committed by
Mathias Agopian
parent
30979124b5
commit
bc40ff7f19
@@ -2016,7 +2016,7 @@ I(f(n, v, \alpha)) \equiv \frac{1}{N}\sum_{\forall \, i \in image} f(l_i,
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\end{equation}$$
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However, in practice we're using _importance sampling_ which needs to take the $pdf$ of the distribution
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into account and adds a term $\frac{\left<\VoH\right>}{D(h_i, \alpha)\left<\NoH\right>}$.
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into account and adds a term $\frac{4\left<\VoH\right>}{D(h_i, \alpha)\left<\NoH\right>}$.
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See Importance Sampling For The IBL section:
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$$\begin{equation}\label{iblImportanceSampling}
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@@ -3486,15 +3486,15 @@ The _important direction samples_ $l_i$ are calculated as the reflection of $v$
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**do not** have the same PDF as $h_i$. The PDF of a transformed distribution is given by:
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$$\begin{equation}
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p(T_r(x)) = p(x) |J(T_r)|
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p(T_r(x)) = p(x) |J(T_r)|^{-1}
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\end{equation}$$
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Where $|J(T_r)|$ is the determinant of the Jacobian of the transform. In our case we're considering
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the transform from $h_i$ to $l_i$ and the determinant of its Jacobian is given in \ref{iblPDF}.
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$$\begin{equation}\label{iblPDF}
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p(l,v,\Theta) = D(h,\alpha) \left< \NoH \right> |J_{h \rightarrow l}| \\
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|J_{h \rightarrow l}| = \frac{1}{4 \left< \VoH \right>}
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p(l,v,\Theta) = D(h,\alpha) \left< \NoH \right> |J_{h \rightarrow l}|^{-1} \\
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|J_{h \rightarrow l}| = 4 \left< \VoH \right>
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\end{equation}$$
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### Choosing important directions
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