1. Registration of Point Cloud Data from a Geometric Optimization Perspective
Niloy J. Mitra1, Natasha Gelfand1, Helmut Pottmann2, Leonidas J. Guibas1
1 Stanford University
2 Vienna University of Technology

2. Registration Problem Given Q P model data Assume Q is a part of P.
Two point cloud data sets P (model) and Q (data) sampled from surfaces P and Q respectively. Q P
model
data
Assume Q is a part of P.
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3. Registration Problem Given Goal Q P data model
Two point cloud data sets P and Q.
Goal
Register Q against P by minimizing the squared distance between the underlying surfaces using only rigid transforms. Q P
data
model
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4. Registration Problem Given Goal Contributions
Two point cloud data sets P and Q.
Goal
Register Q against P by minimizing the squared distance between the underlying surfaces using only rigid transforms.
Contributions
use of second order accurate approximation to the squared distance field.
no explicit closest point information needed.
proposed algorithm has good convergence properties.
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5. Related Work Iterated Closest Point (ICP) Matching point clouds
point-point ICP [ Besl-McKay ]
point-plane ICP [ Chen-Medioni ]
Matching point clouds
based on flow complex [ Dey et al. ]
based on geodesic distance [ Sapiro and Memoli ]
MLS surface for PCD [ Levin ]
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6. Notations
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7. Notations
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8. Squared Distance Function (F)x
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9. Squared Distance Function (F)x
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10. Registration Problem Rigid transform that takes points
Our goal is to solve for,
An optimization problem in the squared distance field of P, the model PCD.
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11. Registration Problem Our goal is to solve for, Optimize for R and t.
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12. Overview of Our Approach
Construct approximate such that, to second order.
Linearize a.
Solve to get a linear system.
Apply a to data PCD (Q) and iterate.
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13. Registration in 2D
Quadratic Approximant
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14. Registration in 2D Quadratic Approximant Linearize rigid transform
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15. Registration in 2D Quadratic Approximant Linearize rigid transform
Residual error
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16. Depends on F+ and data PCD (Q).
Registration in 2D
Minimize residual error
Depends on F+ and data PCD (Q).
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17. Registration in 2D Minimize residual error Solve for R and t.
Apply a fraction of the computed motion
F+ valid locally
Step size determined by Armijo condition
Fractional transforms [Alexa et al. 2002]
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18. Minimize to get a linear system
Registration in 3D
Quadratic Approximant
Linearize rigid transform
Residual error
Minimize to get a linear system
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19. Approximate Squared Distance
valid in the neighborhood of x
Two methods for estimating F
d2Tree based computation
On-demand computation
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20. F(x, P) using d2Tree [ Leopoldseder et al. 2003]
A kd-tree like data structure for storing approximants of the squared distance function.
Each cell (c) stores a quadratic approximant as a matrix Qc.
Efficient to query.
[ Leopoldseder et al. 2003]
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21. F(x, P) using d2Tree
A kd-tree like data structure for storing approximants of the squared distance function.
Each cell (c) stores a quadratic approximant as a matrix Qc.
Efficient to query.
Simple bottom-up construction
Pre-computed for a given PCD.
Closest point information implicitly embedded in the squared distance function.
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22. Example d2trees 2D 3D
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23. Approximate Squared DistanceY
For a curve Y,
[ Pottmann and Hofer 2003 ]
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24. Approximate Squared Distance
For a curve Y,
For a surface F,
[ Pottmann and Hofer 2003 ]
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25. On-demand Computation
Given a PCD, at each point p we pre-compute,
a local frame
normal
principal direction of curvatures
radii of principal curvature (r1 and r2)
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26. On-demand Computation
Given a PCD, at each point p we pre-compute,
a local frame
normal
principal direction of curvatures
radii of principal curvature (r1 and r2)
Estimated from a PCD using local analysis
covariance analysis for local frame
quadric fitting for principal curvatures
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27. On-demand Computation
Given a point x,
nearest neighbor (p) computed using approximate nearest neighbor (ANN) data structure
where j =
d/(d-j) if d < 0
otherwise.
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28. Iterated Closest Point (ICP)
Find correspondence between P and Q.
closest point (point-to-point).
tangent plane of closest point (point-to-plane).
Solve for the best rigid transform given the correspondence.
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29. ICP in Our Framework Point-to-point ICP (good for large d)
Point-to-plane ICP (good for small d)
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30. Convergence Properties
Gradient decent over the error landscape Gauss -Newton Iteration
Zero residue problem (model and data PCD-s match) Quadratic Convergence
For fractional steps, Armijo condition used Damped Gauss-Newton Iteration Linear convergence
can be improved by quadratic motion approximation (not currently used)
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31. Convergence Funnel
Set of all initial poses of the data PCD with respect to the model PCD that is successfully aligned using the algorithm.
Desirable properties
broad
stable
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32. Convergence Funnel Translation in x-z plane. Rotation about y-axis.
Converges
Does not converge
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33. Convergence Funnel Plane-to-plane ICP Our algorithm
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34. Convergence Rate I
Bad Initial Alignment
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35. Good Initial Alignment
Convergence Rate II
Good Initial Alignment
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36. Partial Alignment
Starting Position
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37. Partial Alignment
After 6 iterations
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38. Partial Alignment Different sampling density After 6 iterations
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39. Future Work Partial matching Global registration Non-rigid transforms
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40. Questions?
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