We present a fast, scalable algorithm to generate high-quality blue noise point distributions of arbitrary density functions. At its core is a novel formulation of the recently-introduced concept of capacityconstrained Voronoi tessellation as an optimal transport problem. This insight leads to a continuous formulation able to enforce the capacity constraints exactly, unlike previous work. We exploit the variational nature of this formulation to design an efficient optimization technique of point distributions via constrained minimization in the space of power diagrams. Our mathematical, algorithmic, and practical contributions lead to high-quality blue noise point sets with improved spectral and spatial properties.

In this contribution, we introduce a new error-diffusion scheme that produces higher quality results. The algorithm is faster than the universally used Floyd-Steinberg algorithm, while maintaining its original simplicity. The efficiency of our algorithm is based on a deliberately restricted choice of the distribution coefficients. Its pleasing nearly artifact-free behavior is due to the off-line minimization process applied to the basic algorithm's parameters (distribution coefficients). This minimization brings the Fourier spectra of the selected key intensity levels as close as possible to the corresponding "blue noise" spectra. The continuity of the algorithm's behavior across the full range of intensity levels is achieved thanks to smooth interpolation between the distribution coefficients corresponding to key levels. This algorithm is applicable in a wide range of computer graphics applications, where a color quantization algorithm with good visual properties is needed.

Figure 1: A high dynamic range 1024×512 environment map [Debevec 98] sampled with 3000 point lights. In this image, importance density is represented by the lightness of the background. It took 0.064 seconds on a 2.6 GHz P4 to generate this point set. Similar results using a hardware accelerated Lloyd relaxation [Hoff et al. 1999] required 1 second, while Structured Importance Sampling [Agarwal et al. 2003] took 1393 seconds. AbstractThis paper presents a novel method for efficiently generating a good sampling pattern given an importance density over a 2D domain. A Penrose tiling is hierarchically subdivided creating a sufficiently large number of sample points. These points are numbered using the Fibonacci number system, and these numbers are used to threshold the samples against the local value of the importance density. Pre-computed correction vectors, obtained using relaxation, are used to improve the spectral characteristics of the sampling pattern. The technique is deterministic and very fast; the sampling time grows linearly with the required number of samples. We illustrate our technique with importance-based environment mapping, but the technique is versatile enough to be used in a large variety of computer graphics applications, such as light transport calculations, digital halftoning, geometry processing, and various rendering techniques.

Figure 1: A high-quality animated production model (Ptex T-rex model c Walt Disney Animation Studios.) rendered in real time under directional and environment lighting using LEADR mapping on an NVidia GTX 480 GPU. The surface appearance is preserved at all scales, using a single shading sample per pixel. Combined with adaptive GPU tessellation, our method provides the fastest, seamless, and antialiased progressive representation for displaced surfaces. AbstractWe present Linear Efficient Antialiased Displacement and Reflectance (LEADR) mapping, a reflectance filtering technique for displacement mapped surfaces. Similarly to LEAN mapping, it employs two mipmapped texture maps, which store the first two moments of the displacement gradients. During rendering, the projection of this data over a pixel is used to compute a noncentered anisotropic Beckmann distribution using only simple, linear filtering operations. The distribution is then injected in a new, physically based, rough surface microfacet BRDF model, that includes masking and shadowing effects for both diffuse and specular reflection under directional, point, and environment lighting. Furthermore, our method is compatible with animation and deformation, making it extremely general and flexible. Combined with an adaptive meshing scheme, LEADR mapping provides the very first seamless and hardware-accelerated multi-resolution representation for surfaces. In order to demonstrate its effectiveness, we render highly detailed production models in real time on a commodity GPU, with quality matching supersampled ground-truth images.