Spherical Harmonic Lighting

1
Spherical Harmonic Lighting

• Jaroslav Krivánek

2
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

3
I) Function Approximation
4
Function Approximation

• G(x) ... function to approximate
• B1(x), B2(x), Bn(x) basis functions
• We want
• Storing a finite number of coefficients ci gives
  an approximation of G(x)

5
Function Approximation

• How to find coefficients ci?
• Minimize an error measure
• What error measure?
• L2 error

6
Function Approximation

• Minimizing EL2 leads to
• Where(function scalar product)

7
Function Approximation

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

8
II) Spherical Harmonics
9
Spherical Harmonics

• Spherical function approximation
• Domain I unit sphere S
• directions in 3D
• Approximated function G(?,f)
• Basis functions Yi(?,f) Yl,m(?,f)
• indexing i l (l1) m

10
Spherical Harmonics
band 0 (l0)
band 1 (l1)
band 2 (l2)
11
Spherical Harmonics

• K normalization constant
• P Associted Legendre polynomial
• Orthonormal polynomial basis on (0,1)
• In general
• Yl,m(?,f) K . ?(f) . Pl,m(cos ?)
• Yl,m(?,f) is separable in ? and f

12
Function Approximation with SH

• napproximation order
• There are n2 harmonics for order n

13
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

14
Product Integral with SH

• Simplified indexing
• Yi Yl,m
• i l (l1) m
• 2 functions represented by SH
• Integral of F(?).G(?) is the dot product of Fs
  and Gs SH coefficients

15
Product Integral with SH
16
Product Integral with SH

• Fundamental property for graphics
• Proof

17
III) Illumination from environment maps
18
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?

19
Illumination from environment maps

• SH representation for lighting BRDF
• Rotation

20
III) Hemispherical harmonics
21
Hemispherical harmonics

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

22
Hemispherical harmonics
Shifting
23
Hemispherical harmonics
SH Yl,m(?,f) K . ?(f) . Pl,m(cos ?) HSH
Hl,m(?,f) K . ?(f) . Pl,m(2cos ?-1)
24
Hemispherical Harmonics

• video

25
III) Radiance caching
26
Radiance Caching

• Irradiance caching Ward88
• Diffuse indirect illumination is smooth
• Sample only sparsely, cache and interpolate later
• 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

27
Radiance Caching

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