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

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

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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

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Traditional Lighting - Wikipedia

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Lighting a scene

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

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

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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.

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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 ).

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

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

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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?

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

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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')]

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

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Replace the

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

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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

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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.

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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

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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

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Artists place volumes Automatic sampling within volume produces many voxels

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

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