An Efficient Representation for Irradiance Environment Maps. Stanford University SIGGRAPH 2001

Transcription

1 An Efficient Representation for Irradiance Environment Maps Ravi Ramamoorthi Pat Hanrahan Stanford University SIGGRAPH 2001

2 Irradiance Environment Maps R N Incident Radiance (Illumination Environment Map) Irradiance Environment Map

3 Diffuse surfaces Distant illumination Assumptions No shadowing, interreflection Hence, Irradiance is a function of surface normal E( n) = L( ω)( n ω) dω Ω( n)

4 Diffuse Reflection B( x, n) = ρ( x) E( n) Radiosity (image intensity) Reflectance (albedo/texture) Irradiance (incoming light) = quake light map

5 Computing Irradiance Classically, hemispherical integral for each pixel Incident Radiance Irradiance Lambertian surface is like low pass filter Frequency-space analysis

6 Spherical Harmonics 0 l m 1 Ylm( θ, ϕ) 1 y z x 2... xy yz 3z 2 1 zx x 2 y

7 Spherical Harmonic Expansion Expand lighting (L), irradiance (E) in basis functions + l L( θ, φ) L Y ( θ, φ) = l= 0 m= l + l l= 0 m= l lm lm E( θ, φ) E Y ( θ, φ) = lm lm =

8 Analytic Irradiance Formula Lambertian surface acts like low-pass filter π 2 π /3 E = AL lm l lm A l π /4 l A l 2 ( 1) 1 l! = 2 π ( 2)( 1) 2 (!) l l 2 l+ l l 2 leven

9 9 Parameter Approximation Exact image Order 0 1 term RMS error = 25 % 0 l m Y (, ) lm θ ϕ 1 y z x 2 2 xy yz 3z 1 zx x y

10 9 Parameter Approximation Exact image Order 1 4 terms RMS Error = 8% 0 l m Y (, ) lm θ ϕ 1 y z x 2 2 xy yz 3z 1 zx x y

11 9 Parameter Approximation Exact image Order 2 9 terms RMS Error = 1% For any illumination, average error < 3% [Basri Jacobs 01] l m y 2 xy yz 3z 1 z Y (, ) lm θ ϕ x zx x y 2 2

12 Computing Light Coefficients Compute 9 lighting coefficients L lm 9 numbers instead of integrals for every pixel Lighting coefficients are moments of lighting 2π L L( θ, φ) Y ( θ, φ) sinθ dθdφ lm π = θ= 0φ= 0 lm Weighted sum of pixels in the environment map L = envmap[ pixel] basisfunc [ pixel] lm lm pixels( θφ, )

13 Comparison Incident illumination 300x300 Irradiance map Texture: 256x256 Hemispherical Integration 2Hrs Irradiance map Texture: 256x256 Spherical Harmonic Coefficients 1sec Time Time

14 Rendering We have found the SH coefficients for irradiance which is a spherical function. Given a spherical coordinate, we want to calculate the corresponding irradiance quickly. E( θ, φ) = Aˆ L l, m l lm Y lm ( θ, φ )

15 Rendering Irradiance approximated by quadratic polynomial En ( ) = cl + 2cL + 2cL + 2cL + cl x 2 1 1y 2 10z 5 20(3z cL 1 2 2xy 2 cl 1 21xz 2cL 1 2 1yz+ cl 1 22( x y ) ) + E( n) = t n Mn 4x4 matrix (depends linearly on coefficients L lm ) Surface Normal vector column 4-vector x y z 1

16 Hardware Implementation En ( ) = t nmn Simple procedural rendering method (no textures) Requires only matrix-vector multiply and dot-product In software or NVIDIA vertex programming hardware surface float1 irradmat (matrix4 M, float3 v) { float4 n = {v, 1} ; return dot(n, M*n) ; }

17 Complex Geometry Assume no shadowing: Simply use surface normal y

18 Lighting Design Final image sum of 3D basis functions scaled by L lm Alter appearance by changing weights of basis functions

19 Results

20 Theory Summary Analytic formula for irradiance Frequency-space: Spherical Harmonics To order 2, constant, linear, quadratic polynomials 9 coefficients (up to order 2) suffice Practical Applications Efficient computation of irradiance Simple procedural rendering New representation, many applications

21 Precomputed Radiance Transfer for Real-Time Rendering in Dynamic, Low-Frequency Lighting Environments Peter-Pike Sloan, Microsoft Research Jan Kautz, MPI Informatik John Snyder, Microsoft Research SIGGRAPH 2002

22 r r L( s) = % lb i i( s) Basic idea v r V( s r ) r r r r r r r R( v) = L( s) V( s) f( s, v) H N ( s) ds = % lt i i N r r r r r r r R( v) H () slb i= i( max( s) Preprocess V( s) f ( Nsv,,0) ) Hfor N( sds ) all i r ( r ) ( r l B sv s ) f ( r sv, r ) H ( r s ) r Rv ( ) % i i ds = % ( ) N

23 Use 25 bases... Precomputation Basis 16 Basis 17 illuminate result Basis 18...

24 Diffuse No Shadows/Inter Shadows Shadows+Inter

25 Glossy No Shadows/Inter Shadows Shadows+Inter Glossy object, 50K mesh Runs at 3.6/16/125fps on 2.2Ghz P4, ATI Radeon 8500

26 Arbitrary BRDF Anisotropic BRDFs Other BRDFs Spatially Varying

27 Volumes Diffuse volume: 32x32x32 grid Runs at 40fps on 2.2Ghz P4, ATI 8500 Here: dynamic lighting