Transcript A Hierarchical Method for Aligning Warped Meshes

A Hierarchical Method for
Aligning Warped Meshes
Leslie Ikemoto1,
Natasha Gelfand2,
Marc Levoy2
1UC
Berkeley, formerly Stanford
2Stanford University
Scan Alignment Pipeline
Pairwise
alignment
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Global
relaxation
A Hierarchical Method for Aligning
Warped Meshes
Scan
merging
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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
ni
Point constraints
from ICP
qi
pi
2) Apply transform and repeat
Rigid scans
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A Hierarchical Method for Aligning
Warped Meshes
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The Digital Michelangelo
Statue Scanner
Calibrated motions: pitch (yellow), pan (blue),
horizontal translation (orange)
• Large
• High resolution (0.25 mm)
• Reconfigurable
• Deployed in the field
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Uncalibrated motions: vertical translation (red),
remounting the scan head, reconfiguring the
scanner, moving the entire scanner
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Warped Meshes
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Registration Errors
ICP
Global Registration
Correct calibration
Correct calibration
0.13 mm avg. err.
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Incorrect calibration
1.81 mm avg. err.
Incorrect calibration
A Hierarchical Method for Aligning
Warped Meshes
4 mm
misalignment
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Model Generated
Spacing of range
samples = 0.5 mm
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Warped Meshes
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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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Warped Meshes
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Compensating Warp
Smooth warp
Approximate with a piecewise rigid
model of overlapping sub-meshes
Create pieces hierarchically
13 original scans
R, T
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84 sub-meshes
R1..8, T1..8
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Warped Meshes
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Proposed Pipeline
Initial guess
Find most
misaligned
pair of scans
Loop until error below
threshold
Dice into
pieces
Pairwise
alignment
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Global
registration
Pairwise
alignment
Global
registration
A Hierarchical Method for Aligning
Warped Meshes
Scan
merging9
Design Criteria for Dicing a Scan
• Isotropic warp
– Keep even aspect ratio
Arbitrary cutting planes
• Overlap with neighbors
– Needed for alignment
– Use size to control
tendency to warp
• Sufficiently
constraining features
for alignment
No features
– Pre-analyze meshes for ICP
stability
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Warped Meshes
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Determining Placement
of Cutting Planes
Overlap size affects “hinge stiffness”
Smaller overlap
Lower squared error
Larger overlap
Mesh dicing scheme
Higher squared error
• Even aspect ratio: dice along longest dimension of
oriented bounding-box
• Overlap: determined empirically (30% of oriented
bounding box)
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Warped Meshes
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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
1 rotation
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3 rotations
1 rotation, 1 translation
1 translation
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Warped Meshes
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Sampling Technique
• Sample to constrain transformations
during alignment [Gelfand et al. 3DIM03]
Translation in
the plane
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Rotation in
the plane
A Hierarchical Method for Aligning
Warped Meshes
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
4 meshes
31
3:09
0:02
1200
37 scanner
sweeps
31
10:33
0:02
25,800
2:15
75 submeshes
35.8
34:00
0:04
26,200
2:30
117 submeshes
40.2
21:00
0:05
63,510
3:30
160 submeshes
46.1
20:00
0:06
123,110
3:00
197 submeshes
52.2
17:00
0:13
204,268
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A Hierarchical Method for Aligning
Warped Meshes
Dicing
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Model Generated
After
Original
warping
Average error = 0.8
1.15mm
mm
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Warped Meshes
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More Results
Avg. err. =
0.8
0.4 mm
Double lines
After
Original
warping
Blurry lines
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Warped Meshes
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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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Warped Meshes
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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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Warped Meshes
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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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Warped Meshes
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Acknowledgements
Our sponsors…
• National Science Foundation Research
Grant IIS-0113427
• Stanford University President’s Fund
Also thanks to…
• Digital Michelangelo team
• Forma Urbis team
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Warped Meshes
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