Smooth Geometry Images

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Smooth Geometry Images

• Frank Losasso, Hugues Hoppe, Scott Schaefer, Joe
  Warren

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Overview

• Provide a simple representation using a single
  uniform bi-cubic B-spline

Multiple Patches
Single Patch
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Geometry Images
Gu et al 2002

• Sample arbitrary surface using a regular 2D grid
• Connectivity is implicit

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Gu et al 2002
cut
parametrize
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Gu et al 2002
cut
sample
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Gu et al 2002
cut
store
render
r,g,b x,y,z
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General cut
Gu et al 2002

• Supports surfaces of arbitrary genus
• But, boundary has complicated topology requires
  sideband

a
a
a
a
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Our Approach X-Cut

• Special type of cut curve
• Make a X-cut centered at x
• Unfold domain into a square image
• Creates simple boundary symmetries

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The X-Cut
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Spherical Remeshing
image I
domain D
sphere S
mesh M
demo
Praun and Hoppe 2003
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How To Obtain Control Points

• B-spline is approximating ? should not directly
  sample surface
• Instead, use least-squares fitting

?
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How To Obtain Smoothness

• Application of boundary rules
• Pad image to recreate 1-ring around all vertices

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How To Obtain Smoothness

• All vertices regular, except boundary midpoints

generally not C1
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Bi-cubic Subdivision on GPU

• 4 operators, stored as fragment programs

limit
bilinearsubdivision
meshaveraging
tangent
repeat
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Bi-cubic Subdivision on GPU

• 4 operators, stored as fragment programs

limit
bilinearsubdivision
meshaveraging
tangent
repeat
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Rendering

• Interpret as vertex array (OpenGL extension)
• Render using triangles

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Discrete Subdivision Levels

• Both sub-sampling and subdivision are easily
  implemented

(2k1) x (2k1)
Original Image
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Continuous Subdivision Levels

• Prevent popping when changing subdivision levels

gk
gk1
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Continuous Subdivision Levels

• Prevent popping when changing subdivision levels

linearsubdivision
1-?

?
gk?
meshaveraging
bilinearsubdivision
gk
gk1
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Real-Time Demo
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Displacement Mapping
simulation
GPU
CPU
33x33
GPU
scalardisplacementmap
257x257
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Performance Results
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Drawbacks and Limitations

• Genus cannot be gt0
• Surface rippling

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Summary

• Closed smooth surface using single patch
• Stored as geometry image
• Simple and efficient GPU evaluation
• Continuous level-of-detail
• Displacement mapping

5x5
9x9
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Eye candy
65x65
33x33
65x65