Spherical Harmonic Lighting Jaroslav Kivnek Overview Function approximation

Spherical Harmonic Lighting Jaroslav Křivánek

Overview • Function approximation • Spherical harmonics • Some other time § Illumination from environment maps • BRDF representation by spherical harmonics • Spherical harmonics rotation § Hemispherical harmonics § Radiance Caching § Precomputed Radiance Transfer • Clustered Principal Component Analysis • Wavelet Methods

I) Function Approximation

Function Approximation • G(x). . . function to approximate • B 1(x), B 2(x), … Bn(x) … basis functions • We want • Storing a finite number of coefficients ci gives an approximation of G(x)

Function Approximation • How to find coefficients ci? • § Minimize an error measure What error measure? § L 2 error

Function Approximation • Minimizing EL 2 leads to • Where (function scalar product)

Function Approximation • Orthonormal basis • If basis is orthonormal then • we want our bases to be orthonormal

II) Spherical Harmonics

Spherical Harmonics • Spherical function approximation • Domain I = unit sphere S § = directions in 3 D • Approximated function: G(θ, φ) • Basis functions: Yi(θ, φ)= Yl, m(θ, φ) § indexing: i = l (l+1) + m

Spherical Harmonics band 0 (l=0) Y 0, 0 band 1 (l=1) Y 1, -1 Y 2, -2 Y 2, -1 Y 1, 0 Y 2, 0 Y 1, 1 Y 2, 2 band 2 (l=2)

Spherical Harmonics • K … normalization constant • P … Associted Legendre polynomial • § Orthonormal polynomial basis on (0, 1) In general: Yl, m(θ, φ) = K. Ψ(φ). Pl, m(cos θ) § Yl, m(θ, φ) is separable in θ and φ

Function Approximation with SH • n…approximation order • There are n 2 harmonics for order n

Function Approximation with SH • Spherical harmonics are ORTHONORMAL • Function projection § Computing the SH coefficients § Usually evaluated by numerical integration • Low number of coefficients low-frequency signal

Product Integral with SH • Simplified indexing § Yi= Yl, m § i = l (l+1) + m • 2 functions represented by SH • Integral of F(ω). G(ω) is the dot product of F’s and G’s SH coefficients

Product Integral with SH F(ω) = fi Yi(ω) G(ω)F(ω)dx = fi G(ω) =gi Yi(ω) gi

Product Integral with SH • Fundamental property for graphics • Proof

III) Illumination from environment maps

Direct Lighting • Illumination integral at a point • How it simplifies for a parallel directional light • Environment maps § Approximate specular reflection § Lighting does not depend on position § General illumination integral for an environment § § map How it simplifies for a specular BRDF What if the BRDF is not perfectly specular?

Illumination from environment maps • SH representation for lighting & BRDF • Rotation

III) Hemispherical harmonics

Hemispherical harmonics • New set of basis functions • Designed for representing hemispherical functions • Definition similar to spherical harmonics

Hemispherical harmonics Shifting

Hemispherical harmonics SH: Yl, m(θ, φ) = K. Ψ(φ). Pl, m(cos θ) HSH: Hl, m(θ, φ) = K. Ψ(φ). Pl, m(2 cos θ-1) (0, 0) (1, -1) (2, -2) (1, 0) (2, -1) (2, 0) (1, 1) (2, 2)

Hemispherical Harmonics • video

III) Radiance caching

Radiance Caching • Irradiance caching [Ward 88] § Diffuse indirect illumination is smooth § Sample only sparsely, cache and interpolater • Low-frequency view BRDF § Indirect illumination smooth as well § But the illumination is view dependent § Irradiance does not describe view dependence § Cache radiance instead of irradiance § RADIANCE CACHING

Radiance Caching • Incoming radiance representation • BRDF representation • Interpolation • Alignment • Gradients • Video