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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 Density Function. A $pdf(x)$ gives the probability of a random sample generated being $x$. Long Answer: To start, let's review what Monte Carlo Integration ...

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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 common technique in this case), you have to divide the contribution of each sample by your PDF. This is well exampled here. Note also that in the mentioned ...

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When shading a point on an opaque surface, you need to gather incoming light and weight it with the bidirectional reflectance distribution function (BRDF) of the material. The naive approach is to distribute samples equally over the hemisphere and probe all directions equally for incoming light. This is called uniform sampling (Fig. 1). While this works in ...

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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 sample and calculate the average of the samples. However, this "naive" Monte Carlo integration is said to "converge slowly", i.e. you need a large number of ...

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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 methods are tailored to problems of the form: $$I = \int_{\Omega}{f(x)d\mu(x)} \approx \sum_{k=0}^{N-1}w(x_k)f(x_k)$$ Note that this is not the case for the ...

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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 Sampling for Direct Illumination. Steerable Importance Sampling In this paper they propose a method for sampling the product of the clamped cosine component ...

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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, so then you divide by the current number of frames. To be theoretically "correct", you should also be weighting those samples by $2\pi$, as that's the solid ...

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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 formulate the problem mathematically and analyze it. I do this mainly for myself, so I make it clear for my self but I hope that OP and others will find it useful. ...

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PBRT v3 Page 799: Veach determined it empirically.

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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 distribution. As the projection preserves stratification, you need only use stratified sampling of the disc to get a stratified cosine distribution.

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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 care about the mathematical explanation of why using MIS to combine estimators is valid. I'll have to start with what the purpose of MIS is. The general idea is ...

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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 more than 1024 samples) in order to get it to converge, ie get rid of the noise. As per 2) (from comment below) The 64 primary rays will not hit the same object ...

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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〉in the LD term as well as a different weighting 1/(∑Ni〈n·l〉). These empirical terms have been introduce by Karis to allows to improve the reconstructed ...

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It seems that the answer to my question is that my approach inherently can't work. After doing thinking about it some more and researching existing renderers, none seem to implement what I'm doing, and I think the noise comes from contributions from lights other than the one randomly picked will not be estimated with the PDF. To do it correctly I would need ...

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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 reflection integral is split into direct and indirect illumination and they are estimated separately. The direct illumination can be estimated using more sampling ...

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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 distributed samples. The mapping will keep the strata separated. Usually one sample per stratum is used and the number of strata is set according to the total amount of ...

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In general, the idea of importance sampling is to distribute samples in a way that matches the function being integrated (or more practically, matching some factor in it). This reduces the variance of the output values, allowing the Monte Carlo integration to converge faster. In volume scattering, when a medium is homogeneous, we have the Beer–Lambert law, ...

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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 the BRDF as 0. In fact, you would only need to trace a single reflection ray. A small amount of blur means that a given camera ray will see a small area of the ...

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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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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, by the law of reflection. So, for given incident and outgoing directions, only those microfacets whose normals are aligned with the half-vector are "active". ...

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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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Sampling is typically done in local coordinates first and then mapped to world space. You don't have to change the formula for this, but $\mathbf{p}'$ needs to be in world space when you compute the solid-angle-to-area Jacobian determinant. For a sphere for instance, you would have a warping function that maps two canonical uniform random numbers to a 3D ...

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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 distribution is nonzero everywhere that you want to integrate). Different distributions will give different amounts of variance though. The trick with importance ...

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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 corresponding to this sampling approach that returns the corresponding PDF: 1 over the surface area. So it is not always 1. This is just the most basic way to ...

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One of the first things that many people get wrong with MIS of direct lighting is that you have to always consider the same light source for both light sampling and BSDF sampling. For example, if you sampled Light $L_i$ during light sampling and the ray spawned from BSDF sampling hit $L_j$, you cannot mix them together since the probabilities cannot be used ...

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Edit: I suddenly understood the issue. The noise in the top right corner of (b) is because it's struggling to find the tiny fraction of angles in which the BSDF is not extremely close to zero. Yes your reasoning here is correct. Taken to the extreme if the top most surface was perfectly mirror (dirac delta BRDF), then it would never* sample the right ray ...

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