Light Transport, the Right Way

The above equation and geometry for \(\Phi_1\) is just:

\[ \Phi_1 = \int_{\A{0}} \int_{\A{1}} L_e(\pt_1 \rightarrow \pt_0) \cdot \Gterm{0}{1} \cdot d\A{1} d\A{0} \]

Notice that there are no BRDF \({\color{colorbrdf} f_r(\cdots)}\) terms because there are no interior nodes.

If we want, we can convert this to an integral over only the left-most surface by converting the sampling of the other surface \(\A{1}\) into a sampling over solid angle \(\Omega(\pt_0)\) from the first surface.

Recall the following identity from differential geometry:

\[ d\A{k+1} = d\omega(\pt_k) \frac{ || \pt_{k+1} - \pt_k ||^2 }{ | \cosdir{k+1}{k} | } \]

We simply apply this to \(d\A{1}\), giving:

\begin{align*} \require{cancel} \Phi_1 &= \int_{\A{0}} \int_{\Omega(\pt_0)} L_e(\pt_1 \rightarrow \pt_0) \cdot {\color{colorgeom} \frac{ | \cosdir{0}{1} | \cancel{| \cosdir{1}{0} |} }{ \cancel{|| \pt_1 - \pt_0 ||^2} }} \cdot d\omega(\pt_0) \frac{ \cancel{|| \pt_1 - \pt_0 ||^2} }{ \cancel{| \cosdir{1}{0} |} } d\A{0}\\ &= \int_{\A{0}} \int_{\Omega(\pt_0)} L_e(\pt_1 \rightarrow \pt_0) \cdot | \cosdir{0}{1} | \cdot d\omega(\pt_0) d\A{0} \end{align*}

This is nothing but the measurement equation!

The equation and geometry for \(\Phi_2\) is:

\[ \Phi_2 = \int_{\A{0}} \int_{\A{1}} \int_{A(\pt_2)} L_e(\pt_2 \rightarrow \pt_1) \cdot \Gterm{0}{1} \cdot \fr{0}{1}{2} \cdot \Gterm{1}{2} \cdot d A(\pt_2) d\A{1} d\A{0} \]

Again, if we want the angular form, we can apply the same identity as in the previous example, twice; once each on \(\A{1}\) and \(A(\pt_2)\) to get \(\Omega(\pt_0)\) and \(\Omega(\pt_1)\), respectively. This gives:

\begin{align*} \Phi_2 &= \int_{\A{0}} \int_{\Omega(\pt_0)} \int_{\Omega(\pt_1)} L_e(\pt_2 \rightarrow \pt_1) \cdot {\color{colorgeom} \frac{ | \cosdir{0}{1} | \cancel{| \cosdir{1}{0} |} }{ \cancel{|| \pt_1 - \pt_0 ||^2} }} \cdot \fr{0}{1}{2} \cdot\\ &\quad\quad\quad {\color{colorgeom} \frac{ | \cosdir{1}{2} | \cancel{| \cosdir{2}{1} |} }{ \cancel{|| \pt_2 - \pt_1 ||^2} }} \cdot d\omega(\pt_1) \frac{\cancel{|| \pt_2 - \pt_1 ||^2}}{\cancel{| \cosdir{2}{1} |}} d\omega(\pt_0) \frac{\cancel{|| \pt_1 - \pt_0 ||^2}}{\cancel{| \cosdir{1}{0} |}} d\A{0}\\ &= \int_{\A{0}} \int_{\Omega(\pt_0)} \int_{\Omega(\pt_1)} L_e(\pt_2 \rightarrow \pt_1) \cdot | \cosdir{0}{1} | \cdot \fr{0}{1}{2} \cdot | \cosdir{1}{2} | \cdot d\omega(\pt_1) d\omega(\pt_0) d\A{0} \end{align*}

This can be easily rearranged into:

\[ \Phi_2 = \int_{\A{0}} \int_{\Omega(\pt_0)} \left[ \int_{\Omega(\pt_1)} L_e(\pt_2 \rightarrow \pt_1) \cdot | \cosdir{1}{2} | \cdot \fr{0}{1}{2} \cdot d\omega(\pt_1) \right] \cdot | \cosdir{0}{1} | \cdot d\omega(\pt_0) \cdot d\A{0} \]

The part inside the brackets \([\cdots]\) is nothing but the rendering equation! The outside part is the measurement equation, again.

To get the final result for one-bounce pathtracing, you compute \(\Phi = \Phi_1 + \Phi_2\).

For an arbitrary number \(n\) of bounces, we get back again the original definition for \(\Phi_n\) given at the top of the page.

Again, if we want the angular form, as in the previous examples, we apply the same identity, for the first \(n\) surfaces. This results in:

\begin{align*} \Phi_n &= \int_{\A{0}} \ub{ \int_{\Omega(\pt_0)} \int_{\Omega(\pt_1)} \cdots \int_{\Omega(\pt_{n-1})} }{n} L_e(\pt_n \rightarrow \pt_{n-1}) \left( \prod_{k=0}^{n-1} {\color{colorgeom} \frac{ | \cosdir{k}{k+1} | \cancel{| \cosdir{k+1}{k} |} }{ \cancel{|| \pt_{k+1} - \pt_k ||^2} }} \right) \cdot\\ &\quad\quad\quad \left( \prod_{k=1}^{n-1} \fr{k-1}{k}{k+1} \right) \left( \prod_{k=0}^{n-1} d\omega(\pt_k) \frac{ \cancel{|| \pt_{k+1} - \pt_k ||^2} }{ \cancel{| \cosdir{k+1}{k} |} } \right) d\A{0}\\ &= \int_{\A{0}} \ub{ \int_{\Omega(\pt_0)} \int_{\Omega(\pt_1)} \cdots \int_{\Omega(\pt_{n-1})} }{n} L_e(\pt_n \rightarrow \pt_{n-1}) \left( \prod_{k=0}^{n-1} | \cosdir{k}{k+1} | \right) \left( \prod_{k=1}^{n-1} \fr{k-1}{k}{k+1} \right) \cdot\\ &\quad\quad\quad \ub{ d\omega(\pt_{n-1}) d\omega(\pt_{n-2}) \cdots d\omega(\pt_0) }{n} \cdot d\A{0}\\ &= \int_{\A{0}} \ub{ \int_{\Omega(\pt_0)} \int_{\Omega(\pt_1)} \cdots \int_{\Omega(\pt_{n-1})} }{n} L_e(\pt_n \rightarrow \pt_{n-1}) \left( \prod_{k=1}^{n-1} \ub{ | \cosdir{k}{k+1} | \cdot \fr{k-1}{k}{k+1} }{F_k} \right) \cdot\\ &\quad\quad\quad \ub{ d\omega(\pt_{n-1}) d\omega(\pt_{n-2}) \cdots d\omega(\pt_1) }{n-1} \cdot | \cosdir{0}{1} | \cdot d\omega(\pt_0) \cdot d\A{0}\\ &= \int_{\A{0}} \int_{\Omega(\pt_0)} \left[ \ub{ \int_{\Omega(\pt_1)} \cdots \int_{\Omega(\pt_{n-1})} }{n-1} L_e(\pt_n \rightarrow \pt_{n-1}) \left( \prod_{k=1}^{n-1} F_k \cdot d\omega(\pt_k) \right) \right] | \cosdir{0}{1} | \cdot d\omega(\pt_0) \cdot d\A{0} \end{align*}

This is also a standard result.

In bidirectional pathtracing, we generate one subpath from the eye (\(\pt_0,\pt_1,\cdots,\pt_e\)) and one subpath from the light (\(\qt_0,\qt_1,\cdots,\qt_l\)). These are then connected in the middle to form a single path (\(\pt_0,\pt_1,\cdots,\pt_e,\qt_l,\qt_{l-1},\cdots,\qt_0\)).

We relabel this so that it's a contiguous range (\(\pt_0,\pt_1,\cdots,\pt_s,\pt_{s+1},\cdots,\pt_n\)), where the split is between \(\pt_s\) and \(\pt_{s+1}\). When we do this, the exact original area equation from the top of the page applies (we're sampling over area either way, regardless of which subpath a new node was first connected to)!

For the angular form, however, we have to be a little more careful. Point \(\pt_s\) is sampled, as in standard pathtracing, from point \(\pt_{s-1}\). However, point \(\pt_{s+1}\) is sampled in the other direction, from \(\pt_{s+2}\). Therefore, the angles between \(\pt_s\) and \(\pt_{s+1}\) do not matter in the sampling (although they do matter for the geometry term that remains to connect them).

The equation and geometry are:

\begin{align*} \Phi_n &= \int_{\A{0}} \ub{ \int_{\Omega(\pt_0)} \int_{\Omega(\pt_1)} \cdots \int_{A(\pt_{n-1})} }{n} L_e(\pt_n \rightarrow \pt_{n-1}) \cdot\\ &\quad\quad\quad \left( \prod_{k=1}^{n-1} \fr{k-1}{k}{k+1} \right) \left( \prod_{k=0}^{s-1} {\color{colorgeom} \frac{ | \cosdir{k}{k+1} | \cancel{| \cosdir{k+1}{k} |} }{ \cancel{|| \pt_{k+1} - \pt_k ||^2} }} d\omega(\pt_k) \frac{ \cancel{|| \pt_{k+1} - \pt_k ||^2} }{ \cancel{| \cosdir{k+1}{k} |} } \right) \cdot\\ &\quad\quad\quad {\color{colorgeom} \frac{ | \cosdir{s}{s+1} | | \cosdir{s+1}{s} | }{ || \pt_{s+1} - \pt_s ||^2 }} \left( \prod_{k=s+2}^n {\color{colorgeom} \frac{ \cancel{| \cosdir{k-1}{k} |} | \cosdir{k}{k-1} | }{ \cancel{|| \pt_k - \pt_{k-1} ||^2} }} d\omega(\pt_k) \frac{ \cancel{|| \pt_{k-1} - \pt_k ||^2} }{ \cancel{| \cosdir{k-1}{k} |} } \right) d\A{0}\\ &= \int_{\A{0}} \ub{ \int_{\Omega(\pt_0)} \int_{\Omega(\pt_1)} \cdots \int_{A(\pt_{n-1})} }{n} L_e(\pt_n \rightarrow \pt_{n-1}) \left( \prod_{k=1}^{n-1} \fr{k-1}{k}{k+1} \right) \cdot\\ &\quad\quad\quad \left( \prod_{k=0}^{s-1} | \cosdir{k}{k+1} | \cdot d\omega(\pt_k) \right) \Gterm{s}{s+1} \left( \prod_{k=s+2}^n | \cosdir{k}{k-1} | \cdot d\omega(\pt_k) \right) d\A{0} \end{align*}