Help Understanding Luminance Calculation Process

[KLoux]

Hello all,

I have a system that consists of:

• A projector with a known position, orientation, and brightness (luminous flux, I guess - units are lumens),
• A lens with known distortion characteristics (i.e. a known light ray vector for every point on the projector's image-generating panel)
• A projection surface with known geometry (not necessarily a planar surface) and assumed reflectivity, R
• An eye point

I am trying to calculate the luminance at the eye point for a small light ray emanating from the projector. I am defining the "small light ray" by choosing four corners of a square on the projector's panel. The geometry is no problem - I am calculating the corresponding vectors for the light leaving the lens and their intersection with the projection surface (call these points $p_1$through $p_4$). My question is related to the process of using this information to arrive at a luminance value.

I am using the following process:

1. Calculate luminous flux for the small light ray ($\phi_{V0}$) by dividing the area of the square on the panel by the total panel area and multiplying by projector luminous flux ($\phi_{V0} = \frac{A_0}{A} \phi_V$) in lumens.
2. Average the angles of incidence where each of the four light direction vectors intersect the projection surface and evaluate the cosine of this angle ($\cos(\theta_0)$).
3. Calculate the luminous flux of the light reflecting off of the projection surface (perpendicular to the surface) by multiplying the quantities above ($\phi_{V1} = \phi_{V0} \cos(\theta_0)$ in lumens).
4. Average the viewing angles of incidence (vectors from the eye point to $p_1$ through $p_4$) and evaluate the cosine of this angle ($\cos(\theta_1)$).
5. Calculate the solid angle subtended from the eye point toward the area bounded by $p_1$ through $p_4$ ($\Omega$) in steradians.
6. Average the vectors from the eye point to $p_1$ through $p_4$; this average direction vector defines the normal to a plane. Project the area of the projection surface bounded by $p_1$ through $p_4$ onto this plane and compute the area ($A_P$) in m².
7. Compute the luminance as $L_V = \frac{\phi_{V1} R \cos(\theta_1)}{\Omega A_P}$ in cd/m².

I've tried to illustrate these steps in the following images.
Steps 2 and 3:

Step 4:

Step 5:

Step 6:

I am assuming that the "small light ray" is small enough that the edges of the points of intersection with the projection surface can be connected by straight lines without great loss of accuracy. I intended to also assume perfect diffusion of light by the projection surface. I believe this is done by including the two cosine terms, but I haven't found examples that go from light source to surface to eye point, so I'm not 100% sure if including two cosine terms is correct. Several sources have stated that for a perfect diffuser, luminance is the same in all directions (implying that maybe I shouldn't need the second cosine term), but the accompanying images seem to show that the direction is important (i.e. different magnitude vectors in different directions), like this image (from https://www.bksv.com/media/doc/18-231.pdf):

I know from measurements on the physical system that I should expect the luminance to be ~10 cd/m^2, but my calculations yield much larger numbers (off by ~3 OOM). I believe the arithmetic is correct, so I assume that the error is in my understanding of luminance. Maybe I'm using the wrong solid angle or projected area?

Thanks,

Kerry

 

[eljose79]

Dear Kerry,

Thank you for sharing your detailed process for calculating luminance at a specific point in a projection system. It seems like you have a good understanding of the various components involved, such as the projector, lens, projection surface, and eye point. Your assumptions of perfect diffusion and straight-line connections for the small light ray are also valid.

From your description, it seems like you are on the right track with your calculations. However, I would like to offer a few suggestions that may help you to refine your process and potentially address the discrepancy in your results.

Firstly, when calculating the luminous flux for the small light ray, you mentioned using the area of the square on the panel and the total panel area. It may be beneficial to also consider the brightness (luminous flux) of the individual pixels on the panel, as this could affect the overall luminous flux of the small light ray.

Secondly, in step 4, you mentioned averaging the viewing angles of incidence and evaluating the cosine of this angle. It may be more accurate to calculate the luminous flux for each individual viewing angle and then average the results. This could account for any variations in luminous flux at different viewing angles.

Lastly, in step 6, when calculating the solid angle subtended from the eye point, make sure to take into account the size and shape of the projection surface. If the surface is not a perfect plane, this could affect the calculation of the solid angle and the resulting luminance value.

I hope these suggestions are helpful to you and that you are able to refine your process and arrive at a more accurate luminance value. Keep up the good work!