Spherical Parameterization and Remeshing Emil Praun, University of Utah Hugues Hoppe, Microsoft Research

Motivation: Geometry Images [Gu et al. ’02] 3D geometry completely regular sampling geometry image 257 x 257; 12 bits/channel

Geometry Images [Gu et al. ’02] No connectivity to store No connectivity to store Render without memory gather operations Render without memory gather operations –No vertex indices –No texture coordinates Regularity allows use of image processing tools Regularity allows use of image processing tools Motivation: Geometry Images

Spherical Parametrization geometry image 257 x 257; 12 bits/channel Genus-0 models: no a priori cuts

Contribution Our method: genus-0  no constraining cuts Less distortion in map; better compression New applications: morphing morphing GPU splines GPU splines DSP DSP

Process

Outline 1.Spherical parametrization 2.Spherical remeshing 3.Results & applications

Spherical Parametrization Goals: robustness robustness good sampling good sampling sphere S mesh M [Sander et al. 2001] [Hormann et al. 1999] [Sander et al. 2002] [Hoppe 1996]  coarse-to-fine  stretch metric  coarse-to-fine  stretch metric [Kent et al. ’92] [Haker et al. 2000] [Alexa 2002] [Grimm 2002] [Sheffer et al. 2003] [Gotsman et al. 2003]

Coarse-to-Fine Algorithm Convert to progressive mesh Parametrize coarse-to-fine Maintain embedding & minimize stretch

Before Vsplit: No degenerate/flipped  No degenerate/flipped   1-ring kernel  Apply Vsplit: No flips if V inside kernel V Coarse-to-Fine Algorithm

Before Vsplit: No degenerate/flipped  No degenerate/flipped   1-ring kernel  Apply Vsplit: No flips if V inside kernel Optimize stretch: No degenerate  (they have  stretch) V Coarse-to-Fine Algorithm

Traditional Conformal Metric Preserve angles but “area compression” Bad for sampling using regular grids

Stretch Metric [Sander et al. 2001] [Sander et al. 2002] Penalizes undersampling Better samples the surface

Regularized Stretch Stretch alone is unstable Add small fraction of inverse stretch withoutwith

Outline 1.Spherical parametrization 2.Spherical remeshing 3.Results & applications

Domains And Their Sphere Maps tetrahedron octahedron cube

Domain Unfoldings

Boundary Constraints

Spherical Image Topology

Outline 1.Spherical parametrization 2.Spherical remeshing 3.Results & applications

Example Results

Results

David Model courtesy of Stanford University

Timing Results Model # faces Time Cow23,216 7 min. 7 min. David60,000 8 min. Bunny69, min. Horse96, min. Gargoyle200, min. Tyrannosaurus200, min. Pentium IV, 3GHz, initial code

Timing Results Model # faces Time Cow23, sec. David60, sec. Bunny69, min. Horse96, min. Gargoyle200,000 4 min. Tyrannosaurus200,000 Pentium IV, 3GHz, optimized code

Rendering interpret domain render tessellation

Level-of-Detail Control n=1 n=2 n=4 n=8 n=16 n=32 n=64

Morphing Align meshes & interpolate geometry images

Geometry Compression Image wavelets Boundary extension rules Boundary extension rules –spherical topology –Infinite C 1 lattice* Globally smooth parametrization* Globally smooth parametrization* *(except edge midpoints)

Compression Results 12 KB3 KB1.5 KB

Compression Results

Smooth Geometry Images 33x33 geometry image C 1 surface GPU 3.17 ms [Losasso et al. 2003] ordinary uniform bicubic B-spline

Summary original spherical parametrization geometry image remesh

Conclusions Spherical parametrization Guaranteed one-to-one Guaranteed one-to-one New construction for geometry images Specialized to genus-0 Specialized to genus-0 No a priori cuts  better performance No a priori cuts  better performance New boundary extension rules New boundary extension rules –Effective compression, DSP, GPU splines, …

Future Work Explore DSP on unfolded octahedron 4 singular points at image edge midpoints 4 singular points at image edge midpoints Fine-to-coarse integrated metric tensors Faster parametrization; signal-specialized map Faster parametrization; signal-specialized map Direct D  S  M optimization Consistent inter-model parametrization