Veach thesis - formula question

[spectral]

Hi all,

I'm currently looking at the Veach thesis and at the page 223 there is the following formula :

path_formulation_example.png (19.64 KiB) Viewed 7363 times

where

geometric_term.png (3.06 KiB) Viewed 7363 times

What is surprising me in this example is that the light is attenuated by the distance... of course it is normal to use G(x0 -> x1) (To convert the emission from the whole light source surface to the point x1).

But I don't understand why we continue to use G for the point x2 ? (and not simply use the solid-angle instead) ?

Thx

[ingenious]

Because this function is integrated with (product) area measure, not solid angle measure. When you estimate that integral using Monte Carlo, and sample path vertices in a random walk via directional sampling, the pdfs of the resulting vertices, when converted to the area measure, have the same G term, which then cancels out. The conversion from solid angle to area measure is given by formula (8.10), which includes most of the terms in G.

[spectral]

Thanks Ingenious,

It mean that if I move my camera at (by example) 10x the current distance, the image will become darker ? (because G(x2->x3)) ?!!

[ingenious]

It mean that if I move my camera at (by example) 10x the current distance, the image will become darker ? (because G(x2->x3)) ?!!

Whether the image will become darker depends on what you're looking at If you're looking at a diffuse surface with a wider extent than your viewing frustum, the image won't become darker. The path throughput, i.e. the inner part of f_j(x), without Le and We, will become smaller, and thus the whole f_j(x) will be smaller for a single path. But then also We (the camera importance function) will have larger support, so "more paths" will contribute to the image, and you'll get the same brightness in the end.

[spectral]

Thanks,

I have miss the "We" term, but how to compute it ?

[ingenious]

I have miss the "We" term, but how to compute it ?

That's your pixel filter. You don't need to do anything special. When the camera moves back, it sees more points, hence the larger support of We. In another thread, I gave a link to Dietger's notes on sampling techniques, which covers bidirectional path tracing.

[spectral]

As usual, thanks Ingenious for your answers...

I have also take a look in pbrt and Dietger thesis (very well explained).
Everything is clear now... and even easy. This formulation sounds very practical... I'll give some try...

Thanks

[ingenious]

This formulation sounds very practical...

And it is extremely useful and insightful theoretically. Not that many people appreciate it though.

[spectral]

Sure,

What's useful is that we can separate the rendering integral in several one, so we can mix them and solve each with different technique !
I even have some ideas for my own algorithm and constraint

It will be funny to play with

Thanks

[Igors]

The G formula is a simplified solid angle calculation. But it works well only if light "far enough" (comparing to light area). If x-difference is small it does not produce correct results