Nbickford
commented
1 year ago Here's one way of thinking about the 1/pi term: imagine a diffuse, white plane. If I shine light on this plane, the amount of light reflected - summing over all directions it might scatter to - should be the same as the incoming amount of light. In other words, the material model should be energy-conserving.
Mathematically, 1/pi is the right factor to make this hold for the Lambertian normal distribution function; if we multiplied by a greater number, the material would reflect more light than it received, and if it was smaller, the material would appear darker than it should be. (The white furnace test is one way to verify that one's normalization factor for a material model is correct.)
In more detail, energy conservation requires that for any incoming light direction (above the surface), the integral over all outgoing light directions of
max(0, dot(outgoing light direction, normal)) * brdf(incoming light direction, outgoing light direction)
is 1. (The dot product comes from https://en.m.wikipedia.org/wiki/Rendering_equation. See also https://en.m.wikipedia.org/wiki/Bidirectional_reflectance_distribution_function#Physically_based_BRDFs). The Lambertian BRDF corresponds to BRDF(incoming, outgoing) := 1/pi.
In practice, we'd like to have an algorithm to estimate the integral in the rendering equation. For perfect material sampling, we'd choose outgoing light directions with a probability density function equal to the integrand (ignoring the visibility and lighting factors^1). This is possible for the Lambertian BRDF, and the algorithm in the code chooses outgoing directions with pdf = cos theta/pi (where cos theta is the dot product in the equation above).
However, with more complex BRDFs (e.g. GGX), perfectly sampling the BRDF like this isn't always possible. So, usually material model code implements
- A function to evaluate the BRDF
- An algorithm to produce samples that match the BRDF relatively well
- A function that returns the probability density function of that algorithm choosing a certain direction.
Then when choosing samples, we weight them by cos(n, out) BRDF(in, out)/PDF(in, out). This method is known as importance sampling: https://en.m.wikipedia.org/wiki/Importance_sampling .
Finally, if you've read through vk_mini_path_tracer, you might have noticed
that vk_mini_path_tracer never mentions PDFs. This is because I hid this
detail: I always chose material models that could be perfectly sampled, and
so always had cos(n, out) BRDF(in, out)/PDF(in, out) == 1. (Ray Tracing
in One Weekend uses the same idea in Book 1, and introduces PDFs in Book
3.) Vk_raytracing_tutorial_KHR gives more of the story around material
models, in this sense.
[^1]: Ignoring these visibility and lighting factors makes sampling the BRDF easy - but means that our sampling pattern won't match the true value of the rendering equation's integrand. The results will still be correct, but they'll be noisy (if we could somehow sample the integrand perfectly, we'd have no noise, but this is usually impossible except for the simplest of scenes). Techniques like ReSTIR incorporate visibility terms, and lighting terms are often used in sampling in combined techniques like multiple importance sampling.
--Neil
On Sun, May 21, 2023, 00:35 莫成莫成你最棒 @.***> wrote:
the cosine_weight important sampling' pdf is cos_theta / PI why here is 1/PI really Confused
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https://github.com/nvpro-samples/vk_raytracing_tutorial_KHR/blob/8d4ba64b1816ef020bebe33298d496b305df2aca/ray_tracing_gltf/shaders/pathtrace.rchit#LL114C8-L114C9
the cosine_weight important sampling' pdf is cos_theta / PI why here is 1/PI really Confused