Computing the film plane distance in perspective camera ?

[xacond00]

I was looking into implementation of perspective camera, and noticed "importance" function doesn't scale with film plane distance squared at all.
Every other implementation from scientific papers defines the pdf/importance as:
(film_distance)^2 / (A * cos(theta)^3)
but yours doesn't ?
Specifically I mean these paper and their related sources:
https://bfraboni.github.io/mvpt19/index.html
https://bfraboni.github.io/eg2022/index.html
Could you explain why there might be such a discrepancy ?

Only thing that comes to mind, is that the film distance in Mitsuba 3 is always normalized to one ? But I can't really understand why, and how ? The "film" dimensions in that case cannot be physically correct ? Ie. when I look at bounds of generated samples in camera space, they fall into around (-1,1) at z_near distance, even though shouldn't they physically fit into a 35mm film ?
But then again, the papers also define the film plane area and distance related to resolution and not really any physical dimensions...

Should I just consider the m_normalization to correspond to the whole d^2 / A term then ?
Or how should I go around computing the film distance in Mitsuba ? I've tried:
m_film_dist = 0.5f / dr::tan(dr::deg_to_rad(m_x_fov * 0.5f));,
however it's missing the scale in the nominator, and I have no idea how it should be related to the computed film area:

Point3f pmin(m_sample_to_camera * Point3f(0.f, 0.f, 0.f)), pmax(m_sample_to_camera * Point3f(1.f, 1.f, 0.f));  
m_image_rect.reset();
m_image_rect.expand(Point2f(pmin.x(), pmin.y()) / pmin.z());
m_image_rect.expand(Point2f(pmax.x(), pmax.y()) / pmax.z());  
m_normalization  = 1.f / m_image_rect.volume();  

Any help is much appreciated. And sorry for bothering again, I really cannot find any sources, which might explain this topic a little bit better...

[njroussel]

Hi @xacond00

I'm unfamiliar with those works, so take the following with a pinch of salt - there might be some explanations that are specific to whatever setup they are doing/using in their work.

Fundamentally, where the film physically lies on the camera's z axis is irrelevant (assuming you're scaling it to keep the correct fov). As you've pointed out, we chose d=1 in PerspectiveCamera::importance(), so the measure should now be 1 / (A * cos(theta)^3). The 1/A term of that measure is indeed computed in m_normalization. Semantically, it should be equivalent to inverse area occupied by the film in camera space. So to compute it, we must figure out where the top-left corner and bottom right corners of the film are in camera space. The trick is to to use their respective sample space equivalent, i.e (0, 0, 0) and (1,1,1) and map them to camera space using m_sample_to_camera. From there we can compute the area using the last 4 lines of the snippet you copied.
I guess that the important bit is that m_sample_to_camera is coherent with the seemingly arbitrary choice of d=1. If you have a look at Transform::perspective() it is explicitly built such that points are projected onto a plane at z=1 in camera space, hence d=1.
I hope this helped.

[xacond00]

where the film physically lies on the camera's z axis is irrelevant (assuming you're scaling it to keep the correct fov

I think this is what I wanted to hear. The film distance stays constant, while the whole film scales instead to account for fov.

So to compute it, we must figure out where the top-left corner and bottom right corners of the film are in camera space.

Yes, this is the one part I understood. However I got really confused by the "other sources" using the actual film resolution to compute the area, instead of some "projected" plane area, which was indeed quite weird.

I think the film_distance^2 measure was used there to account for the film area being scaled with the film resolution, thus my main concern was that my film distance, has the same relative scale to the film plane area. Which it apparently does, when the film plane distance is indeed normalized to 1 and film area scales according to the fov, instead of the other way around.

One other thing, in thin lens camera, the film distance should be just equal to m_focus_distance, right ? As I see, in the sample_ray method the point is scaled to fit onto a "focal plane", instead of a film plane:

// Sampled position on the focal plane
Point3f focus_p = near_p * (m_focus_distance / near_p.z());

I'm forever grateful, and will contribute to the project, once my research is done.

[njroussel]

One other thing, in thin lens camera, the film distance should be just equal to m_focus_distance, right ?

Yes, I believe so.