Variational blue noise sampling

Blue noise point sampling is one of the core algorithms in computer graphics. In this paper, we present a new and versatile variational framework for generating point distributions with high-quality blue noise characteristics while precisely adapting to given density functions. Different from previous approaches based on discrete settings of capacity-constrained Voronoi tessellation, we cast the blue noise sampling generation as a variational problem with continuous settings. Based on an accurate evaluation of the gradient of an energy function, an efficient optimization is developed which delivers significantly faster performance than the previous optimization-based methods. Our framework can easily be extended to generating blue noise point samples on manifold surfaces and for multi-class sampling. The optimization formulation also allows us to naturally deal with dynamic domains, such as deformable surfaces, and to yield blue noise samplings with temporal coherence. We present experimental results to validate the efficacy of our variational framework. Finally, we show a variety of applications of the proposed methods, including nonphotorealistic image stippling, color stippling, and blue noise sampling on deformable surfaces. © 1995-2012 IEEE.published_or_final_versio

A PDE approach to centroidal tessellations of domains

We introduce a class of systems of Hamilton-Jacobi equations that characterize critical points of functionals associated to centroidal tessellations of domains, i.e. tessellations where generators and centroids coincide, such as centroidal Voronoi tessellations and centroidal power diagrams. An appropriate version of the Lloyd algorithm, combined with a Fast Marching method on unstructured grids for the Hamilton-Jacobi equation, allows computing the solution of the system. We propose various numerical examples to illustrate the features of the technique

基于重心Delaunay三角剖分的蓝噪声点采样算法

为了生成带有高质量蓝噪声性质的采样分布,提出一种基于重心Delaunay三角剖分的点采样算法.该算法将Delaunay三角剖分与1-邻域三角片重心相结合,迭代地将每个采样点移至其1-邻域三角片的重心处并更新采样点之间的拓扑连接关系;重心通过给定的密度函数计算得出.实验结果表明,本文算法在运行效率与鲁棒性方面均有一定优势.国家自然科学基金(61472332);;福建省自然科学基金(2018J01104

A Hierarchical Approach for Regular Centroidal Voronoi Tessellations

International audienceIn this paper we consider Centroidal Voronoi Tessellations (CVTs) and study their regularity. CVTs are geometric structures that enable regular tessellations of geometric objects and are widely used in shape modeling and analysis. While several efficient iterative schemes, with defined local convergence properties, have been proposed to compute CVTs, little attention has been paid to the evaluation of the resulting cell decompositions. In this paper, we propose a regularity criterion that allows us to evaluate and compare CVTs independently of their sizes and of their cell numbers. This criterion allows us to compare CVTs on a common basis. It builds on earlier theoretical work showing that second moments of cells converge to a lower bound when optimising CVTs. In addition to proposing a regularity criterion, this paper also considers computational strategies to determine regular CVTs. We introduce a hierarchical framework that propagates regularity over decomposition levels and hence provides CVTs with provably better regularities than existing methods. We illustrate these principles with a wide range of experiments on synthetic and real models

A Systematic Review of Algorithms with Linear-time Behaviour to Generate Delaunay and Voronoi Tessellations

Triangulations and tetrahedrizations are important geometrical discretization procedures applied to several areas, such as the reconstruction of surfaces and data visualization. Delaunay and Voronoi tessellations are discretization structures of domains with desirable geometrical properties. In this work, a systematic review of algorithms with linear-time behaviour to generate 2D/3D Delaunay and/or Voronoi tessellations is presented

Gap Processing for Adaptive Maximal Poisson-Disk Sampling

In this paper, we study the generation of maximal Poisson-disk sets with varying radii. First, we present a geometric analysis of gaps in such disk sets. This analysis is the basis for maximal and adaptive sampling in Euclidean space and on manifolds. Second, we propose efficient algorithms and data structures to detect gaps and update gaps when disks are inserted, deleted, moved, or have their radius changed. We build on the concepts of the regular triangulation and the power diagram. Third, we will show how our analysis can make a contribution to the state-of-the-art in surface remeshing.Comment: 16 pages. ACM Transactions on Graphics, 201