighborhoods. Our approach reduces the memory requirements of textured point clouds by one order of magnitude, while retaining the possibility to efficiently render the compressed data.

## Quad Layouts via Constrained T-Mesh Quantization

We 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.

» Show BibTeX

@article{Lyon:2021:Quad,

title = {Quad Layouts via Constrained T-Mesh Quantization},

author = {Lyon, Max and Campen, Marcel and Kobbelt, Leif},

journal = {Computer Graphics Forum},

volume = {40},

number = {2},

year = {2021}

}

## Fast Exact Booleans for Iterated CSG using Octree-Embedded BSPs

We present octree-embedded BSPs, a volumetric mesh data structure suited for performing a sequence of Boolean operations (iterated CSG) efficiently. At its core, our data structure leverages a plane-based geometry representation and integer arithmetics to guarantee unconditionally robust operations. These typically present considerable performance challenges which we overcome by using custom-tailored fixed-precision operations and an efficient algorithm for cutting a convex mesh against a plane. Consequently, BSP Booleans and mesh extraction are formulated in terms of mesh cutting. The octree is used as a global acceleration structure to keep modifications local and bound the BSP complexity. With our optimizations, we can perform up to 2.5 million mesh-plane cuts per second on a single core, which creates roughly 40-50 million output BSP nodes for CSG. We demonstrate our system in two iterated CSG settings: sweep volumes and a milling simulation.

@article{NEHRINGWIRXEL2021103015,

title = {Fast Exact Booleans for Iterated CSG using Octree-Embedded BSPs},

journal = {Computer-Aided Design},

volume = {135},

pages = {103015},

year = {2021},

issn = {0010-4485},

doi = {https://doi.org/10.1016/j.cad.2021.103015},

url = {https://www.sciencedirect.com/science/article/pii/S0010448521000269},

author = {Julius Nehring-Wirxel and Philip Trettner and Leif Kobbelt},

keywords = {Plane-based geometry, CSG, Mesh Booleans, BSP, Octree, Integer arithmetic},

abstract = {We present octree-embedded BSPs, a volumetric mesh data structure suited for performing a sequence of Boolean operations (iterated CSG) efficiently. At its core, our data structure leverages a plane-based geometry representation and integer arithmetics to guarantee unconditionally robust operations. These typically present considerable performance challenges which we overcome by using custom-tailored fixed-precision operations and an efficient algorithm for cutting a convex mesh against a plane. Consequently, BSP Booleans and mesh extraction are formulated in terms of mesh cutting. The octree is used as a global acceleration structure to keep modifications local and bound the BSP complexity. With our optimizations, we can perform up to 2.5 million mesh-plane cuts per second on a single core, which creates roughly 40-50 million output BSP nodes for CSG. We demonstrate our system in two iterated CSG settings: sweep volumes and a milling simulation.}

}

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