$$L( x'\to x'' )=L_{e}( x'\to x')+\int_{A }L_{i}( x \to x' )f_{s}(x \to x' \to x'')G(x \leftrightarrow x')dx $$
into the measurement equation:
$$I = \int_{A_{lens}} \int_{A_{film}} W_{e}(x_{film}\rightarrow x_{lens})L(x_{film}\rightarrow x_{lens}) G(x_{film}\leftrightarrow x_{lens}) dx_{film} dx_{lens}$$
recursive expand it, we get:
\(I = \sum_{k=1}^{\infty}\int_{A^{k+1}}L_{e}(x_{0}\rightarrow x_{1})G(x_{0}\leftrightarrow x_{1})\prod_{i=1}^{k-1}f_{s}(x_{i-1} \rightarrow x_{i} \rightarrow x_{i+1})G(x_{i} \leftrightarrow x_{i+1})\)
\(\:\:\:\:\:\:\:\:W_{e}(x_{k-1} \rightarrow x_{k}) dA(x_{0})...dA(x_{k}) \)
\(= \int_{A^{2}}L_{e}(x_{0}\rightarrow x_{1})G(x_{0}\leftrightarrow x_{1})W_{e}(x_{0} \rightarrow x_{1})dA(x_{0})dA(x_{1})\)
\(+ \int_{A^{3}}L_{e}(x_{0}\rightarrow x_{1})G(x_{0}\leftrightarrow x_{1})f_{s}(x_{0} \rightarrow x_{1} \rightarrow x_{2})G(x_{1}\leftrightarrow x_{2})\)
\(\:\:\:\:\:\:\:\:W_{e}(x_{1} \rightarrow x_{2})dA(x_{0})dA(x_{1})dA(x_{2})\)
\(+ \int_{A^{4}}L_{e}(x_{0}\rightarrow x_{1})G(x_{0}\leftrightarrow x_{1})f_{s}(x_{0} \rightarrow x_{1} \rightarrow x_{2})G(x_{1}\leftrightarrow x_{2})f_{s}(x_{1} \rightarrow x_{2} \rightarrow x_{3})\)
\(\:\:\:\:\:\:\:\:G(x_{2}\leftrightarrow x_{3})W_{e}(x_{2} \rightarrow x_{3})dA(x_{0})dA(x_{1})dA(x_{2})dA(x_{3})\)
\(+\:\: ............\)
In English:
measurement result (radiance)
= integrate all measured value with path length 1 to path length infinity
= integrated path length 1 measured value ( direct visible light)
+ integrated path length 2 measured value ( direct lighting )
+ integrated path length 3 measured value ( first bounce lighting )
+ ....
All these paths can be built by tracing ray from camera or from light, or from both and connect together, there are many (be more specific, n + 1 methods if the path contains n vertices) ways to form a path. We can pick out a length 3 (first bounce, 4 vertices) path \(x_{0}x_{1}x_{2}x_{3}\)as case study (for all different lengths it's still the same thing, just feel length 3 is on a sweet spot to cover the idea). Follow Veach's naming convention, \(s\) stands for the path vertices start tracing from light, \(t\) stands for the path vertices start tracing from camera (or eye). For path length 3, s + t = 4, we have 5 kinds of ways to form the path:
\(s = 0, t = 4\) trace ray from camera until it hits the light (naive path tracing)
\(s = 1, t = 3\) trace ray from camera and connect to a sample on light (commonly used path tracing)
\(s = 2, t = 2\) trace ray from both light and camera, connect them together
\(s = 3, t = 1\) trace ray from light and connect to a sample on camera lens ( t = 1 style light tracing )
\(s = 3, t = 1\) trace ray from light and connect to a sample on camera lens ( t = 1 style light tracing )
\(s = 4, t = 0\) trace ray from light until it hits the camera lens (t = 0 style light tracing )
The function to be integrated for path \(x_{0}x_{1}x_{2}x_{3}\) is the same for any s, t combinations:
\(f(\bar{x}) = L_{e}(x_{0}\rightarrow x_{1})G(x_{0}\leftrightarrow x_{1})f_{s}(x_{0} \rightarrow x_{1} \rightarrow x_{2})G(x_{1}\leftrightarrow x_{2})f_{s}(x_{1} \rightarrow x_{2} \rightarrow x_{3})\)
\(\:\:\:\:\:\:\:\:G(x_{2}\leftrightarrow x_{3})W_{e}(x_{2} \rightarrow x_{3})\)
The pdf to form the path is different from one combination to another combination though. In Veach97 the pdf usually measured in "projection solid angle" space. It is defined as:
$$pdf_{\omega^{\perp}}(x, \omega) = pdf_{\omega}(x, \omega) / \cos\theta$$
and since we can transform pdf from solid angle space to area space as:
$$pdf_{A}(x, {x}') = pdf_{\omega}(x, \omega)\cos\theta_{{x}'}/d(x\leftrightarrow{x}')^{2}$$
The introduction of projection solid angle brings the geometry term \(G\) into battle:
$$pdf_{A}(x, {x}') = pdf_{\omega^{\perp}}(x, \omega)G(x\leftrightarrow{x}')$$
With this projection solid angle \(pdf_{\omega^{\perp}}\), the \(pdf_{A}\) for different s, t combination can be written as:
\(pdf_{A}(\bar{x}_{s=0,t=4}) =pdf_{A}(x_{3})pdf_{\omega^{\perp}}(x_{3}\rightarrow x_{2})G(x_{2}\leftrightarrow x_{3}) pdf_{\omega^{\perp}}(x_{2}\rightarrow x_{1})G(x_{1}\leftrightarrow x_{2})\)
\(\:\:\:\:\:\:\:\:pdf_{\omega^{\perp}}(x_{1}\rightarrow x_{0})G(x_{0}\leftrightarrow x_{1})\)
\(pdf_{A}(\bar{x}_{s=1,t=3}) =pdf_{A}(x_{3})pdf_{\omega^{\perp}}(x_{3}\rightarrow x_{2})G(x_{2}\leftrightarrow x_{3}) pdf_{\omega^{\perp}}(x_{2}\rightarrow x_{1})G(x_{1}\leftrightarrow x_{2})\)
\(\:\:\:\:\:\:\:\:pdf_{A}(x_{0})\)
\(pdf_{A}(\bar{x}_{s=2,t=2}) =pdf_{A}(x_{3})pdf_{\omega^{\perp}}(x_{3}\rightarrow x_{2})G(x_{2}\leftrightarrow x_{3}) pdf_{\omega^{\perp}}(x_{1}\leftarrow x_{0})G(x_{0}\leftrightarrow x_{1})\)
\(\:\:\:\:\:\:\:\:pdf_{A}(x_{0})\)
\(pdf_{A}(\bar{x}_{s=3,t=1}) =pdf_{A}(x_{3})pdf_{\omega^{\perp}}(x_{2}\leftarrow x_{1})G(x_{1}\leftrightarrow x_{2}) pdf_{\omega^{\perp}}(x_{1}\leftarrow x_{0})G(x_{0}\leftrightarrow x_{1})\)
\(\:\:\:\:\:\:\:\:pdf_{A}(x_{0})\)
\(pdf_{A}(\bar{x}_{s=4,t=0}) =pdf_{\omega^{\perp}}(x_{3}\leftarrow x_{2})G(x_{2}\leftrightarrow x_{3})pdf_{\omega^{\perp}}(x_{2}\leftarrow x_{1})G(x_{1}\leftrightarrow x_{2}) \)
\(\:\:\:\:\:\:\:\:pdf_{\omega^{\perp}}(x_{1}\leftarrow x_{0})G(x_{0}\leftrightarrow x_{1})pdf_{A}(x_{0})\)
Divide \(f(\bar{x})\) by \(pdf_A\), we got five different Monte Carlo estimators \(C_{s,t}^{*}\)for path \(x_{0}x_{1}x_{2}x_{3}\):
\(C_{0,4}^{*} =\frac{ L_{e}(x_{0}\rightarrow x_{1})f_{s}(x_{0} \rightarrow x_{1} \rightarrow x_{2})f_{s}(x_{1} \rightarrow x_{2} \rightarrow x_{3})W_{e}(x_{2} \rightarrow x_{3}) }{pdf_{A}(x_{3})pdf_{\omega^{\perp}}(x_{3}\rightarrow x_{2})pdf_{\omega^{\perp}}(x_{2}\rightarrow x_{1})pdf_{\omega^{\perp}}(x_{1}\rightarrow x_{0})}\)
\(C_{1,3}^{*} =\frac{ L_{e}(x_{0}\rightarrow x_{1})G(x_{0}\leftrightarrow x_{1})f_{s}(x_{0} \rightarrow x_{1} \rightarrow x_{2})f_{s}(x_{1} \rightarrow x_{2} \rightarrow x_{3})W_{e}(x_{2} \rightarrow x_{3}) }{pdf_{A}(x_{3})pdf_{\omega^{\perp}}(x_{3}\rightarrow x_{2}) pdf_{\omega^{\perp}}(x_{2}\rightarrow x_{1})pdf_{A}(x_{0})}\)
\(C_{2,2}^{*} =\frac{ L_{e}(x_{0}\rightarrow x_{1})f_{s}(x_{0} \rightarrow x_{1} \rightarrow x_{2})G(x_{1}\leftrightarrow x_{2})f_{s}(x_{1} \rightarrow x_{2} \rightarrow x_{3})W_{e}(x_{2} \rightarrow x_{3}) }{pdf_{A}(x_{3})pdf_{\omega^{\perp}}(x_{3}\rightarrow x_{2}) pdf_{\omega^{\perp}}(x_{1}\leftarrow x_{0})pdf_{A}(x_{0})}\)
\(C_{3,1}^{*} =\frac{ L_{e}(x_{0}\rightarrow x_{1})f_{s}(x_{0} \rightarrow x_{1} \rightarrow x_{2})f_{s}(x_{1} \rightarrow x_{2} \rightarrow x_{3})G(x_{2}\leftrightarrow x_{3})W_{e}(x_{2} \rightarrow x_{3}) }{pdf_{A}(x_{3})pdf_{\omega^{\perp}}(x_{2}\leftarrow x_{1}) pdf_{\omega^{\perp}}(x_{1}\leftarrow x_{0})pdf_{A}(x_{0})}\)
\(C_{4,0}^{*} =\frac{ L_{e}(x_{0}\rightarrow x_{1})f_{s}(x_{0} \rightarrow x_{1} \rightarrow x_{2})f_{s}(x_{1} \rightarrow x_{2} \rightarrow x_{3})W_{e}(x_{2} \rightarrow x_{3}) }{pdf_{\omega^{\perp}}(x_{3}\leftarrow x_{2})pdf_{\omega^{\perp}}(x_{2}\leftarrow x_{1}) pdf_{\omega^{\perp}}(x_{1}\leftarrow x_{0})pdf_{A}(x_{0})}\)
all geometry terms got cancelled out except the one that connect end points between light path and eye path. The 5 estimators show beautiful symmetry between each opposite side (s1t3 vs s3t1, s0t4 vs s4t0 ). In fact, Veach made all path length cases into a generalized equation and it is one of the largest building block for bidirectional path tracing:
For a path \(\bar{x}_{s,t} = y_{0}...y_{s-1}z_{t-1}...z_{0}\), the unweighted contribution can be computed as \(C_{s,t}^{*} = \alpha_{s}^{L}c_{s,t}\alpha_{t}^{E}\), where light path factor:
\(\alpha_{0}^{L} = 1\)
\(\alpha_{1}^{L} = L_{e}^{(0)}(y_{0}) / P_{A}(y_{0})\)
\(\alpha_{i}^{L} = \frac{f_{s}(y_{i-3} \rightarrow y_{i-2} \rightarrow y_{i-1})}{P_{\omega^{\perp}}(y_{i-2} \rightarrow y_{i-1})} \alpha_{i-1}^{L}\) for \(i \geq 2\)
and eye path factor:
\(\alpha_{0}^{E} = 1\)
\(\alpha_{1}^{E} = W_{e}^{(0)}(z_{0}) / P_{A}(z_{0})\)
\(\alpha_{i}^{E} = \frac{f_{s}(z_{i-3} \rightarrow z_{i-2} \rightarrow z_{i-1})}{P_{\omega^{\perp}}(z_{i-2} \rightarrow z_{i-1})} \alpha_{i-1}^{E}\) for \(i \geq 2\)
and connection factor:
\(c_{0,t} = L_{e}(z_{t-1} \rightarrow z_{t-2})\)
\(c_{s,0} = W_{e}(y_{s-2} \rightarrow y_{s-1})\)
\(c_{s,t} = f_{s}(y_{s-2} \rightarrow y_{s-1} \rightarrow z_{t-1})G(y_{s-1} \leftrightarrow z_{t-1}) f_{s}(y_{s-1} \rightarrow z_{t-1} \rightarrow \rightarrow z_{t-2} ) \) for \(s, t > 0\)
One notation looks a bit weird is that \(L_{e}^{(0)}\) and \(W_{e}^{(0)}\). Veach97 call this "spatial component" and define the radiance/importance as:
\(L_{e}(y_{0} \rightarrow y_{1}) = L_{e}^{(0)}(y_{0})L_{e}^{(1)}(y_{0} \rightarrow y_{1})\)
\(W_{e}(z_{1} \rightarrow z_{0}) = W_{e}^{(0)}(z_{0})W_{e}^{(1)}(z_{1} \rightarrow z_{0})\)
\(f_{s}(y_{-1} \rightarrow y_{0} \rightarrow y_{1}) = L_{e}^{(1)}(y_{0} \rightarrow y_{1})\)
\(f_{s}(z_{1} \rightarrow z_{0} \rightarrow z_{-1}) = W_{e}^{(1)}(z_{1} \rightarrow z_{0})\)
The purpose of this notation, according to Veach, is to reduce the number of special cases that need to be considered, by interpreting the directional component of emission as a BSDF. Though personally, I don't feel this particular tempting to me. Writing a branch VS writing another bsdf interface for light and camera, I prefer a branch.
My fingers are aching by typing latex equation now and it's kinda dry to go through the math, but at this point we have all the stuff I need to implement light tracing : importance, filter, path contribution \(C_{s,t}^{*}\). The next post will focus on the light tracing implementation, and some (not that academic) ways to verify the result.
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