Spherical Maps with the Near-Equal Solid-Angle

Property

Liang WanTien-Tsin Wong

The Chinese University of Hong Kong

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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, …

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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

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Three Spherical Maps

RhombicdodecahedronHEALPix Isocube

Equal-area Similar distortion

Six-face,Equal-area

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Base Faces

Isocube

Cubemap HEALPix

6 12

12 6

Rhombicdodecahedron

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Sample Distribution

Rhombicdodecahedron

Isocube

Cubemap HEALPix

Uneven sampli

ng

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Rhombicdodecahedron

Shape Distortion

Isocube

Cubemap HEALPix

Shapedistorti

on

Similar distorti

on

Not equal-area

Quadrilateral

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Property Summary

Cubemap HEALPixRhombic

Dodecahedron Isocube

Base facenumber

6 12 12 6

UniformDistribution

Uneven Even Fair Even

Equal-area property

Not equal Equal Near Equal

Texelshape

Quad. Quad. Quad. Quad./ Tri.

Base PolyhedronSpherical

partitioningPolyhedron

Spherical

partitioning

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Property Summary

Cubemap HEALPixRhombic

Dodecahedron Isocube

Base facenumber

6 12 12 6

UniformDistribution

Uneven Even Fair Even

Equal-area property

Not equal Equal Near Equal

Texelshape

Quad. Quad. Quad. Quad./ Tri.

Base PolyhedronSpherical

partitioningPolyhedron

Spherical

partitioning

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HEALPix Construction Originated in astrophysics

Curvilinear partitioning of the sphere

3/2z kz 4

3

22

1

lz 4

3

22

1

lk,2

ktz 13

ltz 113

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HEALPix Mapping01 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;0708 x=dir.x; y=dir.z; z=dir.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 zone15 if ( za < 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 zone26 tmp= sqrt(3.0f - za); 27 // If in south pole zone28 south = ( z < 0 );29 tt = tmp * tf;30 tmp= 1.0 - tmp;31 tf = tmp * south;32 res= float3(tn+8*south, tmp-tf+tt, tf+tt);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

Cubemap HEALPixRhombic

Dodecahedron Isocube

Base facenumber

6 12 12 6

UniformDistribution

Uneven Even Fair Even

Equal-area property

Not equal Equal Near Equal

Texelshape

Quad. Quad. Quad. Quad./ Tri.

Base PolyhedronSpherical

partitioningPolyhedron

Spherical

partitioning

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Rhombic Dodecahedron Construction Great circle subdivision

10 5

8

69 10

11

27 3

4

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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 filter

ing

What we desire? 6 faces so as to fully utilize the hardware cubemap Retain good properties

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Property Summary

Cubemap HEALPixRhombic

Dodecahedron Isocube

Base facenumber

6 12 12 6

UniformDistribution

Uneven Even Fair Even

Equal-area property

Not equal Equal Near Equal

Texelshape

Quad. Quad. Quad. Quad./ Tri.

Base PolyhedronSpherical

partitioningPolyhedron

Spherical

partitioning

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Isocube Construction Spherical partitioning

3/2z 2/12

k

)12(3

2 lz

zk 13/2

2

3

41 lz

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Isocube Mapping01 float3 R2Q( float3 R )02 { 03 float2 I;04 float3 Q;05 float4 coef;06 float phi, y, ya, bequ,quar;0708 // Compute azimuth angle and convert it in the range [0,4)09 phi = 2*atan2(R.z, R.x)/PI;10 phi+= step(phi, -0.5) * 4;1112 // Decide whether the pixel is in the equatorial region13 y = R.y * 1.5;14 ya = abs(y);15 bequ= step(ya, 1.);1617 // Convert R → I18 I.x = sqrt(3 - 2*ya);19 I.x = lerp(I.x, 1, bequ);20 I.y = phi * I.x;2122 // Map I → Q23 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);2829 return Q;30 }

QR

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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

RhombicDodecahedron

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Environment Mapping

Cubemap Isocube

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Rendering Comparison

Cubemap

Demo

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Discussions

Performance comparison

All three maps are resampled from a high-resolution cubemap

Cubemap HEALPixRhombic

Dodecahedron Isocube

Timing (fps) 232.6 65.9 56.5 168.2

Discrepancy Figure. Discrepancy Comparison

Stretch Figure. Stretch Variance Comparison

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

1. K. M. Górski, E. Hivon, and B. D. Wandelt, Analysis issues for large CMB data sets. In Proc. of the MPA/ESO Conference on Evolution of Large-Scale Structure: from Recombination to Garching, 1998

2. T.T. Wong, L. Wan, C.S. Leung, and P.M. Lam, Real-time environment mapping with equal solid-angle spherical quad-map, Shader X4: Lighting & Rendering, Edited by W. Engel, Charles River Media, 2006

3. L. Wan, T.T. Wong, and C.S. Leung, Spherical Q2tree for sampling dynamic environment sequences, in Proc. of Eurographics Symposium on Rendering 2005 (EGSR 2005), Konstanz, Germany, pp. 21-30, 2005

Isocube1. L. Wan, T.T. Wong, and C.S. Leung, 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/spheremap/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

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Discrepancy comparison

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Stretch variance comparison