<p>I'm reading a graphics book and it talks about BRDFs being the ratio between differential outgoing radiance and differential irradiance. The formula it gives is 
f(l, v) = dL(v)/dE(l)
Where l is the light vector and v is the outgoing direction vector, and L is outgoing radiance and E is incoming irradiance. </p>

<p>But then the book says that for non area lights like directional and point lights, the BRDF can be expressed without differentials:
f(l, v) = L(v) / E(l). 
How is this possible? The first formula seems to be a derivative, how do we get to this formula? What is the difference between non area and area lights that would allow us to arrive at this simplification? </p>

CC BY-SA 4.0
2020-02-10T06:22:35.193Z
What book is that? Directional and point lights are described through Dirac deltas.
lightxbulb
CC BY-SA 4.0
2020-02-11T21:51:43.437Z
@lightxbulb Real-Time Rendering 3rd edition. Chapter 7 I think, in the BRDF section.
Yoko
CC BY-SA 4.0
2020-02-12T05:01:53.390Z
Can you quote the relevant part?
lightxbulb
CC BY-SA 4.0
2022-05-12T06:29:08.043Z
I'm really sorry, I never got a notif for your reply! I don't expect you to answer after all this time but, it was page 223, formula 7.19 and formula 7.20. It said "The precise definition of the BRDF is the ratio between differential
outgoing radiance and differential irradiance", and then it shows  f(l, v) = dL(v)/dE(l) 

Then it says: "The discussion in this chapter is restricted to shading with non-area
light sources such as point or directional lights. In this case, the BRDF definition can be expressed in a non-differential form" and gives f(l, v) = L(v) / E(l)
Yoko
CC BY-SA 4.0
2022-05-12T06:39:40.740Z
On second read of this, and you mentioning the Dirac delta (I didn't really understand what those were back in 2020) I think I get what the book is saying now. Since point/directional lights don't have a distribution of incoming light rays around some solid angle, but rather a singular direction where all of the light is concentrated, it doesn't make sense to model the BRDF over the differential irradiance/radiance, which would be over a solid angle. That's why we just use the proportion of incoming light to outgoing light directly for point/directional lights.
Yoko
CC BY-SA 4.0
2022-05-12T07:52:15.063Z
A Dirac delta "consumes" an integral: $\int_{\Omega}g(x)\delta(x-y)\,dx = g(y)$. So instead of having to estimate an integral numerically, you just have to point sample a function instead. Note that the Dirac delta is not a function, it is a measure or distribution. If you have a reproducing kernel hilbert space then you can use a reproducing kernel instead of a Dirac delta (the importance of this is theoretical, in practice you get the same code). See Christian Lessig's dissertation for more details on the latter.
lightxbulb
<lighting><brdf>
Formula for non area light BRDFs
Yoko
# Formula for non area light BRDFs

(Asked by: username_0 on 2020-02-10)

I'm reading a graphics book and it talks about BRDFs being the ratio between differential outgoing radiance and differential irradiance. The formula it gives is
f(l, v) = dL(v)/dE(l)
Where l is the light vector and v is the outgoing direction vector, and L is outgoing radiance and E is incoming irradiance.

But then the book says that for non area lights like directional and point lights, the BRDF can be expressed without differentials:
f(l, v) = L(v) / E(l).
How is this possible? The first formula seems to be a derivative, how do we get to this formula? What is the difference between non area and area lights that would allow us to arrive at this simplification?

**Tags:** lighting, brdf

> **Comments:**
> **username_1**: What book is that? Directional and point lights are described through Dirac deltas.  
> **username_0**: @username_1 Real-Time Rendering 3rd edition. Chapter 7 I think, in the BRDF section.  
> **username_1**: Can you quote the relevant part?  
> **username_0**: I'm really sorry, I never got a notif for your reply! I don't expect you to answer after all this time but, it was page 223, formula 7.19 and formula 7.20. It said "The precise definition of the BRDF is the ratio between differential outgoing radiance and differential irradiance", and then it shows  f(l, v) = dL(v)/dE(l)   Then it says: "The discussion in this chapter is restricted to shading with non-area light sources such as point or directional lights. In this case, the BRDF definition can be expressed in a non-differential form" and gives f(l, v) = L(v) / E(l)  
> **username_0**: On second read of this, and you mentioning the Dirac delta (I didn't really understand what those were back in 2020) I think I get what the book is saying now. Since point/directional lights don't have a distribution of incoming light rays around some solid angle, but rather a singular direction where all of the light is concentrated, it doesn't make sense to model the BRDF over the differential irradiance/radiance, which would be over a solid angle. That's why we just use the proportion of incoming light to outgoing light directly for point/directional lights.  
> **username_1**: A Dirac delta "consumes" an integral: $\int_{\Omega}g(x)\delta(x-y)\,dx = g(y)$. So instead of having to estimate an integral numerically, you just have to point sample a function instead. Note that the Dirac delta is not a function, it is a measure or distribution. If you have a reproducing kernel hilbert space then you can use a reproducing kernel instead of a Dirac delta (the importance of this is theoretical, in practice you get the same code). See Christian Lessig's dissertation for more details on the latter.  

---

## 0 answers

