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

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Presentation on theme: "Spherical Maps with the Near- Equal Solid-Angle Property Liang Wan Tien-Tsin Wong The Chinese University of Hong Kong."— Presentation transcript:

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

2 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 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

4 4 Cubemap  Properties  6 base faces  Fast look-up  Hardware cubemap  Drawbacks  Not uniformly distributed  Not equal-area  Distortion at corners

5 5 Three Spherical Maps Rhombic dodecahedron HEALPix Isocube Equal-area Similar distortion Six-face, Equal-area

6 6 Base Faces Isocube CubemapHEALPix Rhombic dodecahedron

7 7 Sample Distribution Rhombic dodecahedron Isocube Cubemap HEALPix Uneven samplin g

8 8 Rhombic dodecahedron Shape Distortion Isocube Cubemap HEALPix Shape distortio n Similar distortion Not equal- area Quadrilatera l

9 9 Property Summary CubemapHEALPix Rhombic Dodecahedron Isocube Base face number Uniform Distribution UnevenEvenFairEven Equal-area property Not equalEqualNearEqual Texel shape Quad. Quad./ Tri. BasePolyhedron Spherical partitioning Polyhedron Spherical partitioning

10 10 Property Summary CubemapHEALPix Rhombic Dodecahedron Isocube Base face number Uniform Distribution UnevenEvenFairEven Equal-area property Not equalEqualNearEqual Texel shape Quad. Quad./ Tri. BasePolyhedron Spherical partitioning Polyhedron Spherical partitioning

11 11 HEALPix Construction  Originated in astrophysics  Curvilinear partitioning of the sphere

12 12 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; 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 zone 15 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 zone 26 tmp= sqrt(3.0f - za); 27 // If in south pole zone 28 south = ( z < 0 ); 29 tt = tmp * tf; 30 tmp= tmp; 31 tf = tmp * south; 32 res= float3(tn+8*south, tmp-tf+tt, tf+tt); 33 } 34 return res; 35 }

13 13 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

14 14 Property Summary CubemapHEALPix Rhombic Dodecahedron Isocube Base face number Uniform Distribution UnevenEvenFairEven Equal-area property Not equalEqualNearEqual Texel shape Quad. Quad./ Tri. BasePolyhedron Spherical partitioning Polyhedron Spherical partitioning

15 15 Rhombic Dodecahedron Construction  Great circle subdivision

16 16 Rhombic Dodecahedron  Characteristics  All texels are distorted similarly  Identical base faces  Geodesic property

17 17 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

18 18 Property Summary CubemapHEALPix Rhombic Dodecahedron Isocube Base face number Uniform Distribution UnevenEvenFairEven Equal-area property Not equalEqualNearEqual Texel shape Quad. Quad./ Tri. BasePolyhedron Spherical partitioning Polyhedron Spherical partitioning

19 19 Isocube Construction  Spherical partitioning

20 20 Isocube Mapping 01 float3 R2Q( float3 R ) 02 { 03 float2 I; 04 float3 Q; 05 float4 coef; 06 float phi, y, ya, bequ,quar; // 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; // Decide whether the pixel is in the equatorial region 13 y = R.y * 1.5; 14 ya = abs(y); 15 bequ= step(ya, 1.); // Convert R → I 18 I.x = sqrt(3 - 2*ya); 19 I.x = lerp(I.x, 1, bequ); 20 I.y = phi * I.x; // 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); return Q; 30 } Q R

21 21 Isocube  Characteristics  Equal area  6 faces  Extremely fast look-up  Drawback  Distortion in polar regions

22 22 Environment Mapping HEALPix Rhombic Dodecahedron

23 23 Environment Mapping Cubemap Isocube

24 24 Rendering Comparison Cubemap Demo

25 25 Discussions  Performance comparison  All three maps are resampled from a high-resolution cubemap CubemapHEALPix Rhombic Dodecahedron Isocube Timing (fps) DiscrepancyFigure. Discrepancy Comparison StretchFigure. Stretch Variance Comparison The timing test context: object with 106,466 vertices, Pentium IV 2.6 GHz CPU, nVidia GeForceFX 6800 Ultra.

26 26 Potential Applications  Equal-area, uniform sampling  OmniMax video HEALPix, isocube  Similar distortion  Shadow mapping HEALPix, rhombic dodecahedron  Hemicube Cubemap, isocube

27 27 References  HEALPix  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  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  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 , 2005  Isocube  L. Wan, T.T. Wong, and C.S. Leung, Isocube: Exploiting the Cubemap Hardware, IEEE Transactions on Visualization and Computer Graphics, to appear

28 28 Webpage  The updated document can be found in the website pheremap.html  The demo code is free to download from the link heremap.html

29 29 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)

30 30 Q & A

31 31 Discrepancy comparison

32 32 Stretch variance comparison


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