GDC 2005

1
Sphere Maps with the Near-Equal Solid-Angle
Property
Liang Wan Tien-Tsin Wong The Chinese
University of Hong Kong
2
Spherical Maps

• Represent the surrounding environment
• Applications
• Environment mapping
• Precomputed radiance transfer (PRT)
• Image-based relighting
• Spherical maps
• Cubemap, longitude/latitude map, dual paraboloid
  map, sphere map,

3
Criteria

• Uniform distribution
• How uniformly the samples are distributed?
• Equal-area property
• Whether the texels span the same solid-angle
• Stretch
• Measure the mapping distortion of texel shapes
• Query efficiency
• Speed of querying a point in the map
• Base face number
• 6-face map fits nicely into the hardware

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Cubemap

• Properties
• 6 base faces
• Fast look-up
• Hardware cubemap
• Drawbacks
• Not uniformly distributed
• Not equal-area
• Distortion at corners

5
Three Spherical Maps
Equal-area
Similar distortion
Six-face,Equal-area
6
Base Faces
6
12
Cubemap
HEALPix
12
6
Rhombic dodecahedron
Isocube
7
Sample Distribution
Uneven sampling
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Shape Distortion
Quadrilateral
Not equal-area
Shape distortion
Similar distortion
Isocube
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Property Summary
10
Property Summary
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HEALPix Construction

• Originated in astrophysics
• Curvilinear partitioning of the sphere

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HEALPix Mapping
01 float3 hpmap(float3 dir) 02 03 float
fn, u, v, iu, iv, x, y, z, t 04 float tt, tn,
tf, za, tmp, zone, south 06 float3 res 07 08
xdir.x ydir.z zdir.y 09 t
atan2(-y, x)/1.571 10 t step(t, 0.f)
4.0 11 za 3.0 abs(z) 12 tf
modf(t, tn) 13 // Equatorial or polar zone 15
if ( za lt 2.0 ) 16 // Equatorial zone
17 tt t 0.5 18 tmp z 0.75
19 u modf(tt tmp, iu) 20 v
modf(tt - tmp, iv) 21 fn min(iu, iv) 22
fn 4 (sign(iv-iu) - floor(fn/4))4 23
res float3(fn, u, v) 24 else 25
// Polar zone 26 tmp sqrt(3.0f - za) 27
// If in south pole zone 28 south ( z lt
0 ) 29 tt tmp tf 30 tmp 1.0 -
tmp 31 tf tmp south 32 res
float3(tn8south, tmp-tftt, tftt) 33 34
return res 35
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HEALPix

• Characteristics
• Equal area
• A hierarchical structure
• Samples on parallel small circles
• Facilitate spherical harmonic transform
• Drawback
• Texels in different base faces have different
  shapes

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Property Summary
15
Rhombic Dodecahedron Construction

• Great circle subdivision

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Rhombic Dodecahedron

• Characteristics
• All texels are distorted similarly
• Identical base faces
• Geodesic property

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HEALPix Rhombic Dodecahedron

• Drawbacks
• Both consist of 12 base faces
• Tailor-made programs for texture lookup
• Difficult for mipmap construction and tri-linear
  filtering
• What we desire?
• 6 faces so as to fully utilize the hardware
  cubemap
• Retain good properties

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Property Summary
19
Isocube Construction

• Spherical partitioning

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Isocube Mapping
01 float3 R2Q( float3 R ) 02 03 float2 I 04
float3 Q 05 float4 coef 06 float phi, y, ya,
bequ,quar 07 08 // Compute azimuth angle and
convert it in the range 0,4) 09 phi
2atan2(R.z, R.x)/PI 10 phi step(phi, -0.5)
4 11 12 // Decide whether the pixel is in the
equatorial region 13 y R.y 1.5 14 ya
abs(y) 15 bequ step(ya, 1.) 16 17 // Convert
R ? I 18 I.x sqrt(3 - 2ya) 19 I.x
lerp(I.x, 1, bequ) 20 I.y phi I.x 21 22
// Map I ? Q 23 quar floor(phi 0.5) 24
coef texRECT(signTBL, float2(quar, 0)) 25 Q.x
dot(coef.xy, I) 26 Q.y lerp(sign(y), y,
bequ) 27 Q.z dot(coef.zw, I) 28 29 return
Q 30
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Isocube

• Characteristics
• Equal area
• 6 faces
• Extremely fast look-up
• Drawback
• Distortion in polar regions

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Environment Mapping
HEALPix
Rhombic Dodecahedron
23
Environment Mapping
Isocube
Cubemap
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Cubemap v.s. HEALPix
Cubemap
HEALPix
25
Cubemap v.s. Rhombic Dodecahedron
Cubemap
Rhombic Dodecahedron
26
Cubemap v.s. Isocube
Cubemap
Isocube
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Discussions

• Performance comparison
• All three maps are resampled from a
  high-resolution cubemap

The timing test context object with 106,466
vertices, Pentium IV 2.6 GHz CPU, nVidia
GeForceFX 6800 Ultra.
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Potential Applications

• Equal-area, uniform sampling
• OmniMax video
• HEALPix, isocube
• Similar distortion
• Shadow mapping
• HEALPix, rhombic dodecahedron
• Hemicube
• Cubemap, isocube

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References

• HEALPix
• GÓRSKI K. M., HIVON E., WANDELT B. D. Analysis
  issues for large CMB data sets. In Proceedings of
  the MPA/ESO Conference on Evolution of
  Large-Scale Structure from Recombination to
  Garching, 1998
• Wong T.T., Wan L., Leung C.S., and Lam P.M.,
  Real-time environment mapping with equal
  solid-angle spherical quad-map, Shader X4
  Lighting Rendering, Edited by W. Engel, Charles
  River Media, 2006
• Wan L., Wong T.T., and Leung C.S., Spherical
  Q2tree for sampling dynamic environment
  sequences, in Proceedings of Eurographics
  Symposium on Rendering 2005 (EGSR 2005),
  Konstanz, Germany, pp. 21-30, 2005
• Isocube
• Wan L., Wong T.T., and Leung C.S., Isocube
  Exploiting the Cubemap Hardware, IEEE
  Transactions on Visualization and Computer
  Graphics, to appear

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Webpage

• The updated document can be found in the website
• http//www.cse.cuhk.edu.hk/ttwong/papers/spherem
  ap/spheremap.html
• The demo code is free to download from the link
• http//www.cse.cuhk.edu.hk/ttwong/demo/spheremap
  /spheremap.html

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Credits Acknowledgments

• Rhombic dodecahedron is a joint work with
  Chi-Wing Fu (HKUST) and Chi-Sing Leung (CityU)
• Isocube is a joint work with Chi-Sing Leung
  (CityU)
• Thanks to Lai-Sze Ng and Ping-Man Lam for
  implementing part of the codes
• Thanks to Xuemiao Xu for capturing some of the
  panoramas
• This work is supported by Research Grants Council
  of the Hong Kong Special Administrative Region,
  under RGC Earmarked Grants (Project No.
  CUHK416806)

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Q A
33
Discrepancy comparison
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Stretch variance comparison