Closed Form Transmittance in Heterogeneous Media Using Cosine Noise
We present an analytically integrable noise function with similar computational cost and quality to Perlin’s noise. We show how to evaluate transmittance integrals through our proposed noise function in a closed form. Such evaluation requires only two samples of our noise function. In contrast, previous methods require a number of samples that is proportional to the resolution of the noise to evaluate a transmittance integral. We also propose a distance importance sampling method for our noise function, which avoids the limitations of delta tracking. We compare our method to delta tracking. As the resolution of the noise increases exponentially with respect to the number of octaves of noise, our method becomes much faster. Additionally, with distance importance sampling, the probability density function of the samples can be calculated analytically, allowing for fast multiple importance sampling. We also discuss the limitations of our method regarding the shape of participating media and provide alternative approaches to overcome these limitations. Finally, we propose approximated solutions that allow for the use of our method in real-time applications at the cost of small bias.