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Imagine a surface which radiates light equally in all directions from all points

if we measure the radiance of the surface from two positions

one opposite the surface

and one at a 45 degree angle

would the measured radiance be the same?

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I will confess this one left me confused for a while. Yes, the radiance is the same, assuming the emitter is at the same distance (it's not clear from the diagram). The reason for this is because the emitter is always at normal incidence to the detector's view angle.

There's this neat theorem called conservation of radiance which is really helpful in puzzling situations like these, which states that radiance measured at the emitter is the same as radiance measured at the detector, assuming no radiance is lost between emitter and detector. In this case, it becomes obvious the radiance should be the same, as in both cases the emitters are identical up to rotation (which conserves areas, distances, and basically everything you could possibly care about when working with radiance).

There are also some helpful notes under the Wikipedia Talk page for radiance, see the "cosine term" paragraph. The key point is that the cosine term in the definition of radiance does not use the detector's surface normal but the emitter's surface normal. Notably, the sentence "The cosine factor in the denominator reflects the fact that the apparent size of the source goes to zero as your angle of view approaches 90°." highlights this. In your case, the apparent size of the source is quite clearly constant due to the nature of your setup. This can get horribly confusing especially when you see computer graphics papers such as the rendering equation which apparently use the cosine term with the detector's surface normal, but they make entirely different assumptions.

Another, perhaps more intuitive way of deducing this result is by looking at the units of radiance: watts per meter squared per steradian. Also known as power per area per solid angle. Just to be clear, "power" is the power of the emitter, "area" is the surface area of the emitter, and "solid angle" is the solid angle subtended by the emitter from the detector's point of view. Now, for both cases considered:

- power is constant, since the emitter is obviously still emitting the same amount of light in both cases

- area is constant, as the emitter's surface area hasn't changed

- solid angle is constant, because the emitter is held facing the detector at normal incidence in both cases

Therefore, measured radiance is the same in both cases.

That said, I could be wrong on this. Please feel free to correct me if I made a mistake.

EDIT: actually, I don't even know anymore. It's all foggy

EDIT 2: no, I think, radiance should really be interpreted as emitted intensity (power / area) per solid angle (direction). There, that's better  still mighty confused though,

Edited by Bacterius

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Radiometry is a puzzle, but using this special non-Lambertian surface makes things easier

I believe the _flux_ incident to the detector is equal in both cases

but the measured _radiance_ is 1.0 / 0.707 = 1.414 times higher for the angled detector

because from the angled detector's point of view the emitter looks smaller (it has smaller projected area)

which means you have the same amount of flux in a smaller area which increases flux density .... and therefore "brightness"

this is due to the confusing projected area term in the radiance equation:

radiance = flux / solid_angle / projected_area

and THAT is why Lambertian reflectance needs a cosine law term ... to balance out the increase in radiance due to viewing angle

that is why Lambertian surface gives constant radiance when viewing angle changes

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(edit) rubbish

Edited by skytiger

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I think it's best to always talk about radiance with respect to a imaginary surface perpendicular to the chosen direction. In all settings I ever encountered you were free to choose the surface normal, so why not chose the easiest configuration, in which the confusing cosine term simply disappears.

Assuming the receiver is left and the emitter right in your images:
Incident radiance (say) at the center of the receiver (x) coming from the direction pointing to the center of the emitter (w) is the same in both configurations. It's also the same as the exitant radiance from the center of the emitter toward the center of the receiver.
Irradiance at every point of the receiver is smaller for the angled configuration. The solid angle covered by the emitter is the same, but in the case of irradiance you are not free to choose the normal of the surface (since you asking specifically about irradiance of a concrete surface). So in this case the cosine factor is not 1 and it'll give you a smaller value.
Power collected in total by the receiver is smaller, too (obviously, if irradiance at every point is).

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Irradiance at every point of the receiver is smaller for the angled configuration. The solid angle covered by the emitter is the same, but in the case of irradiance you are not free to choose the normal

I do not understand exactly what you mean by this. Could you elaborate a bit?

The way I thought about it: imagine a hemisphere around the receiver center point's normal. In the first configuration, the emitting surface would be close to the Zenith of this hemisphere. In the second configuration, the surface would be closer to the Azimuth and thus the cosinus factor would be different.

Is this what you mean?

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I do not understand exactly what you mean by this. Could you elaborate a bit?

The way I thought about it: imagine a hemisphere around the receiver center point's normal. In the first configuration, the emitting surface would be close to the Zenith of this hemisphere. In the second configuration, the surface would be closer to the Azimuth and thus the cosinus factor would be different.
Is this what you mean?

Yes, that is correct. The incident beam would be smaller due to the cosine factor, hence irradiance decreases by a corresponding amount. That is not radiance, though, but irradiance.

Assuming the receiver is left and the emitter right in your images:

That was my assumption as well. Perhaps this was not the expected interpretation?

Edited by Bacterius

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The emitter is on the left

and the receivers are on the right

the blue represents the solid angle subtended by the receivers

the magenta shows the area and the projected area of the emitter

they are meant to be infinitesimally small differential areas - so the direction is from the center of emitter to the center of receiver

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The emitter is on the left

and the receivers are on the right

the blue represents the solid angle subtended by the receivers

the magenta shows the area and the projected area of the emitter

they are meant to be infinitesimally small differential areas - so the direction is from the center of emitter to the center of receiver

Well in that case, yes, radiance is lower. Think about what happens when you aim the emitter perpendicular to the receiver, the projected area goes to zero and no light moves towards the receiver so radiance is zero, as expected. So a lower emitter projected area for the emitter -> lower radiance, by the factor you stated in your first post.

The blue illustration in the diagram was confusing, though. And it's important to not confuse the cosine term in the emitter's projected area with the cosine term for the irradiance's incidence area, they are not the same!

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Irradiance at every point of the receiver is smaller for the angled configuration. The solid angle covered by the emitter is the same, but in the case of irradiance you are not free to choose the normal

I do not understand exactly what you mean by this. Could you elaborate a bit?

The way I thought about it: imagine a hemisphere around the receiver center point's normal. In the first configuration, the emitting surface would be close to the Zenith of this hemisphere. In the second configuration, the surface would be closer to the Azimuth and thus the cosinus factor would be different.
Is this what you mean?

That's what I meant. Although it wasn't quite correct of me to speak of 'the' cosine factor when a finite solid angle is involved, because there are many cosine factors; one for each direction in the direction bundle that makes up the solid angle. But each factor is smaller in (b) than the corresponding factor in (a), so it's save to say that the irradiance will be lower due to 'the' cosine factor.

The emitter is on the left
and the receivers are on the right

the blue represents the solid angle subtended by the receivers
the magenta shows the area and the projected area of the emitter

they are meant to be infinitesimally small differential areas - so the direction is from the center of emitter to the center of receiver

Well in that case, yes, radiance is lower. Think about what happens when you aim the emitter perpendicular to the receiver, the projected area goes to zero and no light moves towards the receiver so radiance is zero, as expected. So a lower emitter projected area for the emitter -> lower radiance, by the factor you stated in your first post.

The blue illustration in the diagram was confusing, though. And it's important to not confuse the cosine term in the emitter's projected area with the cosine term for the irradiance's incidence area, they are not the same!

I have to disagree, in my opinion you'll approximately get the same results.
- Radiance is the same. You look up from the receiver with in a single direction. Either you see the emitter, then you get the radiance it emits, or you don't see the emitter, then you get zero. The orientation of the emitter doesn't matter since it's defined to emit equally in all directions.
- Irradiance is lower in b than in a. This time not because of the cosine factor(s), but because of the smaller solid angle.
- Power also lower.

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I have to disagree, in my opinion you'll approximately get the same results.
- Radiance is the same. You look up from the receiver with in a single direction. Either you see the emitter, then you get the radiance it emits, or you don't see the emitter, then you get zero. The orientation of the emitter doesn't matter since it's defined to emit equally in all directions.
- Irradiance is lower in b than in a. This time not because of the cosine factor(s), but because of the smaller solid angle.
- Power also lower.

Yes, I agree with everything you say. Good point about the orientation of the emitter.

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emitter doesn't matter since it's defined to emit equally in all directions

Yes, but it's defined to emit LIGHT equally in all directions

If the incident radiance was the same in both cases the surface would be Lambertian ... but it isn't

From wiki:

In optics, Lambert's cosine law says that the radiant intensity or luminous intensity observed from an ideal diffusely reflecting surface or ideal diffuse radiator is directly proportional to the cosine of the angle ? between the observer's line of sight and the surface normal

My surface above is NOT Lambertian as the radiant intensity is not proportional to the cosine ... it is constant

In the radiance equation we are dividing by cosine, as the view angle approaches 90º the radiance approaches infinity ...

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In the radiance equation we are dividing by cosine, as the view angle approaches 90º the radiance approaches infinity ...

The cosine is counter-balanced by the other terms in the formula (namely radiant flux). Radiance clearly doesn't tend to infinity as the angle approaches grazing incidence. See the Wikipedia talk page for details, someone has asked the same question.

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I strongly disagree

The talk article says "The radiant flux for physical sources falls off at least as fast with angle as cos(?)"

"for physical sources" he is saying that _in real life_ it never happens ...

That is why I created the non-Lambertian geometry above
the surface is not a "physical source" it is constructed deliberately to isolate the cosine / projected area term in the radiance equation
and expose it for what it is - highly confusing!

What happens at 90° is not described in the radiance equation ... it is due to occlusion ... there is no radiance because you can not see the surface

The key to understanding radiance is understanding what happens when the viewing angle _approaches_ 90° ... the radiance approaches infinity

This is due to flux density:

Edited by skytiger

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Here is another gamedev topic that agrees with me (very messily though ...)

quoting:

That's right. Radiance gives the transmitted radiant power through the given solid angle per unit of projected area. That is why you need to divide by the projected area given by cos(theta)*dA. At a grazing angle the projected area approaches zero, hence the radiant power per unit of projected area approaches infinity.

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I can't quite follow your argument with this strange 'non-Lambertian geometry', but if you define your emitter in such a way that it emits inifinite radiance at grazing angles then you'll see infinite radiance when you look at it at grazing angles. But this has nothing to do with the general definition of radiance. It is also not a very useful definition.

Anyway, I just wanted to say two things for other people who may be reading this:
1. Bacterius is absolutely right about the cosine factor.
2. Incident radiance at a point x from a direction w has nothing to with whether the surface on which x lies is Lambertian or not. Incident radiance is called incident radiance because it's measured before it interacts with the surface. It is also idependent of the surface orientation, as described above.

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I don't think you understand anything I have said

It is an isotropic surface - it radiates intensity equally in all directions

Lambertian surfaces do *not* radiate intensity equally in all directions - they radiate intensity proportional to cosine(viewing_angle)

As the viewing angle increases the projected area of the emitter decreases and the radiance increases

When you COMBINE Lambertian reflectance's cosine term with radiance's 1/cosine term you get CONSTANT RADIANCE

You seem to believe that both radiance *and* intensity are constant!

your point that "radiance is independent of surface orientation" is simply completely wrong

radiance can ONLY be measured with respect to surface orientation!

Edited by skytiger

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To reach agreement on this point we need to agree:

(1) that incident power is the same for both detectors
(2) that solid angle subtended by the detector is the same for both detectors
(3) that the area of the emitter does not change
(4) that the projected area of the emitter is smaller for the angled detector (by factor of 0.707)
(5) that radiance = flux / solid_angle / projected_area MUST be greater for the angled detector (by factor of 1.0 / 0.707 = 1.414)

radiance = flux / solid_angle / projected_area

Given flux of 1.0 and solid_angle of 1.0:

For my isotropic surface:

non-angled
radiance = flux / solid_angle / 1.0 = 1.0
angled
radiance = flux / solid_angle / 0.707 = 1.414        // radiance varies

*If* it was Lambertian: (the flux would be reduced due to cosine term)

non-angled
radiance = flux / solid_angle / 1.0 = 1.0
angled
radiance = flux * 0.707 / solid_angle / 0.707 = 1.0   // radiance is constant

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I don't think you understand anything I have said

I think I understood some of it, but some things elude me. And I'm probably not alone.

your point that "radiance is independent of surface orientation" is simply completely wrong
radiance can ONLY be measured with respect to surface orientation!

One last remark, again, mostly for the grinning and/or confused bystanders:
Indeed, you can measure radiance only with respect to surface orientation. It's even better: you can choose any surface orientation you like. The point is you'll always get the same result. If you tilt the surface normal of the _imaginary_ surface away from the direction in which you're probing for radiance the _irradiance_ will get lower. Less energy per unit area. Because of the tilt. Now the cute little cosine factor in the denominator comes in, counterbalances this (exactly like Bacterius said), and brings the result back up again. So, the surface orientation you choose for 'measuring' radiance doesn't matter. In other words: radiance is independent of surface orientation.

1. You look only in a single direction instead of the whole hemisphere (that's dw)
2. You go to projected area from oriented (or real) area, in order to detatch yourself from any concrete surfaces (that's cos(t))

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Can you please point out which parts of my above post are wrong?

I have carefully reread everything that has been said and still think I'm right ...

(I also have read many articles that seem to agree with my point of view)

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Mistakes I perceived:

the emitter is always at normal incidence to the detector's view angle

no it isn't

radiance measured at the emitter is the same as radiance measured at the detector

this doesn't mean the radiance is the same when measured from a DIFFERENT ANGLE
it means the radiance doesn't change over DISTANCE

also it means radiance is the same for both combinations of terms:

- solid angle of emitter and projected area of detector
- solid angle of detector and projected area of emitter

The key point is that the cosine term in the definition of radiance does not use the detector's surface normal but the emitter's surface normal

it can use either, see my point above

apparent size of the source is quite clearly constant due to the nature of your setup

no it isn't, the apparent size of the source is SMALLER for the angled detector

you calculated radiance using the following terms:

- power is constant, since the emitter is obviously still emitting the same amount of light in both cases
- area is constant, as the emitter's surface area hasn't changed
- solid angle is constant, because the emitter is held facing the detector at normal incidence in both cases

but you missed PROJECTED AREA (or the cosine term)

Therefore, measured radiance is the same in both cases

not once you factor in the projected area term you missed

but for all practical purposes ™ it’s a point and a direction

no it isn't
you can't just ignore the differential projected area and solid angles ... otherwise what's the point in defining radiance at all?
you CAN simplify like this inside a video game using the simplified Lambertian dot product lighting we use ... but not in Radiometry and definitely not in the example I gave!
that is actually the WHOLE POINT of my post, to clarify why this is the case!

Assuming the receiver is left and the emitter right in your images
Irradiance at every point of the receiver is smaller for the angled configuration

agreed
but the emitter is on the left
so irradiance at the detector is the same
as the flux is the same and the detector area is the same

Think about what happens when you aim the emitter perpendicular to the receiver, the projected area goes to zero and no light moves towards the receiver so radiance is zero, as expected

when the emitter is perpendicular there is nothing to discuss, nothing to measure ...
it is the same as if I removed the emitter completely!
you are making the assumption that because the radiance is zero when perpendicular
that it must change from 1 to 0 as the viewing angle increases
this makes no sense
if instead you considered what happened at a viewing angle of 89.99999999 degrees - then you would have seen my point!

The orientation of the emitter doesn't matter since it's defined to emit equally in all directions

it doesn't matter when considering intensity or power ... it DOES matter when measuring radiance
as the differential projected area is LESS

Edited by skytiger

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If you carefully read these 2 articles

http://en.wikipedia.org/wiki/Lambert%27s_cosine_law

See the section "Details of equal brightness effect"

You will see that Lambertian cosine * Radiance's 1/cosine _combine_ to give constant radiance

Obviously a 1/cosine term tends to INFINITY when the angle tends to 90°

Also the wikipedia Talk comments are agreeing with ME

you are misinterpreting them

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And another explanation:

In the figure on the right, if the source is Lambertian and equally bright in all directions, then the amount of energy that falls each the detectors is not the same for each of the detectors, but depends on the angle ? as cos?.
[Note: A source that is equally bright in all direction does not emit the same amount of energy in all directions.  When determining the brightness of a small source we measure the energy that falls on a detector subtending a given solid angle at the source and then divide by the apparent size (area) of the source.  A small source emitting the same amount of energy into the given solid angle as a larger source is brighter than the larger source.]

Note:

A small source emitting the same amount of energy into the given solid angle as a larger source is brighter than the larger source

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Maybe if you stopped attacking everyone and turning this into a "I'm right and you're wrong" argument, people would actually be able to discuss this to benefit everyone's understanding. And it is entirely possible to misinterpret the definitions in such a way that everything is still consistent, but incorrect with respect to the accepted definitions, so the distinction between "right" and "wrong" is meaningless in itself - the real question should be correct with respect to what? Radiometry is a very confusing field where everyone seems to use different definitions (I think this thread is a living example).

But as far as I am concerned, any physical quantity which goes to infinity like this is implausible and absolutely useless as it cannot be effectively measured. Regarding the Wikipedia comments - a physical source is defined as a source with an area (i.e. not a point light source). Now Srleffler says:

"The radiant flux for physical sources falls off at least as fast with angle as cos(?)."

And the radiant flux term in the radiance definition is at the top of the fraction, effectively cancelling the cosine term in the denominator. Radiance does not approach infinity as the angle approaches 90 degrees. The 1 / cosine term does, obviously, but the radiant flux term approaches zero at a rate at least inversely proportional to that. Radiant flux is the amount of radiant power crossing some imaginary cross-section (so, energy per unit of time). And guess what - the radiant flux considered here must pass through the projected area of the light source. And guess what again - this projected area is proportional to cos(theta). So radiant flux is proportional to cos(theta). Lambert's Law! The number of photons/sec (power) directed into any wedge is proportional to the area of the wedge!

And for the record, I see your point about intensity per unit solid angle. But your interpretation that radiance tends to infinity seems inconsistent with other equations, such as, say, the rendering equation. With your current interpretation of "infinite radiance", perceived brightness would increase without bounds when looking at anything that emits or reflect light in your direction (not just a light source) at close to 90 degree angles. That is clearly not the case.

apparent size of the source is quite clearly constant due to the nature of your setup

no it isn't, the apparent size of the source is SMALLER for the angled detector

Nice strawman. But that was my first post where I assumed emitter was on the right and receiver was on the left. Since you said later on that you meant something else, this is obviously not relevant anymore, and I acknowledged it. Good job quoting my post out of context, it is really helping this thread move forward.

----

Either way, your tone and attitude are not making me want to reply to you, and I believe I speak for the other people in the thread as well. You created this thread asking a question. Now it seems you know all the answers, and are literally challenging anyone to prove you wrong. Now it is possible I am wrong. It is possible you are wrong. It is possible everyone in this thread is wrong, or everyone is right (and just talking past one another). It could just be because of some conceptual difference between the definition of radiance as used in computer graphics (where "radiance" might in fact mean "radiant exitance", for instance, which would not surprise me) and in physics (which you are using). You can probably come up with a set of definitions which makes any statement correct. That doesn't mean they will be of any practical use for whatever you are trying to do. Different applications sometimes call for different definitions (which should probably have different names, to avoid confusion, but sometimes that doesn't happen)

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If radiance, which is a measure of energy passing through some part of space, approaches infinity as the viewing angle approaches 90º, then we could build infinite energy machines by placing an LED panel and a photovoltaic panel at 89.999 degress relative to each other.

A small source emitting the same amount of energy into the given solid angle as a larger source is brighter than the larger source

That should just be common sense following from geometry though.
e.g. a big and a small emitter on the same plane, viewed by two detectors on a different angled plane. For the left detector in the image to read the same amount of energy as the right detector, then the small/left source will have to be outputting much more energy per m2 than the big/right source, because more area of the right source is visible to the detector.

If radiance is watts per steridian per area, and area increases with angle, then received watts should decrease as angle increases, because you divide by area. So, we should think that these energy measurements approach zero as the angle increases, because the area approaches infinity...

However, it's important to note that as the detector angle increases and the area visible to the detector increases, the solid angle occupied by the emitter also decreases, which balances things out somewhat.

e.g. above, the top detector, which is at a more shallow angle can see a larger area, but emitter is a smaller fraction of it's visible area. If we draw the wedge taken up by the emitter, it shrinks as the visible area grows.

You will see that Lambertian cosine * Radiance's 1/cosine _combine_ to give constant radiance

Obviously a 1/cosine term tends to INFINITY when the angle tends to 90°

If there's a term that approaches infinity, but the final result remains constant... then there must be another term that approaches zero to balance it out? And then it goes without saying that if radiance remains constant, then radiance can't approach infinity? So which is it, constant or increasing with angle?

Edited by Hodgman