<p>According to Peter Shirely <a href="http://www.cs.virginia.edu/~jdl/bib/globillum/mis/shirley96.pdf" rel="nofollow noreferrer">paper</a>, one pdf(probability density function) can be defined for the union of the light sources and pick only one using a random number from this density:</p>

<p>$$p(x) = \alpha_1 P_1(x) + (1-\alpha_1) P_2(x)$$</p>

<p>This has been referred to as mixture densities. where $\alpha_1 \in [0,1]$ and $P_1$ is a pdf for selecting a point on the $L_1$ geometry (usually $1/A$). If the random number is less than $\alpha_1$ then the total estimate is $(\text{estimated }L_1)/\alpha_1$. </p>

<p>This has been extended for N:
$$p(x) = \alpha_1 P_1(x) + \alpha_2 P_2(x) + \cdots + \alpha_n P_n(x)$$</p>

<p>My question is how one pick two or multiple light sources, instead of just one, from the above density and what the estimate would be in that case.</p>

CC BY-SA 3.0
2017-02-14T23:18:15.520Z
Not sure I'm understanding your question. Are you asking how do you sample from the mixture of $n$ lights? If you want to sample two or more lights, you would just repeat the procedure to sample a single light (with fresh random numbers), right? Or is there something else you're asking?
Nathan Reed
CC BY-SA 3.0
2017-02-16T12:08:29.090Z
Thanks Nathan, you are correct you as to do the procedure again for multiple lights. But I wonder if the above mixture density allows one to pick two or more samples from the distribution.
ali
CC BY-SA 3.0
2017-02-19T20:54:59.087Z
Are you worried about anything in particular with just repeating the process multiple times? Were you concerned about sometimes choosing the same light more than once?
trichoplax is on Codidact now
CC BY-SA 3.0
2017-02-20T12:53:42.807Z
@trichoplax. No. I just want to understand mathematically how one choose multiple samples from a merge density.
ali
<raytracing><lighting><global-illumination><sampling><monte-carlo>
Picking multiple light sources from a mixture densities
ali
# Picking multiple light sources from a mixture densities

(Asked by: username_0 on 2017-02-14)

According to Peter Shirely [paper](http://www.cs.virginia.edu/~jdl/bib/globillum/mis/shirley96.pdf), one pdf(probability density function) can be defined for the union of the light sources and pick only one using a random number from this density:

$$p(x) = \alpha\_1 P\_1(x) + (1-\alpha\_1) P\_2(x)$$

This has been referred to as mixture densities. where $\alpha\_1 \in [0,1]$ and $P\_1$ is a pdf for selecting a point on the $L\_1$ geometry (usually $1/A$). If the random number is less than $\alpha\_1$ then the total estimate is $(\text{estimated }L\_1)/\alpha\_1$.

This has been extended for N:
$$p(x) = \alpha\_1 P\_1(x) + \alpha\_2 P\_2(x) + \cdots + \alpha\_n P\_n(x)$$

My question is how one pick two or multiple light sources, instead of just one, from the above density and what the estimate would be in that case.

**Tags:** raytracing, lighting, global-illumination, sampling, monte-carlo

> **Comments:**
> **username_1**: Not sure I'm understanding your question. Are you asking how do you sample from the mixture of $n$ lights? If you want to sample two or more lights, you would just repeat the procedure to sample a single light (with fresh random numbers), right? Or is there something else you're asking?  
> **username_0**: Thanks username_1, you are correct you as to do the procedure again for multiple lights. But I wonder if the above mixture density allows one to pick two or more samples from the distribution.  
> **username_2**: Are you worried about anything in particular with just repeating the process multiple times? Were you concerned about sometimes choosing the same light more than once?  
> **username_0**: @username_2. No. I just want to understand mathematically how one choose multiple samples from a merge density.  

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## 0 answers

