David Neubelt

Skytiger,

Let me clarify my post above. I'm not disagreeing with your math that radiance is higher as the emission area grows larger while the solid angle stays constant.

However we can't think of radiance physically in those terms, imagine a scenario where you have a spectroradiometer and you were calibrating a TV. On the TV you have a black screen with a picture of a small red square in the center emitting red light. If you point the spectroradiometer directly at the red square so the red square fills the view of your measuring device then you will get the full radiance. Now, if you start angling the gun closer to the tv so the red square still fills the view then the spectroradiomer will see more of the red square (because the area the gun sees is now larger) and the radiance will go higher. This would be expected (assuming the TV emits photons equivalently in all directions).

However, if you go past 45 degrees to 70 degrees and suddenly not all of the red square and some of the black part of the screen is seen by the spectroradiometer because the angle is so oblique then now the radiance will start to decrease.

If I was able to jam the spectroradiometer into the TV so its parallel to the red square the radiance would fall to 0 because it sees none of the red square.

What happens in our example is the solid angle that is subtended by our red surface element decreases, the flux decreases and the projected area decreases as the gun becomes more oblique to the TV and the radiance decreases to zero.

That only makes sense if you let the emitter area increase unbounded but physical devices are bounded in area so it doesn't make sense to think in those terms.

-= Dave

The solid angle the detector subtends is constant in all my diagrams

 

because the distance between detector and emitter (center points) is equal

and the detector is always normal to the light

 

In my first post the solid angle the emitter subtends does decrease with angle

But in the post above the solid angles subtended by both emitter and detector are constant

In the first post the emitter area is constant, in the last post the emitter *projected* area is constant

 

Either way the result is the same, radiance increases with viewing angle

 

because radiance = flux / solid_angle / projected_area

 

as the solid_angle decreases radiance must increase

as the projected_area decreases radiance must increase

as the flux increases radiance must increase

 

In my diagram above you can measure the plane angle (depicting the solid angle) with a protractor and see it is constant

If the solid angle is constant then the area on the left wall will increase as the angle goes further oblique.

Hodgman

 

very interesting point about the area of the emitter that is visible being greater

however it just confirms the radiance of 1.414

 

here I show the radiance calculation from both points of view, and the result is 1.414 for both:

 

 

 

Your idea that the solid angle changes is wrong ...

As the angle becomes more oblique to the wall on the left the solid angle the detector subtends will decrease. The solid angle will reach it's maximum when the detector and emitter are parallel.

As an aside, the whole 'dot product' notation here is actually mathematically correct, but really confusing for beginners since most people tend to associate it with directions and angles. There are actually no 'directions' involved in SH, since wah waaah wahh wahh wahh frequency domain. You're just multiplying the coefficients and adding the results as you go. Pick apart a dot product, and, hey, there's the same operations.

The basis functions are parameterized in spherical coordinates where the parameters represent a direction.

Personally, I like to think along the same lines that Ashaman73 does as it gives me an easy mental framework to reason about SH with.

I think it's very intuitive to think of SH as a set of lobes in a fixed set of directions. Each basis function adds another set of directions that you can project your signal onto. The directions represent the input angles to the basis function that produce the global maxima's of the polynomal. The spacing between the maxima's represent the resolution of the signal you can represent.

-= Dave