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An Efficient Representation for Irradiance Environment Maps Ravi Ramamoorthi and Pat Hanrahan 2001.

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Presentation on theme: "An Efficient Representation for Irradiance Environment Maps Ravi Ramamoorthi and Pat Hanrahan 2001."— Presentation transcript:

1 An Efficient Representation for Irradiance Environment Maps Ravi Ramamoorthi and Pat Hanrahan 2001

2  What ?  Describes a method of calculating Global Illumination using spherical harmonics

3  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 mapThe irradiance map

4  Traditional Lighting - Wikipedia

5  Lighting a scene

6  What?  A method of approximating light accumulation (Irradiance) at a point in space using spherical harmonics  What?

7  Spherical harmonics  A method of approximating a function over a domain  Similar to Fourier transform Approximating a square wave

8  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.   The integral represents the accumulation of all the light values.

9  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 ).

10  So, let's apply Spherical Harmonics to E(n) !

11  Rewrite E(n) in terms of (θ, ϕ ) ▪ E(n) is in Cartesian coordinates ▪ E(θ, ϕ ) is E(n) converted to polar coordinates

12  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. ▪ Y lm (θ, ϕ ) : The spherical harmonics basis function ▪ E lm : A coefficient for Y lm (θ, ϕ )  How many basis functions?

13  So, let's apply Spherical Harmonics to L(ω) !  Rewrite L(ω) in terms of (θ, ϕ )

14  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')]

15  Simplify the equation!  The derivation of the integrals is quite difficult.  This is the result of the paper:

16  Replace the

17  What does this mean?  Values for drop off quickly l AlAl

18  What is ?  Everything after the 3 rd 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

19  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.

20  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

21  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

22  Artists place volumes  Automatic sampling within volume produces many voxels

23  Interpolate between voxels  Irradiance evaluated as  M is a 4x4 matrix based on coefficients


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