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A Mike Day Paper Exercise
An Efficient Representation for Irradiance Environment MapsRavi Ramamoorthi and Pat Hanrahan2001
An Efficient Representation for Irradiance Environment Maps
What ? Describes a method of calculating Global
Illumination using spherical harmonics
An Efficient Representation for Irradiance Environment Maps
When calculating Global Illumination, high-frequency lighting gets blurred. This implies that we might be able to
ignore higher-order spherical harmonics terms.
The radiance map
The irradiance map
An Efficient Representation for Irradiance Environment Maps
Traditional Lighting
- Wikipedia
An Efficient Representation for Irradiance Environment Maps
Lighting a scene
An Efficient Representation for Irradiance Environment Maps
What? A method of approximating light
accumulation (Irradiance) at a point in space using spherical harmonics
What?
An Efficient Representation for Irradiance Environment Maps
Spherical harmonics A method of approximating a function
over a domain Similar to Fourier transform
Approximating a square wave
An Efficient Representation for Irradiance Environment Maps
What is irradiance? Irradiance is the accumulation of
all lighting values on a half-spherefor a point (given its normal).
It is a function of the normal (n). The integral over ω removes the
dependence on ω.
The integral represents the accumulation of all the light values.
dLE )()()()(
nnn
An Efficient Representation for Irradiance Environment Maps
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).
An Efficient Representation for Irradiance Environment Maps
So, let's apply Spherical Harmonics to E(n)!
dLE )()()()(
nnn
An Efficient Representation for Irradiance Environment Maps
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
dLE )()(),()(
nn
An Efficient Representation for Irradiance Environment Maps
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?
dLYEml
lmlm )()(),()(
,
n
n
An Efficient Representation for Irradiance Environment Maps
So, let's apply Spherical Harmonics to L(ω)!
Rewrite L(ω) in terms of (θ, ϕ)
dLYEml
lmlm )(),(),()(
,
n
n
An Efficient Representation for Irradiance Environment Maps
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')]
dALYEml
lmlm )'(),(),()(
,
n
An Efficient Representation for Irradiance Environment Maps
Simplify the equation! The derivation of the integrals is quite
difficult. This is the result of the paper:
llmlm ALE ˆ
)1,(0
)1(3
2
)()!(2
!
)1)(2(
)1(2
ˆ
22
12
loddl
l
evenll
ll
A
ll
l
l
An Efficient Representation for Irradiance Environment Maps
Replace the
22
12
)!(2
!
)1)(2(
)1(20
3
2
,11
ll
l
lmlmlmlmlm
l
llLEELE
evenloddll
lA
An Efficient Representation for Irradiance Environment Maps
What does this mean?
Values for drop off quickly
22
12
)!(2
!
)1)(2(
)1(20
3
2
,11
ll
l
lmlmlmlmlm
l
llLEELE
evenloddll
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
lA
An Efficient Representation for Irradiance Environment Maps
What is ? 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
lA
An Efficient Representation for Irradiance Environment Maps
What is ? 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.
lA
Application
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
Application
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
Application
Artists place volumes Automatic sampling within volume
produces many voxels
Application
Interpolate between voxels Irradiance evaluated as
M is a 4x4 matrix based on coefficients
MnnnE t)(
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