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If a ray intersect a transparent object (regardless of the order) multiply by the colour to get proper attenuation, then continue. Note that the order does not matter. The other options is to ignore transparent objects and check for occlusion. Only if there's no occlusion intersect the set of transparent objects only without caring about order once again, ...


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Leaving the actual derivation of mathematics aside, I'll try to give a general description of how can we combine the NDF with importance sampling in spherical coordinates. What is NDF ? The problem is the following. Intiutively, we need the NDF for "zooming in" from the realm of macrosurfaces to that of microfacets. Mathematically it is needed for ...


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Think of this way: when integrating uniformly over the hemisphere, it's like you are importance-sampling with a constant pdf of $1/2\pi$. The multiplication by $2\pi$ at the end, then, can be seen as the division by the pdf, factored out of the sum because it's constant. The constant pdf is $1/2\pi$ (as opposed to some other number) because it needs to be ...


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I think this is mainly just giving new names to terms and a bit of algebra. $p^\perp(\omega_o)$ is the name given to the probability density $\frac{dP}{d\sigma^\perp}(\omega_o)$. Similarly $p(\omega_o)$ is the name given to $\frac{dP}{d\sigma}(\omega_o)$. A probability density is the amount of probability ($dP$) per unit area ($d\sigma$), so this is kind of ...


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A common way of combining diffuse and specular brdfs is by using a fresnel equation. Essentially, for some specular materials, the amount reflected and transmitted (passed through the object) depends on the angle you view it. For example water will reflect more if you look at it from one angle, but you can see through it if you look at it from another. A ...


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