A Hierarchical Method for Aligning Warped Meshes

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A Hierarchical Method for Aligning Warped Meshes

• Leslie Ikemoto1,
• Natasha Gelfand2,
• Marc Levoy2

1UC Berkeley, formerly Stanford 2Stanford
University
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Scan Alignment Pipeline
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Alignment Methods

• Pairwise alignment
• Iterated Closest Point (ICP)
• Variant from Chen-Medioni 91

Global relaxation Global registration
1) Compute R, t minimizing distances from pi to
tangent plane at qi
2) Apply transform and repeat
Rigid scans
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The Digital Michelangelo Statue Scanner

• Large
• High resolution (0.25 mm)
• Reconfigurable
• Deployed in the field

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Registration Errors

• ICP

Global Registration
Correct calibration
4 mm misalignment
Correct calibration 0.13 mm avg. err.
Incorrect calibration 1.81 mm avg. err.
Incorrect calibration
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Model Generated
Spacing of range samples 0.5 mm
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Possible Solutions

• Calibrate the scanner better
• Learn warp by self-calibration
• Introduce compensating warp
• fit low-order polynomial to warp
• use piecewise rigid approximation to curved warp

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Compensating Warp

• Smooth warp

Approximate with a piecewise rigid model of
overlapping sub-meshes
Create pieces hierarchically
13 original scans
84 sub-meshes
R, T
R1..8, T1..8
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Proposed Pipeline
Find most misaligned pair of scans
Loop until error below threshold
Initial guess
Dice into pieces
Global registration
Pairwise alignment
Pairwise alignment
Global registration
Scan merging
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Design Criteria for Dicing a Scan

• Isotropic warp
• Keep even aspect ratio
• Overlap with neighbors
• Needed for alignment
• Use size to control
• tendency to warp
• Sufficiently
• constraining features
• for alignment
• Pre-analyze meshes for ICP stability

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Determining Placement of Cutting Planes
Overlap size affects hinge stiffness

• Even aspect ratio dice along longest dimension
  of oriented bounding-box
• Overlap determined empirically (30 of oriented
  bounding box)

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Determining Whether to Dice Using Stability
Analysis

• Sufficiently constraining features stability
  analysis to determine degenerate geometries
  Gelfand et al. 3DIM03

2 translations, 1 rotation
3 rotations
1 rotation, 1 translation
1 translation
1 rotation
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Sampling Technique

• Sample to constrain transformations during
  alignment Gelfand et al. 3DIM03

Translation in the plane
Rotation in the plane
Rotation out of the plane
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Running Times

• Forma Urbis fragment, approximately 85 cm x 120
  cm
• Hardware Intel P4, 2.80 GHz

Meshes Polygons (million) Pair Matching Global Reg. Points Selected Dicing
4 meshes 31 309 002 1200
37 scanner sweeps 31 1033 002 25,800 215
75 sub-meshes 35.8 3400 004 26,200 230
117 sub-meshes 40.2 2100 005 63,510 330
160 sub-meshes 46.1 2000 006 123,110 300
197 sub-meshes 52.2 1700 013 204,268
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Model Generated
Original
After warping
Average error 0.8 mm
Average error 1.15 mm
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More Results
Avg. err. 0.4 mm
Avg. err. 0.8 mm
Double lines
Original
After warping
Blurry lines
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Conclusions

• Alignment method for smoothly warped meshes
• Introduce minimal compensating warp
• Does not require a specific characterization of
  scanner warp
• Relatively simple to implement

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Limitations

• Will not converge if
• scans very noisy
• scans do not have many features
• warp is not smooth

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Future Work

• Fit smooth spline
• yields non-rigid warp
• retrospective scanner
• calibration
• One-to-many stability
• analysis
• Improve measurement
• strategy

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Acknowledgements

• Our sponsors
• National Science Foundation Research Grant
  IIS-0113427
• Stanford University Presidents Fund
• Also thanks to
• Digital Michelangelo team
• Forma Urbis team