An Efficient Representation for Irradiance Environment Maps
Ravi Ramamoorthi and Pat Hanrahan
2001

What ?
 Describes a method of calculating Global
Illumination using spherical harmonics

When calculating Global Illumination, highfrequency lighting gets blurred.
 This implies that we might be able to ignore
higher-order spherical harmonics terms.
The radiance map
The irradiance map

Traditional Lighting
- Wikipedia

Lighting a scene

What?
 A method of approximating light accumulation
(Irradiance) at a point in space using spherical
harmonics

What?

Spherical harmonics
 A method of approximating a function over a
domain
 Similar to Fourier transform
Approximating a square wave

What is irradiance?


Irradiance is the accumulation of
all lighting values on a half-sphere
for a point (given its normal).
It is a function of the normal (n).


The integral over ω removes the
dependence on ω.
E(n)  
(n )

L()(n  )d
The integral represents the accumulation of all the light values.

How can we calculate E(n)?
 Spherical Harmonics!
 Like a Fourier Transform, Spherical Harmonics
translates one function into a sum of basis
functions multiplied by coefficients.
 The basis functions are defined using polar
coordinates (θ, ϕ).

So, let's apply Spherical Harmonics to E(n)!
E(n)  
(n )
L()(n  )d

So, let's apply Spherical Harmonics to E(n)!
 Rewrite E(n) in terms of (θ, ϕ)
▪ E(n) is in Cartesian coordinates
▪ E(θ, ϕ) is E(n) converted to polar coordinates
E ( , )  
(n )
L()(n   )d

So, let's apply Spherical Harmonics to E(n)!
 Spherical harmonics lets us replace the function
E(θ, ϕ) with a summation over l and m.
▪ Ylm(θ, ϕ): The spherical harmonics basis function
▪ Elm: A coefficient for Ylm(θ, ϕ)
 How many basis functions?
E
Y ( , )  
lm lm
l ,m
(n )
L( )(n   )d

So, let's apply Spherical Harmonics to L(ω)!
 Rewrite L(ω) in terms of (θ, ϕ)
E
Y ( , )  
lm lm
l ,m
 (n )
L( , )(n   )d

So, let's apply Spherical Harmonics to L(ω)!
 Replace (n ∙ ω) with the cosine
▪ Remember: a ∙ b = |a| |b| cos(θ)
▪ Note that n and ω are unit vectors so their lengths are 1
▪ Let A(θ') = max[0, cos(θ')]
E
Y ( , )  
lm lm
l ,m
(n )
L( , )A( ' )d

Simplify the equation!
 The derivation of the integrals is quite difficult.
ˆ
 This is the result of the paper: Elm  Llm A
l
l
1

2
 l! 
2 (1)
 (l  2)(l  1)  2l ( l !) 2 
 2 


Aˆl  
2

3



0

(l even)
(l  1)
(l odd, l  1)

Replace the Aˆl
l 1
2
Elm  Llm
3
l  1, odd
Elm  0
l even
l
1
2
 l! 
(1)
Elm  Llm 2
 l l 2
(l  2)(l  1)  2 ( 2 !) 

What does this mean?
l 1
l  1, odd
2
Elm  Llm
3
Elm  0
l even
l
1
2
 l! 
(1)
Elm  Llm 2
 l l 2
(l  2)(l  1)  2 ( 2 !) 
ˆ drop off quickly
 Values for A
l
l
0
1
2
3
4
5
6
7
8
Al
3.142
2.094
0.785
0
-0.131
0
0.049
0
-0.025

What is Aˆl ?
 Everything after the 3rd band of spherical
harmonics contribute little
 You get a good approximation by using only 9
coefficients
▪ l = [0, 1, 2]
-l ≤ m ≤ l
▪ (0, 0), (1, -1), (1, 0), (1, 1), (2, -2), (2, -1), (2, 0), (2, 1), (2, 2)
▪ Technically we get l = 3 as well, since the coefficient is 0

What is Aˆl ?
 The lighting environments they use for their test
cases have an average error of around 1% and a
maximum error of around 5%.
 Their test environments look to be fairly
representative, so the margin for error for our
environments should be similar.

Bounced light calculation is expensive

Light maps only work for static objects

Sampling from voxels would work for
dynamic objects, but needs lots of voxels

9 coefficients for each channel

Spherical harmonic coefficients can be
calculated from environment maps

Irradiance is low frequency so maps do not
need to be especially high res


Artists place volumes
Automatic sampling within volume produces
many voxels


Interpolate between voxels
Irradiance evaluated as E(n)  nt Mn
 M is a 4x4 matrix based on coefficients