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### #ActualMJP

Posted 28 May 2012 - 02:14 AM

Hey Zach,

You actually have those equations wrong. It should be like this:

[attachment=9077:CodeCogsEqn(2).gif]

This is basically saying that your BRDF function describes a ratio between differential output radiance towards the viewer, and differential incident irradiance. It's defined this way because overall what you do with your BRDF is determine the total radiance towards the viewer given the total incident irradiance on the surface. To compute irradiance you integrate incident radiance times cosine theta, which gives you this familiar equation:

[attachment=9076:CodeCogsEqn(1).gif]

In the case of this integral, the differential radiance is just an infinitesimally small portion of the total output radiance in the direction of the viewer. So you can imagine that by integrating you're just making a trip over the hemisphere, and calculating some fraction of the total outgoing radiance by computing the incident radiance from the set of directions given by the differential solid angle.

As for why you use Li and not dLi, you compute incident irradiance by integrating incident radiance * cos(theta) over the hemisphere. From this we can reason that differential irradiance for some infinitesimally small solid angle is equivalent to the incident radiance from that solid angle. You actually have this in the denominator of the first equation I posted:

[attachment=9079:CodeCogsEqn(4).gif]

Does that make sense?

### #2MJP

Posted 28 May 2012 - 02:12 AM

Hey Zach,

You actually have those equations wrong. It should be like this:

[attachment=9077:CodeCogsEqn(2).gif]

This is basically saying that your BRDF function describes a ratio between differential output radiance towards the viewer, and differential incident irradiance. It's defined this way because overall what you do with your BRDF is determine the total radiance towards the viewer given the total incident irradiance on the surface. To compute irradiance you integrate incident radiance times cosine theta, which gives you this familiar equation:

[attachment=9076:CodeCogsEqn(1).gif]

In the case of this integral, the differential radiance is just an infinitesimally small portion of the total output radiance in the direction of the viewer. So you can imagine that by integrating you're just making a trip over the hemisphere, and calculating some fraction of the total outgoing radiance by computing the incident radiance from the set of directions given by the differential solid angle.

As for why you use Li and not dLi, you compute incident irradiance by integrating incident radiance * cos(theta) over the hemisphere. From this we can reason that differential irradiance for some infinitesimally small solid angle is equivalent to the incident radiance from that solid angle. You actually have this in the denominator of the first equation I posted:

[attachment=9078:CodeCogsEqn(3).gif]

Does that make sense?

### #1MJP

Posted 28 May 2012 - 01:48 AM

Hey Zach,

You actually have those equations wrong. It should be like this:

[attachment=9075:CodeCogsEqn.gif]

This is basically saying that your BRDF function describes a ratio between differential output radiance towards the viewer, and differential incident irradiance. It's defined this way because overall what you do with your BRDF is determine the total radiance towards the viewer given the total incident irradiance on the surface. To compute irradiance you integrate incident radiance times cosine theta, which gives you this familiar equation:

[attachment=9076:CodeCogsEqn(1).gif]

So if you can imagine integrating this last equation by summing the infinitely small parts, then for each part you would be calculating the differential output radiance. And we already know that Li * cos(theta) is equivalent to the differential incident irradiance. This is why you don't have dLi, since you're actually integrating incident irradiance and not incident radiance.

Does that make sense?

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