Bidirectional Path Tracing 3 - Importance and measurement equation

The equation that integrate the power flow through focal lens and land on camera lens looks like this:
$$\Phi = \int_{A_{lens}} \int_{\Omega} L(x, \omega_{i}) cos\theta d\omega dx$$
We got the power, but we actually want the average radiance value in specific pixel (so that we can form the image, which is what ray tracing usually looking for). Is there any way that we can convert this power to radiance? Yes :) In Veach97 the equation actually has another extra term, and the equation does not explicitly integrate out power as result. It looks like this:
$$I = \int_{A_{lens}} \int_{\Omega} W_{e}(x, \omega)L(x, \omega_{i}) cos\theta d\omega dx$$
The original paper name this "measurement equation" and that extra term \(W_{e}(x, \omega)\) is named as importance. Measurement equation try to measure certain value related to power, and importance convert the power to that value. The following description is directly copied from Veach Chapter 4.5 (p115):

"For real sensors, \(W_{e}\) is called the flux responsivity of the sensor. The responding units are \(S/Watt\) (Watt is the unit for power), where \(S\) is the unit of sensor response. Depending on the sensor, \(S\) could represent a voltage, current, change in photographic film density, deflection of a meter needle, etc."

Ya that's how scientists write paper...they view problem in a broader term, try to solve voltage, current, change in photographic film density.....all in one equation....but wait! I just want the average radiance in particular pixel! Can you please just give me the \(W_{e}(x, \omega)\) for that?

It sounds silly now but this question is probably the one that trouble me longest while looking in the paper :| , and I think it's worthy to write it down as memo.

Start with measurement equation above:
\(I = \int_{A_{lens}} \int_{\Omega} W_{e}(x, \omega)L(x, \omega_{i}) cos\theta d\omega dx\)
We can transform the \(\Omega\) part solid angle integration to area space integration over the film area with an additional geometry term \(G\)  (like what we did in previous post):
\(I = \int_{A_{lens}} \int_{A_{film}} W_{e}(x_{film}\rightarrow  x_{lens})L(x_{film}\rightarrow  x_{lens}) G(x_{film}\leftrightarrow  x_{lens}) dx_{film} dx_{lens}\)

The \(A_{film}\) term is the world area of film. Again, we use the non real world CG camera model that film stands in front of lens (instead of behind lens like real world camera) and in the focus distance. That is, we treat the focus plane as film like the following image illustrates:

The Monte Carlo estimator for the area space equation looks like:
\(E(I)=\frac{1}{N}\sum_{i= 1}^{N}\frac{W_{e}(x_{film_{i}}\rightarrow  x_{lens_{i}})L(x_{film_{i}}\rightarrow  x_{lens_{i}}) G(x_{film_{i}}\leftrightarrow  x_{lens_{i}})}{pdf_{A}(x_{film_{i}})pdf_{A}(x_{lens_{i}})}\)
We want the \(W_{e}(x_{film_{i}}\rightarrow  x_{lens_{i}})\) term can transfer this measurement result to be radiance \(L(x_{film_{i}}\rightarrow  x_{lens_{i}})\), which gives us the result:
\(W_{e}(x_{film}\rightarrow  x_{lens})=\frac{pdf_{A}(x_{film})pdf_{A}(x_{lens})}{G(x_{film}\leftrightarrow  x_{lens})} = \frac{d(x_{film}\leftrightarrow  x_{lens})^{2}}{A_{film}A_{lens}cos\theta^{2}}\)
If we look in further to the above equation, \(A_{film}\) is proportional to \(d(x_{film}\leftrightarrow  x_{lens})^{2}\), which means distance is not a factor that would affect the result and we can use \(W_{e}(x_{film}\rightarrow  x_{lens})\) directly for \(W_{e}(x, \omega)\) where \(\omega\) is the shooting ray direction from \(x_{lens}\) to \(x_{film}\)

and now we can write some codes :)
A beautiful symmetry between radiance and importance is shaping up once we have importance in hand:
the image can be rendered through camera shooting rays generated by uniformly sampling camera lens and camera film, capturing incoming radiance through the ray and accumulate it to corresponding pixel. At the end, divide accumulated radiance by sample per pixel. This is the parh tracing style (in its simplest form).

the image can also be rendered through lights shooting rays generated by uniformly sampling light surface and hemisphere outgoing directions, capturing incoming importance through the ray, add light ray carrying weight (this weight value \(\alpha\) will be discussed in later post) multiplied importance to pixel corresponding to the incoming importance. At the end, divided accumulated radiance by total number of samples shoot out from light. This method is usually named as light tracing or particle tracing (I use the former naming since particle tracing short as pt is a bit confusing when mention it together with path tracing)

This two methods should converge to the same expectation value when sample numbers crank up (mathematically and theoretically :) we gonna write the light tracer to prove this by our eye! ) I got to say I was mind blown when I learn this symmetry, it's so Zen that kinda like that ancient Chinese philosopher talk ("Now I do not know whether I was then a man dreaming I was a butterfly, or whether I am now a butterfly, dreaming I am a man." - Zhuangzi)

The above method work as it is when each sample only contribute to one pixel (for example, sample (320.4, 510.6) only contributes radiance (320, 511), sample (128.7, 426.3) only contributes radiance (129, 426)), but that's not the case when we have filter kernel that would potentially contribute to multiple pixels per sample...(a simplest example, 2X2 box filter can contribute radiance to 4 pixels per sample), does the "divide by \(N\)(total number of samples shoot out from light)" still works for light tracing when filter jump in? Nope! we need to do some surgery work on filter if we want to get unbiased result, and we will talk about it in the next post :)