A Low-Distortion Map Between Triangle and Square

We introduce a low-distortion map between triangle and square. This mapping yields an area-preserving parameterization that can be used for sampling random points with a uniform density in arbitrary triangles. This parameterization presents two advantages compared to the square-root param-eterization typically used for triangle sampling. First, it has lower distortions and better preserves the blue-noise properties of the input samples. Second, its computation relies only on arithmetic operations (+, *), which makes it faster to evaluate

A Simpler and Exact Sampling Routine for the GGX Distribution of Visible Normals

Heitz and d'Eon introduced importance sampling based on the distribution of visible normals (VNDF) and provided analytic solutions for Beckmann and GGX distributions. In this document, we propose an alternative sampling routine for the GGX VNDF that is simpler and exact. The solution provided by Heitz and d'Eon was approximate for GGX because it required a fitted curve: the derivations were made in slope space and the conditional inverse CDF of GGX does not have a closed form with this parameterization. This improved sampling routine uses the property that for unit roughness alpha = 1 the GGX distribution is a uniform hemisphere. Sampling the GGX VNDF is thus equivalent to sampling the projected area of a hemisphere, which can be done exactly

Cracking the Network Code: Four Principles for Grantmakers

As grantmakers and nonprofits are looking for ways to collaborate more effectively, many are experimenting working with and through networks to achieve greater impact. Because networks are by definition loosely controlled and emergent, understanding how to effectively support them feels like a mystery to many grantmakers.GEO's newest publication sets out to crack the code behind the network mystique. In fact, there is a method to working more efficiently and effectively through networks, and a critical first step for grantmakers is adopting a network mindset, which may require dramatic shifts in attitude and behavior for some. "Cracking the Network Code" outlines four principles that comprise the network mindset, illustrates the principles with a range of examples of networks that have achieved real results, and offers practical questions and recommendations to help grantmakers achieve the benefits and avoid common pitfalls of working through networks

Understanding and controlling contrast oscillations in stochastic texture algorithms using Spectrum of Variance

We identify and analyze a major issue pertaining to all power-spectrum based texture synthesis algorithms – from Fourier synthesis to procedural noise algorithms like Perlin or Gabor noise – , namely, the oscillation of contrast (see Figures 1,2,3,7). One of our key contributions is to introduce a simple yet powerful descriptor of signals, the Spectrum of Variance (not to be confused with the PSD), which, to our surprise, has never been leveraged before. In this new framework, several issues get easy to understand measure and control, with new handles, as we illustrate. We finally show that fixing oscillation of contrast opens many doors to a more controllable authoring of stochastic texturing. We explore some of the new reachable possibilities such as constrained noise content and bridges towards very different families of look such as cellular patterns, points-like distributions or reaction-diffusion

Distributing Monte Carlo Errors as a Blue Noise in Screen Space by Permuting Pixel Seeds Between Frames

International audienceRecent work has shown that distributing Monte Carlo errors as a blue noise in screen space improves the perceptual quality of rendered images. However, obtaining such distributions remains an open problem with high sample counts and high-dimensional rendering integrals. In this paper, we introduce a temporal algorithm that aims at overcoming these limitations. Our algorithm is applicable whenever multiple frames are rendered, typically for animated sequences or interactive applications. Our algorithm locally permutes the pixel sequences (represented by their seeds) to improve the error distribution across frames. Our approach works regardless of the sample count or the dimensionality and significantly improves the images in low-varying screen-space regions under coherent motion. Furthermore, it adds negligible overhead compared to the rendering times

Correlation effect between transmitter and receiver azimuthal directions on the illumination function from a random rough surface

International audienceThe resolution of some problems of electromagnetic scattering from random rough surfaces implies the derivation of the illumination function, especially when the geometrical optics approximation is valid. In current models, the shadowing e ects occurring for both the trans- mitter and the receiver are assumed to be independent of their azimuthal directions. This assumption makes it possible to compute separately the shadowing probabilities in each di- rection. However, if the transmitter and receiver azimuthal directions are close, these proba- bilities become strongly correlated. In such a con guration, the uncorrelation approximation induces an overestimation of the average illumination function up to 10% as well as a discon- tinuity. In this paper, assuming a surface with Gaussian process, this correlation is taken into account. Comparisons of our model with Monte Carlo simulations are made and show a very good agreement, which validates our approach

Generating Random Segments from Non-Uniform Distributions

We investigate the generation of random segments from non-uniform distributions, i.e. we aim at designing methods that generate random segments such that the density of these segments converges towards a target distribution as the number of segments increases. The motivation for this study is that recent research has shown that random segments can yield superior results than random points for Monte Carlo integration. Currently, an important limitation of this research area is that random segments can be generated only from uniform distributions, typically over a rectangular or circular domain. The focus of this presentation is to overcome this limitation by introducing three methods for generating random segments from non-uniform distributions. Our algorithms are inspired by the point-sampling variants of rejection sampling, slice sampling and inverse-CDF sampling. We explain how to apply them on analytic distributions as well as arbitrary distributions such as images

Understanding the Masking-Shadowing Function in Microfacet-Based BRDFs

We give a new presentation of the masking-shadowing functions in microfacet-based BRDFs and answer some common questions about their applications. We use the fact that the masking function (or geometric attenuation factor) is constrained by the visible projected area of the microsurface onto the outgoing direction to derive the properties of the exact masking function. We introduce the distribution of visible normals from the microsurface, whose normalization factor is the masking function, and we show how the common form of microfacet-based BRDFs emerges from this distribution. The consequence of this is that only exact masking functions ensure correct normalization of microfacet-based BRDFs. However, the exact masking function that satisfies these properties can be determined only if a microsurface profile is chosen. Our derivation emphasizes that under the assumptions of their respective microsurface profiles, Smith's and the historical V-cavity masking functions are both exact. However, we show that the V-cavity microsurface is closer to a normal map than a displacement map. This intuition explains why this non-realistic model is responsible for wrong specular highlights at grazing view angles. The insights gained from these observations motivate new research directions in the field of microfacet theory. For instance, we show that masking functions are stretch invariant and we show how this property can be used to derive the masking function for anisotropic microsurfaces in a straightforward way. We also discuss future work such as the incorporation of multiple scattering on the microsurface into BRDF models.Ce document a pour but de répondre à des questions récurrentes concernant la fonction d'ombrage dans les BRDFs à microfacettes. Nous utilisons le fait que la fonction d'ombrage soit contrainte par la surface projetée de la microsurface pour dériver sa forme exacte. Nous introduisons le concept de distribution de normales visibles, dont le coefficient de normalisation est la fonction d'ombrage, et nous montrons comment la BRDF émerge de cette distribution. Notre dérivation montre que, sous les hypothèses habituelles de la théorie des microfacettes, la fonction d'ombrage exacte est celle dérivée par Smith. Nous discutons les propriétés de la fonction de Smith ainsi que ses implications sur la normalisation de la BRDF

Filtering Non-Linear Transfer Functions on Surfaces

International audienceApplying non-linear transfer functions and look-up tables to procedural functions (such as noise), surface attributes, or even surface geometry are common strategies used to enhance visual detail. Their simplicity and ability to mimic a wide range of realistic appearances have led to their adoption in many rendering problems. As with any textured or geometric detail, proper filtering is needed to reduce aliasing when viewed across a range of distances, but accurate and efficient transfer function filtering remains an open problem for several reasons: transfer functions are complex and non-linear, especially when mapped through procedural noise and/or geometry-dependent functions, and the effects of perspective and masking further complicate the filtering over a pixel's footprint. We accurately solve this problem by computing and sampling from specialized filtering distributions on the fly, yielding very fast performance. We investigate the case where the transfer function to filter is a color map applied to (macroscale) surface textures (like noise), as well as color maps applied according to (microscale) geometric details. We introduce a novel representation of a (potentially modulated) color map's distribution over pixel footprints using Gaussian statistics and, in the more complex case of high-resolution color mapped microsurface details, our filtering is view- and light-dependent, and capable of correctly handling masking and occlusion effects. Our approach can be generalized to filter other physical-based rendering quantities. We propose an application to shading with irradiance environment maps over large terrains. Our framework is also compatible with the case of transfer functions used to warp surface geometry, as long as the transformations can be represented with Gaussian statistics, leading to proper view- and light-dependent filtering results. Our results match ground truth and our solution is well suited to real-time applications, requires only a few lines of shader code (provided in supplemental material), is high performance, and has a negligible memory footprint