Understanding Radiometry: Calculating Radiance at Point P (0,0,0)

[mnb96]

Hello,
assuming the definition of radiance in http://web.cs.wpi.edu/~emmanuel/courses/cs563/write_ups/chuckm/chuckm_BRDFs_overview.html" , I want to compute the radiance at the point P=(0,0,0) lying on the xy-plane, given a point-source of light at (0,0,1).

The angle $\theta$ in this scenario is 0 because the light source is placed exactly on the zenith of P, so that integral would reduce to 0 and no light would arrive to the point P, which does not sound correct.

where is the misundersting in all this?

 

[kjl]

Yeah, usually for point lights or directional lights (in computer graphics anyway), you'd use more traditional direct illumination models.

If you want to use irradiance for it you'd need to give your light a non-zero source size, so that you have some non-zero size patches of incoming light from overhead.

 

[mnb96]

Yeah, usually for point lights or directional lights (in computer graphics anyway), you'd use more traditional direct illumination models.

Could you mention one of these illumination models for point-sources?

If you want to use irradiance for it you'd need to give your light a non-zero source size, so that you have some non-zero size patches of incoming light from overhead.

If you replace the point source in (0,0,1) with a spherical source in (0,0,1) you get the same problem. Namely, at the zenith of our point P=(0,0,0) you will find the pole of the sphere, where differential patches have surface-area equal to zero. The point P will indeed receive radiance from other directions but won´t receive any radiance from the pole, which sounds a bit strange because after all the sphere is emitting light also from its pole.

 

[kjl]

Since I don't know exactly what you're trying to do (are you trying to implement a really fancy lighting model as in that paper you mentioned? Or are you just trying to implement a basic lighting model and that web BRDF page is just the first good resource you found?), it's hard to say exactly what you should be doing, but:

Could you mention one of these illumination models for point-sources?

A basic one to start out with is the Phong lighting model, which is a diffuse component (for the matte objects, e.g. paper or a chalkboard are almost all diffuse) and a specular component (for shiny glints off of shiny metal, plastic, etc.) You can google it, but I also found these, both of which look right to me: http://www.nbb.cornell.edu/neurobio/land/oldstudentprojects/cs490-95to96/guo/report.html"

The basic idea is that you're NOT considering a point on a surface and gathering lighting contributions from all angles in the hemisphere that could possibly be contributing light, but that you ARE looping through the lights in the scene and calculating its contribution to the illumination of that point.

If you wanted to get something up as fast as possible, the easiest to do would be just the diffuse contribution, which is simply I*Kd*(L http://en.wikipedia.org/wiki/Dot_product" N), where I is the intensity of the light (set it to 1), Kd is the diffuse constant of whatever material you are trying to set (also set to 1 to be simple), L is the normalized vector from the point you're shading to the light (0,0,1 in your case), and N is the surface normal of your point (also 0,0,1 in your case), so it resolves to (0,0,1) dot (0,0,1) which is 1, so your diffuse response is 1.

If you moved your light source to (1,0,1), then it would resolve to (.707,0,.707) dot (0,0,1), which is .707, so your diffuse response is .707. If the light source moves to (1,0,0), then your diffuse response is 0.

If you replace the point source in (0,0,1) with a spherical source in (0,0,1) you get the same problem. Namely, at the zenith of our point P=(0,0,0) you will find the pole of the sphere, where differential patches have surface-area equal to zero. The point P will indeed receive radiance from other directions but won´t receive any radiance from the pole, which sounds a bit strange because after all the sphere is emitting light also from its pole.

My calculus is too old and rusty to argue the point, but integrals always involve summing an infinite number of slices with 0 size. So yeah, at exactly the pole (0,0,1), there is 0 light coming in, but there is some almost 0 but definitely >0 amount of light coming in from the patch of almost 0 size centered at the pole.

 

[mnb96]

Hi kjl,
thanks for the answer. It was pretty clear.

In the meanwhile I thought and discussed about this problem with other people.
As you said, that issue arises from calculus itself and it is not wrong. The main point however is that in principle it does not make much physical sense to consider point-sources, as the radiance can be defined only for finite solid-angles (which for a point source is always 0).

Concerning the case of the pole of the sphere, I was told that it really works that way: radiance coming from the pole is zero(!).
To be honest, this fact does not convince me at all.

My personal view is that the exact value of radiance coming from the pole cannot be computed with that parametrization of spherical coordinates, as we are dealing with a singularity (the Jacobian in spherical coordinates is zero at the pole).Thanks also for the overview of the other kinds of illumination models.

 

[kjl]

My personal view is that the exact value of radiance coming from the pole cannot be computed with that parametrization of spherical coordinates, as we are dealing with a singularity (the Jacobian in spherical coordinates is zero at the pole).

Ah, see, I told you my calculus was old and rusty. I don't know what a Jacobian is. Maybe for your application, it would be better to rotate that sphere by 90 degrees. e.g. for every shaded point construct your sphere so that the pole lies somewhere on the plane of the surface, so that the singularity/pole is where the light contribution is 0 anyway.