<p>I think this is mainly just giving new names to terms and a bit of algebra.</p>
<p><span class="math-container">$p^\perp(\omega_o)$</span> is the name given to the probability density <span class="math-container">$\frac{dP}{d\sigma^\perp}(\omega_o)$</span>. Similarly <span class="math-container">$p(\omega_o)$</span> is the name given to <span class="math-container">$\frac{dP}{d\sigma}(\omega_o)$</span>. A probability density is the amount of probability (<span class="math-container">$dP$</span>) per unit area (<span class="math-container">$d\sigma$</span>), so this is kind of just expanding out the pdfs into their definitions.</p>
<p>Then we just need the factor <span class="math-container">$\frac{d\sigma(\omega_o)}{d\sigma^\perp(\omega_o)}$</span> which, from the relationship in the 3rd equation, is <span class="math-container">$\frac{1}{|\omega_o \cdot N(x)|}$</span>.</p>
<p>Since the dot product gives the cosine between vectors, <span class="math-container">$\omega_o \cdot N(x)$</span> = <span class="math-container">$\cos(\theta_o)$</span>. I am not sure why the absolute value gets dropped, but it may just be a typo as it reappears in the following equation in the thesis.</p>
<p>The application of calculus here is in putting in the factor <span class="math-container">$\frac{d\sigma(\omega_o)}{d\sigma^\perp(\omega_o)}$</span>, which is the inverse Jacobian, used for transforming probability densities (or other densities) from one domain to another—here, from the &quot;ordinary solid angle&quot; (<span class="math-container">$\sigma$</span>) domain to that of projected solid angle (<span class="math-container">$\sigma^\perp$</span>). Basically an enhanced version of the chain rule.</p>

CC BY-SA 4.0
2021-12-28T23:46:53.450Z
2021-12-28T23:46:53.450Z
Nathan Reed
<p>I'm reading section 8.2 of Eric Veach's thesis, in regards to the relationship between probability densities per solid angle vs projected solid angle:</p>
<p><span class="math-container">$$\frac{dP}{d\sigma^\perp}(w_o) = \frac{dP}{d\sigma}(w_o)\frac{d\sigma(w_o)}{d\sigma^\perp (w_o)}$$</span>
<span class="math-container">$$\implies p^\perp (w_o) = p(w_o)\frac{1}{cos(\theta_o)}$$</span></p>
<p>where we have used the relationship
<span class="math-container">$$d \sigma^\perp(w_o) = |w_o \cdot N(x)|d \sigma(w_o)$$</span></p>
<p>I think this is just a straightforward application of calculus somewhere, but I can't see how to get to the &quot;implies&quot; part. The result makes sense to me intuitively, but I'd like to follow the math. Starting with the first line, then after doing the substitution using the relationship he provides, would you integrate both sides or something? But then aren't they over different domains?</p>

<monte-carlo><importance-sampling>
Relationship between probability densities
James B
# Relationship between probability densities

(Asked by: username_0 on 2021-12-27)

I'm reading section 8.2 of Eric Veach's thesis, in regards to the relationship between probability densities per solid angle vs projected solid angle:

$$\frac{dP}{d\sigma^\perp}(w\_o) = \frac{dP}{d\sigma}(w\_o)\frac{d\sigma(w\_o)}{d\sigma^\perp (w\_o)}$$
$$\implies p^\perp (w\_o) = p(w\_o)\frac{1}{cos(\theta\_o)}$$

where we have used the relationship
$$d \sigma^\perp(w\_o) = |w\_o \cdot N(x)|d \sigma(w\_o)$$

I think this is just a straightforward application of calculus somewhere, but I can't see how to get to the "implies" part. The result makes sense to me intuitively, but I'd like to follow the math. Starting with the first line, then after doing the substitution using the relationship he provides, would you integrate both sides or something? But then aren't they over different domains?

**Tags:** monte-carlo, importance-sampling


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## 1 answers

### Answer by username_1 on 2021-12-28. Score: 1 | Accepted answer

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 just expanding out the pdfs into their definitions.

Then we just need the factor $\frac{d\sigma(\omega\_o)}{d\sigma^\perp(\omega\_o)}$ which, from the relationship in the 3rd equation, is $\frac{1}{|\omega\_o \cdot N(x)|}$.

Since the dot product gives the cosine between vectors, $\omega\_o \cdot N(x)$ = $\cos(\theta\_o)$. I am not sure why the absolute value gets dropped, but it may just be a typo as it reappears in the following equation in the thesis.

The application of calculus here is in putting in the factor $\frac{d\sigma(\omega\_o)}{d\sigma^\perp(\omega\_o)}$, which is the inverse Jacobian, used for transforming probability densities (or other densities) from one domain to another—here, from the "ordinary solid angle" ($\sigma$) domain to that of projected solid angle ($\sigma^\perp$). Basically an enhanced version of the chain rule.


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