Elsevier

Computers & Graphics

Volume 36, Issue 4, June 2012, Pages 232-240
Computers & Graphics

Applications of Geometry Processing
Blue noise sampling of surfaces

https://doi.org/10.1016/j.cag.2012.02.005Get rights and content

Abstract

We present an algorithm to generate point distributions with high-quality blue noise characteristics on discrete surfaces. It is based on the concept of Capacity-Constrained Surface Triangulation (CCST), which approximates the underlying continuous surface as a well-formed triangle mesh with uniform triangle areas. The algorithm takes a triangle mesh and the number of sample points as input, and iteratively alternates between optimization of the geometry (positions) of the points and optimization of their topology (connectivity) until convergence. Since the method is relaxation-based, it allows precise control over the number of sample points. Differential domain analysis shows that the point distribution of CCST exhibits typical blue noise characteristics, superior to other relaxation-based sampling methods and is very efficient compared to other traditional dart-throwing methods. We generalize CCST to non-uniform sampling by incorporating a density function. This can be useful in many geometry processing applications, such as curvature-aware remeshing.

Highlights

► A method efficiently samples points with blue noise property on discrete surfaces. ► The surface triangulation is well-formed with quite uniform triangle areas. ► The vertex set is comparable with other methods in quality of blue noise property. ► The method can easily control the number of sample points. ► It is compatible with arbitrary density to produce non-uniform distribution.

Introduction

The problem of sampling, or point distribution, is ubiquitous in computer graphics [14]. It can be formulated as follows: given a domain S and the number of point samples n, position n points within the domain to form a pattern that satisfies some user-specified preferences. In practice, different applications require different patterns, but it seems that among numerous sampling patterns, the most useful are those which have so-called “blue noise” characteristic, which, in a nutshell, attest to high spatial uniformity and low regularity of the distribution. A good generator of blue noise distributions tends to replace low frequency aliasing with high frequency noise in order to be less visually objectionable [16], thus can be used in applications including rendering, sensing, imaging and geometry processing. One well-known class of sample patterns having the blue noise characteristic is the so-called Poisson Disk distribution [7], where the points are positioned such that the disk with an appropriate radius centered at each point is empty of other points.

Because of its importance, a large volume of work investigating the generation of blue noise sample patterns on the plane exists. However, the problem of efficiently generating such sample patterns on non-planar surfaces is more difficult and much less work focuses on this issue. Most of the existing methods use either surface parameterization to reduce the surface case to the planar case, introducing distortions into the patterns, or extend the classical dart throwing algorithm [8] from the plane to the surface, thus are computationally expensive since geodesic distances must be used. Moreover, in some application domains such as LED displays, the number of “sample points” is required to be explicitly set in order to display an image using a given budget of LEDs, while the variants of the dart throwing algorithm cannot achieve this effect.

To meet the strong demand for an efficient and high-quality surface sampling method affording precise control over the number of points, we present a new relaxation-based approach for blue noise sampling on surfaces. It is an extension of the so-called Capacity-Constrained Delaunay Triangulation (CCDT) method [25], a Delaunay triangulation of a planar domain having as-uniform-as-possible triangle areas. The point distributions generated by CCDT have blue noise characteristics, and the algorithm runs much faster than all other state-of-the-art methods. We generalize the CCDT method to the surface case by restricting the movement of points to the surface, which is approximated by a triangular mesh, and minimizing the variance of triangle areas. We call this new method Capacity-Constrained Surface Triangulation (CCST) due to its uniform-area property. Taking a surface discretized as a piecewise-linear triangle mesh and the number of sample points as input, CCST generates a new triangle mesh lying on the input surface, having uniform area distributions. Similar to CCDT, the CCST algorithm consists of two alternating phases. The first phase optimizes the geometry (positions) of points to equalize the triangle areas with the current connectivity, and the second phase regenerates the topology (connectivity) while keeping the geometry fixed. Using frequency domain analysis, the sampling patterns obtained after convergence of the algorithm are shown to possess typical blue noise characteristics.

Although CCST can be considered a direct extension of the CCDT method, the fundamental difference between the flat two-dimensional plane and the curved 2-manifold surface in three dimensions implies that the most important and difficult part of the CCST algorithm is moving points along the surface. We solve this problem by approximating the local surface around each mesh vertex with an osculating torus, for which the approximation error is second order [18]. The objective energy which we try to optimize when moving the points on the osculating torus is non-quadratic, thus we resort to a second-order Taylor expansion to obtain a quadratic approximation of the energy, reducing the problem to a 2×2 linear system per vertex.

The CCST algorithm can also be extended to non-uniform sampling by incorporating a density function into the objective energy. This is useful for generating point distributions having different sampling densities on different regions of the surface, which are useful in many sampling-based applications, such as curvature-aware sampling.

The contribution of this work is a simple algorithm for generating sample patterns on surfaces with the following advantages:

  • The resulting patterns possess superior blue noise characteristics.

  • The algorithm allows precise control over the number of sample points generated.

  • The algorithm is very fast compared to other relevant methods.

Section snippets

Related work

A blue noise sample pattern contains spatially uniform points while keeping the distribution irregular. In this section we briefly review previous work on both the planar and surface sampling problems, and the methods used to analyze and evaluate the sample quality.

Capacity-Constrained Surface Triangulations (CCST)

In this section, we first give an overview of the CCST algorithm and the notations, and then we elaborate on each step in detail.

CCST for non-uniform surface sampling

The CCST algorithm described in Section 3 generates uniform distributions on surfaces having good blue noise properties. However, in many cases, non-uniform distributions which conform to a given density are required. For instance, surface regions with high curvature may require denser point distributions to achieve a more precise sampling accuracy. In this section we extend the uniform CCST algorithm to the non-uniform case.

Given a piecewise-constant density function ρ(T) defined on triangles

Experimental results

In this section, we show results of the CCST algorithm in both the uniform and non-uniform cases. We compare them with other competing algorithms and measure their quality using differential domain analysis. The efficiency of CCST is also analyzed.

Conclusion

We have presented the CCST algorithm—a novel and efficient approach to generate point distributions on a surface. The algorithm alternates between a geometry optimization phase, which minimizes the variance of triangle capacities, and an edge-flipping based re-triangulation phase, until convergence. When applied to a uniform distribution, the result has been shown to possess superior blue noise characteristics. Our method is the first relaxation-based approach to generate blue noise samples on

Acknowledgments

We thank Rui Wang for providing source code for differential domain analysis of point sets, and Dongming Yan for providing an executable for generating sample patterns on surfaces. We also thank Zhonggui Chen and Renjie Chen for helpful comments during our work. The work of L. Liu was partially supported by the National Natural Science Foundation of China (61070071), the 973 National Key Basic Research Foundation of China (2009CB320801), and the Fundamental Research Funds for the Central

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