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Axis-Normalized Ray-Box Intersection

COMPUTER GRAPHICS FORUM 2025年

作者： Friederichs, F. Benthin, C. Grogorick, S. Eisemann, E. Magnor, M. Eisemann, M. TU Braunschweig Braunschweig Germany Adv Micro Devices Inc Santa Clara CA USA Delft Univ Technol Delft Netherlands

Ray-axis aligned bounding box intersection tests play a crucial role in the runtime performance of many rendering applications, driven not by complexity but mainly by the volume of tests required. While existing solutions were believed to be pretty much optimal in terms of runtime on current hardware, our paper introduces a new intersection test requiring fewer arithmetic operations compared to all previous methods. By transforming the ray we eliminate the need for one third of the traditional bounding-slab tests and achieve a speed enhancement of approximately 13.8% or 10.9%, depending on the compiler. We present detailed runtime analyses in various scenarios.

关键词： CCS Concepts • Computing methodologies → Ray tracing • theory of computation → computational geometry

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LOOPS: LOcally Optimized Polygon Simplification

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COMPUTER GRAPHICS FORUM 2022年第3期41卷 355-365页

作者： Amiraghdam, Alireza Diehl, Alexandra Pajarola, Renato Univ Zurich Dept Informat Zurich Switzerland

Displaying polygonal vector data is essential in various application scenarios such as geometry visualization, vector graphics rendering, CAD drawing and in particular geographic, or cartographic visualization. Dealing with static polygonal datasets that has a large scale and are highly detailed poses several challenges to the efficient and adaptive display of polygons in interactive geographic visualization applications. For linear vector data, only recently a GPU-based level-of-detail (LOD) polyline simplification and rendering approach has been presented which can perform locally-adaptive LOD visualization of large-scale line datasets interactively. However, locally optimized LOD simplification and interactive display of large-scale polygon data, consisting of filled vector line loops, remains still a challenge, specifically in 3D geographic visualizations where varying LOD over a scene is necessary. Our solution to this challenge is a novel technique for locally-optimized simplification and visualization of 2D polygons over a 3D terrain which features a parallelized point-inside-polygon testing mechanism. Our approach is capable of employing any simplification algorithm that sequentially removes vertices such as Douglas-Peucker and Wang-Muller. Moreover, we generalized our technique to also visualizing polylines in order to have a unified method for displaying both data types. The results and performance analysis show that our new algorithm can handle large datasets containing polygons composed of millions of segments in real time, and has a lower memory demand and higher performance in comparison to prior methods of line simplification and visualization.

关键词： CCS Concepts • Human-centered computing → Geographic visualization Visualization techniques • theory of computation → computational geometry • Computing methodologies → Rendering Rasterization

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Multiscale Spectral Manifold Wavelet Regularizer for Unsupervised Deep Functional Maps

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COMPUTER GRAPHICS FORUM 2024年第7期43卷

作者： Liu, Shengjun Meng, Jing Hu, Ling Guo, Yueyu Liu, Xinru Yang, Xiaoxia Wang, Haibo Li, Qinsong Cent South Univ Inst Engn Modeling & Sci Comp Changsha 410083 Hunan Peoples R China Cent South Univ Big Data Inst Changsha 410083 Hunan Peoples R China Hunan First Normal Univ Sch Math & Stat Changsha 410205 Hunan Peoples R China

In deep functional maps, the regularizer computing the functional map is especially crucial for ensuring the global consistency of the computed pointwise map. As the regularizers integrated into deep learning should be differentiable, it is not trivial to incorporate informative axiomatic structural constraints into the deep functional map, such as the orientation-preserving term. Although commonly used regularizers include the Laplacian-commutativity term and the resolvent Laplacian commutativity term, these are limited to single-scale analysis for capturing geometric information. To this end, we propose a novel and theoretically well-justified regularizer commuting the functional map with the multiscale spectral manifold wavelet operator. This regularizer enhances the isometric constraints of the functional map and is conducive to providing it with better structural properties with multiscale analysis. Furthermore, we design an unsupervised deep functional map with the regularizer in a fully differentiable way. The quantitative and qualitative comparisons with several existing techniques on the (near-)isometric and non-isometric datasets show our method's superior accuracy and generalization capabilities. Additionally, we illustrate that our regularizer can be easily inserted into other functional map methods and improve their accuracy.

关键词： CCS Concepts • Computing methodologies → Shape analysis • theory of computation → computational geometry

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Attention And Positional Encoding Are (Almost) All You Need For Shape Matching

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COMPUTER GRAPHICS FORUM 2023年第5期42卷 e14912-e14912页

作者： Raganato, Alessandro Pasi, Gabriella Melzi, Simone Univ Milano Bicoccca Dept Informat Syst & Commun DISCo Milan Italy

The fast development of novel approaches derived from the Transformers architecture has led to outstanding performance in different scenarios, from Natural Language Processing to Computer Vision. Recently, they achieved impressive results even in the challenging task of non-rigid shape matching. However, little is known about the capability of the Transformer-encoder architecture for the shape matching task, and its performances still remained largely unexplored. In this paper, we step back and investigate the contribution made by the Transformer-encoder architecture compared to its more recent alternatives, focusing on why and how it works on this specific task. Thanks to the versatility of our implementation, we can harness the bi-directional structure of the correspondence problem, making it more interpretable. Furthermore, we prove that positional encodings are essential for processing unordered point clouds. Through a comprehensive set of experiments, we find that attention and positional encoding are (almost) all you need for shape matching. The simple Transformer-encoder architecture, coupled with relative position encoding in the attention mechanism, is able to obtain strong improvements, reaching the current state-of-the-art.

关键词： CCS Concepts • Computing methodologies → Shape analysis • theory of computation → computational geometry

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Interpolated corrected curvature measures for polygonal surfaces

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COMPUTER GRAPHICS FORUM 2020年第5期39卷 41-54页

作者： Lachaud, J. O. Romon, P. Thibert, B. Coeurjolly, D. Univ Savoie Mt Blanc Chambery France Univ Gustave Eiffel Champs Sur Marne France Univ Grenoble Alpes Grenoble France Univ Lyon CNRS Lyon France

A consistent and yet practically accurate definition of curvature onto polyhedral meshes remains an open problem. We propose a new framework to define curvature measures, based on the Corrected Normal Current, which generalizes the normal cycle: it uncouples the positional information of the polyhedral mesh from its geometric normal vector field, and the user can freely choose the corrected normal vector field at vertices for curvature computations. A smooth surface is then built in the Grassmannian R-3 x S-2 by simply interpolating the given normal vector field. Curvature measures are then computed using the usual Lipschitz-Killing forms, and we provide closed-form formulas per triangle. We prove a stability result with respect to perturbations of positions and normals. Our approach provides a natural scale-space for all curvature estimations, where the scale is given by the radius of the measuring ball. We show on experiments how this method outperforms state-of-the-art methods on clean and noisy data, and even achieves pointwise convergence on difficult polyhedral meshes like digital surfaces. The framework is also well suited to curvature computations using normal map information.

关键词： CCS Concepts • Computing methodologies → Shape analysis Mesh geometry models • theory of computation → computational geometry

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Properties of Laplace Operators for Tetrahedral Meshes

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COMPUTER GRAPHICS FORUM 2020年第5期39卷 55-68页

作者： Alexa, Marc Herholz, Philipp Kohlbrenner, Maximilian Sorkine-Hornung, Olga TU Berlin Berlin Germany Swiss Fed Inst Technol Zurich Switzerland

Discrete Laplacians for triangle meshes are a fundamental tool in geometry processing. The so-called cotan Laplacian is widely used since it preserves several important properties of its smooth counterpart. It can be derived from different principles: either considering the piecewise linear nature of the primal elements or associating values to the dual vertices. Both approaches lead to the same operator in the two-dimensional setting. In contrast, for tetrahedral meshes, only the primal construction is reminiscent of the cotan weights, involving dihedral angles. We provide explicit formulas for the lesser-known dual construction. In both cases, the weights can be computed by adding the contributions of individual tetrahedra to an edge. The resulting two different discrete Laplacians for tetrahedral meshes only retain some of the properties of their two-dimensional counterpart. In particular, while both constructions have linear precision, only the primal construction is positive semi-definite and only the dual construction generates positive weights and provides a maximum principle for Delaunay meshes. We perform a range of numerical experiments that highlight the benefits and limitations of the two constructions for different problems and meshes.

关键词： CCS Concepts • Mathematics of computing → Mesh generation Discrete optimization • Computing methodologies → Mesh geometry models • theory of computation → computational geometry

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Topology-Aware Surface Reconstruction for Point Clouds

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COMPUTER GRAPHICS FORUM 2020年第5期39卷 197-207页

作者： Bruel-Gabrielsson, Rickard Ganapathi-Subramanian, Vignesh Skraba, Primoz Guibas, Leonidas J. Stanford Univ Stanford CA 94305 USA Unbox AI Stanford CA 94305 USA Queen Mary Univ London London England Jozef Stefan Inst Ljubljana Slovenia

We present an approach to incorporate topological priors in the reconstruction of a surface from a point scan. We base the reconstruction on basis functions which are optimized to provide a good fit to the point scan while satisfying predefined topological constraints. We optimize the parameters of a model to obtain a likelihood function over the reconstruction domain. The topological constraints are captured by persistence diagrams which are incorporated within the optimization algorithm to promote the correct topology. The result is a novel topology-aware technique which can (i) weed out topological noise from point scans, and (ii) capture certain nuanced properties of the underlying shape which could otherwise be lost while performing surface reconstruction. We show results reconstructing shapes with multiple potential topologies, compare to other classical surface construction techniques, and show the completion of real scan data.

关键词： CCS Concepts • theory of computation → computational geometry • Computing methodologies → Shape modeling • Mathematics of computing → Algebraic topology

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LOCALIS: Locally-adaptive Line Simplification for GPU-based Geographic Vector Data Visualization

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COMPUTER GRAPHICS FORUM 2020年第3期39卷 443-453页

作者： Amiraghdam, Alireza Diehl, Alexandra Pajarola, Renato Univ Zurich Dept Informat Zurich Switzerland

Visualization of large vector line data is a core task in geographic and cartographic systems. Vector maps are often displayed at different cartographic generalization levels, traditionally by using several discrete levels-of-detail (LODs). This limits the generalization levels to a fixed and predefined set of LODs, and generally does not support smooth LOD transitions. However, fast GPUs and novel line rendering techniques can be exploited to integrate dynamic vector map LOD management into GPU-based algorithms for locally-adaptive line simplification and real-time rendering. We propose a new technique that interactively visualizes large line vector datasets at variable LODs. It is based on the Douglas-Peucker line simplification principle, generating an exhaustive set of line segments whose specific subsets represent the lines at any variable LOD. At run time, an appropriate and view-dependent error metric supports screen-space adaptive LOD levels and the display of the correct subset of line segments accordingly. Our implementation shows that we can simplify and display large line datasets interactively. We can successfully apply line style patterns, dynamic LOD selection lenses, and anti-aliasing techniques to our line rendering.

关键词： CCS Concepts • Human‐centered computing → Geographic visualization Visualization techniques • theory of computation → computational geometry • Computing methodologies → Rendering Rasterization

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Visibility-Aware Progressive Farthest Point Sampling on the GPU

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COMPUTER GRAPHICS FORUM 2019年第7期38卷 413-424页

作者： Brandt, Sascha Jaehn, Claudius Fischer, Matthias Heide, Friedhelm Meyer Auf Der Paderborn Univ Heinz Nixdorf Inst Paderborn Germany DeepL GmbH Cologne Germany

In this paper, we present the first algorithm for progressive sampling of 3D surfaces with blue noise characteristics that runs entirely on the GPU. The performance of our algorithm is comparable to state-of-the-art GPU Poisson-disk sampling methods, while additionally producing ordered sequences of samples where every prefix exhibits good blue noise properties. The basic idea is, to reduce the 3D sampling domain to a set of 2.5D images which we sample in parallel utilizing the rasterization hardware of current GPUs. This allows for simple visibility-aware sampling that only captures the surface as seen from outside the sampled object, which is especially useful for point-based level-of-detail rendering methods. However, our method can be easily extended for sampling the entire surface without changing the basic algorithm. We provide a statistical analysis of our algorithm and show that it produces good blue noise characteristics for every prefix of the resulting sample sequence and analyze the performance of our method compared to related state-of-the-art sampling methods.

关键词： CCS Concepts Computing methodologies → Rasterization Point‐based models • theory of computation → computational geometry

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