# Path Integral Form with Path Tracing

The description is going to be long and detailed, thanks in advance for your patience!

I am learning the path integral form of light transport equation(LTE) to be able to follow recent rendering papers. The materials I am using are PBRT(chapter 14.4-14.5: link) and SIGGRAPH 2013 course: link.

The path integral form of LTE is:

$$I(x) = \int_{\Omega}f(\overline{x})d\mu(\overline{x})$$

From the SIGGRAPH course's slide, we know that we can estimate $$I$$ by randomly sampling a path, computing its contribution $$f$$, and dividing by its probability $$p$$:

$$\overline{I}(x) = \frac{f(\overline{x})}{p(\overline{x})}$$

From the PBRT's notes and its implementation, it expands the path integral into an infinite sum of integrals on specific lengths of paths. This notation is well shown in the path tracing implementation since the i-th bounce computes the i-th integral and simply summing them up will the final estimate of $$I$$.

$$\overline{I}(x) = \sum_{i=0}^{\infty}\overline{I_{i}}(x)$$

With these two notations based on the path form integral of LTE, my question is how to represent path tracing as the first one(the SIGGRAPH course notation) since most PT implementations I've seen so far is based either on the PBRT's notation(the sum notation) or other notations of LTE.

I have one possible idea that is to follow the formula by generating a path and computing its pdf, but I am not sure about its practical implementation. Should I generate paths for different lengths and do this procedure over and over(since this looks more like the PBRT's notation), or a single path with the correct pdf will suffice?

Or, from another perspective, I think that to apply the SIGGRAPH course's notation in practice, one has to randomly generate paths with different lengths and sum their estimate up since each length corresponds to one term in the expanded notation. Is this correct?

Q1: how to represent path tracing as the first one...

Your materials have brought enough information, the first formula, quoted from your material, presents as

The path integral formulation of light transport formalizes the idea by writing the camera response as an integral over all light transport paths of all lengths in the scene, which given path encompasses

1. the “amount” of light emitted along the path,
2. the light carrying capacity of the path, and
3. the sensitivity of the sensor to light brought along the path.

Can't interpret the formula better

Q2: Should I generate paths for different lengths and do this procedure over and over...

Q3: ...randomly generate paths with different lengths and sum their estimate up since each length...