In this paper, we show that applying a linear transformation---represented by a 3 x 3 matrix---to the direction vectors of a spherical distribution yields another spherical distribution, for which we derive a closed-form expression. With this idea, we can use any spherical distribution as a base shape to create a new family of spherical distributions with parametric roughness, elliptic anisotropy and skewness. If the original distribution has an analytic expression, normalization, integration over spherical polygons, and importance sampling, then these properties are inherited by the linearly transformed distributions.
By choosing a clamped cosine for the original distribution we obtain a family of distributions, which we call
Linearly Transformed Cosines
(LTCs), that provide a good approximation to physically based BRDFs and that can be analytically integrated over arbitrary spherical polygons. We show how to use these properties in a realtime polygonal-light shading application. Our technique is robust, fast, accurate and simple to implement.
× stochastic denoised = Figure 1: Illustration of our formulation. We formulate direct illumination as the product of the unshadowed illumination and the illumination-weighted shadow, and we combine real-time shadowless illumination techniques with raytraced and denoised shadows. This yields a ratio estimator of direct illumination that has low variance and can be denoised without blurring shading details.
Physically based shading is transforming the way we approach production rendering, and simplifying the lives of artists in the process. By adhering to physically based, energy-conserving models, one can easily create realistic materials that maintain their properties under a variety of lighting conditions. In contrast, traditional ad hoc models have required extensive tweaking to achieve the same result. Building upon previous incarnations of the course, we present further research and practical advice on the subject, from film and game production.
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