A Hierarchical Method for Aligning Warped Meshes Leslie Ikemoto 1, Natasha Gelfand 2, Marc Levoy 2 1 UC Berkeley, formerly Stanford 2 Stanford University.

A Hierarchical Method for Aligning Warped Meshes

Leslie Ikemoto1,Natasha Gelfand2,

Marc Levoy2

1UC Berkeley, formerly Stanford2Stanford University

October 9, 2003 A Hierarchical Method for Aligning Warped Meshes

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Scan Alignment Pipeline

Globalrelaxation

Scanmerging

Pairwisealignment


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Alignment MethodsPairwise alignment:

Iterated Closest Point (ICP)[Variant from Chen-Medioni ‘91]

pi

qi

ni

Global relaxation: Global registration

Point constraints from ICP

Rigid scans

1) Compute R, t minimizing distances from pi to tangent plane at qi

2) Apply transform and repeat


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The Digital Michelangelo Statue Scanner

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

Calibrated motions: pitch (yellow), pan (blue), horizontal translation (orange)

Uncalibrated motions: vertical translation (red), remounting the scan head, reconfiguring the scanner, moving the entire scanner


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Registration Errors

ICP Global Registration

Correct calibration0.13 mm avg. err.

Incorrect calibration1.81 mm avg. err.

4 mm misalignment

Correct calibration

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 WarpSmooth warp Approximate with a piecewise rigid

model of overlapping sub-meshes

Create pieces hierarchically

R, T R1..8, T1..8

13 original scans 84 sub-meshes


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Proposed Pipeline

Pairwisealignment

Globalregistration

Find most misaligned

pair of scans

Globalregistration

Dice intopieces

Pairwise alignment

Scanmerging

Loop until error below threshold

Initial guess


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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

No features

Arbitrary cutting planes


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Determining Placement of Cutting Planes

• Even aspect ratio: dice along longest dimension of oriented bounding-box

• Overlap: determined empirically (30% of oriented bounding box)

Smaller overlap

Lower squared error

Larger overlap

Higher squared errorMesh dicing scheme

Overlap size affects “hinge stiffness”


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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 rotation 1 translation


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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

MeshesPolygons (million)

Pair Matching

Global Reg.# Points Selected

Dicing

4 meshes 31 3:09 0:02 1200

37 scanner sweeps

31 10:33 0:02 25,800 2:15

75 sub-meshes

35.8 34:00 0:04 26,200 2:30

117 sub-meshes

40.2 21:00 0:05 63,510 3:30

160 sub-meshes

46.1 20:00 0:06 123,110 3:00

197 sub-meshes

52.2 17:00 0:13 204,268

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


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Model Generated

OriginalAfter warping

Average error = 0.8 mmAverage error = 1.15 mm


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More Results

OriginalAfter warping

Avg. err. =0.4 mm

Avg. err. =0.8 mm

Blurry lines

Double 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 President’s Fund

Also thanks to…• Digital Michelangelo team• Forma Urbis team

A Hierarchical Method for Aligning Warped Meshes Leslie Ikemoto 1, Natasha Gelfand 2, Marc Levoy 2 1 UC Berkeley, formerly Stanford 2 Stanford University.

Documents

hierarchical method

warp slide

repeat slide

alignment global registration

entire scanner slide

stanford university

hinge stiffness slide

alignment methods