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

2. October 9, 2003A Hierarchical Method for Aligning Warped Meshes 2 Scan Alignment Pipeline Global relaxation Scan merging Pairwise alignment

3. October 9, 2003A Hierarchical Method for Aligning Warped Meshes 3 Alignment Methods Pairwise alignment: Iterated Closest Point (ICP) [Variant from Chen-Medioni ‘91] pipi qiqi nini Global relaxation: Global registration Point constraints from ICP Rigid scans 1) Compute R, t minimizing distances from p i to tangent plane at q i 2) Apply transform and repeat

4. October 9, 2003A Hierarchical Method for Aligning Warped Meshes 4 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

5. October 9, 2003A Hierarchical Method for Aligning Warped Meshes 5 Registration Errors ICPGlobal Registration Correct calibration 0.13 mm avg. err. Incorrect calibration 1.81 mm avg. err. 4 mm misalignment Correct calibration Incorrect calibration

6. October 9, 2003A Hierarchical Method for Aligning Warped Meshes 6 Model Generated Spacing of range samples = 0.5 mm

7. October 9, 2003A Hierarchical Method for Aligning Warped Meshes 7 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

8. October 9, 2003A Hierarchical Method for Aligning Warped Meshes 8 Compensating Warp Smooth warpApproximate with a piecewise rigid model of overlapping sub-meshes Create pieces hierarchically R, TR 1..8, T 1..8 13 original scans84 sub-meshes

9. October 9, 2003A Hierarchical Method for Aligning Warped Meshes 9 Proposed Pipeline Pairwise alignment Global registration Find most misaligned pair of scans Global registration Dice into pieces Pairwise alignment Scan merging Loop until error below threshold Initial guess

10. October 9, 2003A Hierarchical Method for Aligning Warped Meshes 10 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

11. October 9, 2003A Hierarchical Method for Aligning Warped Meshes 11 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 error Mesh dicing scheme Overlap size affects “hinge stiffness”

12. October 9, 2003A Hierarchical Method for Aligning Warped Meshes 12 Determining Whether to Dice Using Stability Analysis Sufficiently constraining features: stability analysis to determine degenerate geometries [Gelfand et al. 3DIM03] 2 translations, 1 rotation3 rotations1 rotation, 1 translation 1 rotation 1 translation

13. October 9, 2003A Hierarchical Method for Aligning Warped Meshes 13 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

14. October 9, 2003A Hierarchical Method for Aligning Warped Meshes 14 Running Times Meshes Polygons (million) Pair Matching Global Reg. # Points Selected Dicing 4 meshes313:090:021200 37 scanner sweeps 3110:330:0225,8002:15 75 sub- meshes 35.834:000:0426,2002:30 117 sub- meshes 40.221:000:0563,5103:30 160 sub- meshes 46.120:000:06123,1103:00 197 sub- meshes 52.217:000:13204,268 Forma Urbis fragment, approximately 85 cm x 120 cm Hardware: Intel P4, 2.80 GHz

15. October 9, 2003A Hierarchical Method for Aligning Warped Meshes 15 Model Generated OriginalAfter warping Average error = 0.8 mmAverage error = 1.15 mm

16. October 9, 2003A Hierarchical Method for Aligning Warped Meshes 16 More Results OriginalAfter warping Avg. err. = 0.4 mm Avg. err. = 0.8 mm Blurry lines Double lines

17. October 9, 2003A Hierarchical Method for Aligning Warped Meshes 17 Conclusions Alignment method for smoothly warped meshes –Introduce minimal compensating warp –Does not require a specific characterization of scanner warp –Relatively simple to implement

18. October 9, 2003A Hierarchical Method for Aligning Warped Meshes 18 Limitations Will not converge if: scans very noisy scans do not have many features warp is not smooth

19. October 9, 2003A Hierarchical Method for Aligning Warped Meshes 19 Future Work Fit smooth spline –yields non-rigid warp –retrospective scanner calibration One-to-many stability analysis Improve measurement strategy

20. October 9, 2003A Hierarchical Method for Aligning Warped Meshes 20 Acknowledgements Our sponsors… National Science Foundation Research Grant IIS-0113427 Stanford University President’s Fund Also thanks to… Digital Michelangelo team Forma Urbis team