Real-time Soft Shadows in Dynamic Scenes using Spherical Harmonic Exponentiation

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Real-time Soft Shadows in Dynamic Scenes using
Spherical Harmonic Exponentiation

• Zhong Ren1 Rui Wang1 John Snyder2 Kun Zhou3
  Xinguo Liu3
• Bo Sun4 Peter-Pike Sloan5 Hujun Bao1 Qunsheng
  Peng1 Baining Guo3
• 1 Zhejiang Univ. 2 Microsoft Research 3
  Microsoft Research Asia
• 4 Columbia Univ. 5 Microsoft Corporation

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Fast Soft Shadows in Dynamic Scenes

• Motivation
• realism
• shape/proximity cues
• new look
• Challenges
• large light sources
• complex, dynamic blockers/receivers
• self-shadows cast-shadows

unshadowed
shadowed
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Previous Work in Fast Shadow Rendering
Name Reference Lighting Constraints
shadow buffer/vol. Williams78, point -
accum. buffer Segal92, small area many passes
PRT (SH) Sloan02, low-freq static
PRT (all-freq) Ng03, all-freq static, diffuse
PRT (dynamic) James03,05 low-freq precomp. sequences
LDPRT Sloan05 low-freq local effects
ambient occlusion Bunnel04, DC no casting
shadow fields Zhou05 low-freq few, rigid objs
SHEXP current low-freq many, deform objs
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Previous Work in Fast Shadow Rendering
Name Reference Lighting Constraints
ambient occlusion Bunnel04, DC no casting
shadow fields Zhou05 low-freq few, rigid objs
SHEXP current low-freq many, deform objs

• soft shadows from large-area lights
• dynamic shading
• blocker/receiver motion not precomputed

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Ambient Occlusion vs. SHEXP
ray traced
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SHEXP Summary

• Approximate blockers with spheres
• accumulate over large blockers, not light
  directions
• symmetry simplifies calculation

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SHEXP Summary

• Approximate blockers with spheres
• accumulate over large blockers, not light
  directions
• symmetry simplifies calculation
• Represent low-frequency visibility/lighting in SH

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SHEXP Summary

• Approximate blockers with spheres
• accumulate over large blockers, not light
  directions
• symmetry simplifies calculation
• Represent low-frequency visibility/lighting in SH
• For each receiver point p
• accumulate visibility logarithm over blocker
  spheres
• exponentiate
• shade

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SH ReviewProjection and Reconstruction
s
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SH ReviewProjection and Reconstruction
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SH ReviewProjection and Reconstruction
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SH ReviewProducts
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SH ReviewProducts
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SH ReviewProducts
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SH ReviewShading Ng04
gp
L
H(Np)
lighting
visibility
BRDF

• shade(p) (L, gp, H(Np))

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SH ReviewShading Ng04
gp
L
H(Np)

• shade(p) (L, gp, H(Np))

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Visibility Accumulation (Product Space Zhou05)
g g1g2gm

• m moving blockers
• gi is SH visibility of blocker i
• exact product order m(n-1)1
• truncate to order n after each
• need m-1 expensive SH products

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Visibility Accumulation Example(Product Space)
Product Space Accumulated
64 Spheres
1 Spheres
2 Spheres
3 Spheres
16 Spheres
32 Spheres
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Idea Use Log Space

• SH product expensive O(n5)
• SH add cheap O(n2)
• e.g. SH order n4
• ab ? 673 flops
• ab ? 16 adds!
• suggests rule for positive reals

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Visibility Accumulation(Log Space)
g exp(f 1 f 2 f m)

• applies scalar rule to every visibility direction
• f ilog(gi) is SH log visibility of sphere i
• need m-1 vector adds, one exp
• greatly reduces per-blocker cost ( ? )

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Visibility Accumulation Example(Log Space)
Log Space Accumulated
64 Spheres
1 Spheres
2 Spheres
3 Spheres
16 Spheres
32 Spheres
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Log vs. Product Space
SH Projection
Visibility Function
Log Space Approx.
Product Space Approx.
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SH Exponential Volterra Series

• expand exp(x) in SH projection formula

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SH Exponential Volterra Series

• expand exp(x) in SH projection formula

• yielding sum of powers f p

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SH Exponential Product Series

• exact evaluation horribly expensive!
• order-p tensors
• output order grows with p p(n-1)1
• approximate with product series

• degree p series needs p-1 SH products
• truncate each product back to order n

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Accelerating Product Series

• bigger f ? more series terms in exp( f )
• DC isolation compute DC via scalar exp
• scaling/squaring divide by 2p, exp, square p
  times
• factoring reduce products Higham05

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Optimal Linear Approximation

• linear model (degree 1 series)
• no SH products!
• optimal coefs a and b
• tabulated functions of input magnitude f
• least-squares fit to example visibility functions
• accurate for n 4, f 4.8
• extend magnitude range using scaling/squaring

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SHEXP Method Comparison
log space (SHEXP)
PS-2, 45k
PS-21, 328k
product space, 495k
HYB, 31k
OL, 0
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SHEXP Method Comparison
log space (SHEXP)
PS-2, 45k
PS-21, 328k
product space, 495k

• evaluate on GPU
• workhorse method

HYB, 31k
OL, 0
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SH Logarithm

• computed for sphere blockers as preprocess
• naïve method numerical integration with clipping
• better method series inversion

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Part 2 Rendering System Results
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Approximating Blocker Geometry

• bounding sphere set
• minimize outside volume Wang06
• animate via mean value coordinates Ju05

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Single Sphere Visibility

• tabulate logs of circle visibility, f (q)
• function of angle subtended, q
• canonically oriented around z axis
• rotate on-the-fly

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Single Sphere Visibility(Rotation)

• use rotation rule Sloan05

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Self-Shadowing Replacement Rules
Tp
p
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Sphere Hierarchies
Blocker Hierarchy
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Sphere Hierarchies
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Sphere Hierarchies
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Sphere Hierarchies
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Per-Band Weighting
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Rendering

• Per-cluster computation (on CPU)
• Per-vertex computation (on GPU)

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Per-Cluster Computation (CPU)

• For each cluster
• choose cut
• for each sphere in cut
• compute weight vector
• Pack information into textures
• sphere cut info (center, radius, weight vector)
• vertex info (position, normal, receiver cluster
  id)

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Per-Vertex Computation (GPU)

• For each sphere i in cut
• apply self-shadow replacement rules
• get canonical log visibility vector f (q)
• rotate and apply weights, yielding f i
• accumulate f f i
• Exponentiate using HYB
• total visibility vector g exp( f )
• Shade using (H(N), L, g)

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Demos
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Contributions

• Blocker approximation using bounding sphere sets
• general, deformable objects
• Accumulation of low-frequency blocker visibility
• much faster in log space
• accurate as product space
• Fast evaluation of SH exp/log
• product series approx. to Volterra series
• optimal linear method
• SH log by series inversion

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Limitations

• Low-frequency lighting
• Low-frequency (diffuse) BRDF
• Distant lighting
• true local lighting requires sorting
• demo special case

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Future Work

• Anisotropic blocker models
• Diffuse inter-reflection
• Models for spatial, temporal variation

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Acknowledgements

• Becky Sundling
• video production
• Mingdong Xie
• modeling and animation
• Anon. SIGGRAPH reviewers
• suggestions and comments

and thank you!
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Comparison Log vs. Product Spacen4
SH Projection
Visibility Function
Log Space Approx.
Product Space Approx.
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Comparison Log vs. Product Spacen4,
accumulation at n6
SH Projection
Visibility Function
Log Space Approx.
Product Space Approx.