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69 votes
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What is Importance Sampling?

Short answer: Importance sampling is a method to reduce variance in Monte Carlo Integration by choosing an estimator close to the shape of the actual function. PDF is an abbreviation for Probability ...
RichieSams's user avatar
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12 votes

What is Importance Sampling?

If you have a 1D function $f(x)$ and you want to integrate this function from say 0 to 1, one way to perform this integration is by taking N random samples in range [0, 1], evaluate $f(x)$ for each ...
JarkkoL's user avatar
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12 votes
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Does cosine weighted hemisphere sampling still require NdotL when calculating contribution for indirect light?

You always need to multiply by the cosine term indeed (that's part of the rendering equation). Though when you do indirect diffuse using ray-tracing and thus monte-carol integration (which is the most ...
user18490's user avatar
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8 votes
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Rendering equation in terms of paths rather than directions

This is a very good question. There is a common misconception that Monte Carlo, or integration is applied "recursively" on the rendering equation. That is not what's happening. Numerical integration ...
lightxbulb's user avatar
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7 votes

Path weight for direct light sampling

By "bounces by the BRDF", I assume you mean picking random directions in the hemisphere and weighting by the BRDF, then averaging over those samples. I guess you're accumulating one sample per frame, ...
Nathan Reed's user avatar
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7 votes

Importance Sampling of Environment Maps

This is not a full answer, I would just like to share the knowledge I obtained by studying two of the papers mentioned in the question: Steerable Importance Sampling and Practical Product Importance ...
ivokabel's user avatar
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5 votes

Generation of stratified cosine weighted samples on hemisphere

The classic method is to uniformly sample the disc at the base of your hemisphere and to project your samples upwards on the hemisphere (eg. compute z from x and y). This yields a cosine weighted ...
Olivier's user avatar
  • 1,585
4 votes

Importance Sampling of Environment Maps

Disclaimer: I have no idea what is the state of the art in the environmental map sampling. In fact, I have very little knowledge about this topic. So this will not be complete answer but I will ...
tom's user avatar
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4 votes
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Multiple Importance Sampling in Path tracer produces Dark Images

Throughout my answer I'll sometimes refer to some results in https://sites.fas.harvard.edu/~cs278/papers/veach.pdf by using [MIS,section_number]. You can skip the following derivation if you don't ...
lightxbulb's user avatar
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4 votes
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Power heuristic in multiple importance sampling

PBRT v3 Page 799: Veach determined it empirically.
chaosink's user avatar
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3 votes
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Importance sampling a cosine distribution

You can sample using absolutely any distribution you want, as long as you weight the results by dividing by the pdf of the sampled distribution. It will converge to the right answer (as long as the ...
Nathan Reed's user avatar
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3 votes
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Can't understand the Importance sampling GGX

In microfacet BRDFs, the half-vector is the same as the microfacet normal. The half-vector is exactly the required normal for a microfacet to reflect light from the incident ray to the outgoing ray, ...
Nathan Reed's user avatar
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3 votes
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Path Tracer - sampling dimensions confusion

Only one path per sample. If you had 64 bounces per first hit and 64 per second hit and so forth you'd never get an image. Edit: And that's why you need to sample each pixel so many times (easily ...
beyond's user avatar
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3 votes
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Importance Sampling in Unreal Engine Formula Derivation (Irradiance term)

I just read notes on moving frostbite to pbr and I found the derivation of the method above. So I will just show the derivation here and quote some of the explanation. One can notice an extra〈n·l〉...
kevinyu's user avatar
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3 votes
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Direct Light Sampling produces way too bright images compared to naive diffuse bounces only

I think, the problem lies in dividing the NEE radiance by PDF (inverse area) twice. Since NEE typically uniformly samples an emitter and then samples a point (via area product measure) on that emitter ...
Enigmatisms's user avatar
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2 votes

Efficiently sampling specific surfaces using MIS in path tracing

I will not answer your questions directly, but will try to change your mindset a bit. I hope got your intentions reasonably well... The usual way the path tracing is implemented is that the ...
ivokabel's user avatar
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2 votes
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Generation of stratified cosine weighted samples on hemisphere

If you have a deterministic mapping function which transforms uniformly distributed samples into the desired PDF (cosine shaped in your case), just feed it directly with stratified uniformly ...
ivokabel's user avatar
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2 votes
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Direct illumination, specular surfaces, and MIS

Specular surfaces which use MIS are not perfectly specular like a mirror. They have a small amount of blur, otherwise there is indeed no point in sampling the light as all the samples will evaluate ...
Olivier's user avatar
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2 votes
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Integrating BRDF using Importance Sampling

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 ...
Nathan Reed's user avatar
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2 votes

How can I make the simplest physically based mixed specular-diffuse BRDF (and how to properly sample it)?

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 ...
Peter's user avatar
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2 votes

When sampling direct light, what to do if testing ray been blocked by transparent object?

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 ...
lightxbulb's user avatar
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2 votes
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PBR : Understanding importance sampling

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 ...
Kaan E.'s user avatar
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2 votes
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Importance sampling microfacet BRDF and coordinate frame

The importance sampling methods for the two reflection models you mentioned are different - they pretty much reflect their definition. Cook-Torrance model describes the surface as a collection of ...
Aleš Koblížek's user avatar
2 votes

importance sampling rectangular light

From looking at the equations, it looks to me like alpha must be the angle between the surface normal of the light source, and the ray direction. This is the "projected area" factor: the ...
Nathan Reed's user avatar
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2 votes
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Confusion about the bias during Monte Carlo integration

I think so. I did some numerical experiments using Monte-Carlo intergation. discard no-discard within-range uniform GT $x^2,x\in[0, 2)$ 14.705318 2.66772274 2.6668968 2.666666 $\sin x, x\in[0, \pi)$...
Raven's user avatar
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2 votes
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Cosine weighted hemisphere sampling is a little bit darker and arguably noisier than reference

One thing confuses me in your code. Since you say that you are sampling BRDF and light source together, I figured it might be possible that you did change the sampling methods between ...
Enigmatisms's user avatar
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2 votes
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How to approach implementing Cook-Torrence microfacet brdf?

Let's take a look at what' going on here. The Monte Carlo estimator is given by: $$ \hat{I} = \sum_{i=1}^N\frac{f(X_i)}{p(X_i)} $$ The $f(x)$ (evaluation function) here can be writen as (simplified ...
Enigmatisms's user avatar
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2 votes
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How does a path tracer capture light falloff if it deals exclusively with radiance?

The inverse square law does not apply to radiance. We can compute irradiance as \begin{align} E(x, \Omega) &= \int_{\Omega} L_i(x, \omega_i) \underbrace{\cos\theta_x}_{N_x \cdot \omega_i} \, dA(\...
lightxbulb's user avatar
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1 vote
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Confusion about light PDF

Most of the answer is in the text you linked: Shapes almost always sample uniformly by area on their surface. Therefore, we will provide a default implementation of the Shape::Pdf() method ...
Olivier's user avatar
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