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Registration of Point Cloud Data from a Geometric Optimization Perspective. Niloy J. Mitra 1 , Natasha Gelfand 1 , Helmut Pottmann 2 , Leonidas J. Guibas 1. 1 Stanford University. 2 Vienna University of Technology. Q. P. data. model. Registration Problem. Given. - PowerPoint PPT Presentation
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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
Registration of PCD
Registration Problem
GivenTwo point cloud data sets P (model) and Q (data) sampled from surfaces P and Q respectively.
Q P
modeldata
Assume Q is a part of P.
Registration of PCD
Registration Problem
GivenTwo point cloud data sets P and Q.
GoalRegister Q against P by minimizing the squared distance between the underlying surfaces using only rigid transforms.
Q P
data model
Registration of PCD
Registration Problem
• use of second order accurate approximation to the squared distance field.
• no explicit closest point information needed.
• proposed algorithm has good convergence properties.
GivenTwo point cloud data sets P and Q.
GoalRegister Q against P by minimizing the squared distance between the underlying surfaces using only rigid transforms.
Contributions
Registration of PCD
Related Work
Iterated Closest Point (ICP) 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 ]
Registration of PCD
Notations
}{ ipP
Registration of PCD
Notations
P
Registration of PCD
Squared Distance Function (F)
xP
Registration of PCD
Squared Distance Function (F)
x
2d),( PxF
dP
Registration of PCD
Registration Problem
Pi
i
qF )),((
min
An optimization problem in the squared distance field of P, the model PCD.
. )( ii qq Rigid transform that takes points
Our goal is to solve for,
Registration of PCD
Registration Problem
PitR
i
tRqF ),(,
min
Optimize for R and t.
)()( tR ntranslatio rotation
Our goal is to solve for,
Registration of PCD
Overview of Our Approach
Construct approximate such that, to second order.
Linearize .
Solve
to get a linear system.
Apply to data PCD (Q) and iterate.
),(),( PP xFxF ),( PxF
PitR
i
tRqF ),(,
min
Registration of PCD
Registration in 2D
F Ey Dx CyBxy Ax )( 22 xF
TxQ ][][ 1yx 1yx
• Quadratic Approximant
Registration of PCD
Registration in 2D
)sin(
1)cos(
xt yxx
yt y xy
TxQxF ][][ 1yx 1yx )( • Quadratic Approximant
• Linearize rigid transform
),( tR
of function ] tt [ yx
Registration of PCD
Registration in 2D
xt yxx
yt y xy
• Quadratic Approximant
• Linearize rigid transform
),,(),( yx tt F
QqPi
i
tRq
• Residual error
TxQxF ][][ 1yx 1yx )(
Registration of PCD
Registration in 2D
),,( yx tt• Minimize residual error
21 MM
t
t
y
x
Depends on F+ and data PCD (Q).
Registration of PCD
Registration in 2D
),,( yx tt• 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]
Registration of PCD
Registration in 3D
• Quadratic Approximant
• Linearize rigid transform
),,,,,( zyx ttt
• Residual error
T]1zyx[]1zyx[ )( xQxF
Minimize to get a linear system
Registration of PCD
Approximate Squared Distance
Two methods for estimating F
1. d2Tree based computation
2. On-demand computation
)F(x, P valid in the neighborhood of x
Registration of PCD
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.
[ Leopoldseder et al. 2003]
Registration of PCD
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.
Registration of PCD
Example d2trees
2D 3D
Registration of PCD
Approximate Squared Distance
22
21
22
21 xxxx
ρ)( 1
1
-
x, Fd
d
[ Pottmann and Hofer 2003 ]
For a curve
Registration of PCD
Approximate Squared Distance
22
21
22
21 xxxx
ρ)( 1
1
-
x, Fd
d
[ Pottmann and Hofer 2003 ]
For a curve
23
22
21
23
22
21 xxxxx
ρx
ρ)( 21
21
--
x, Fd
d
d
d
For a surface
Registration of PCD
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 (and
)2e and e( 1
)n(
Registration of PCD
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 (and
)2e and e( 1
)n(
Estimated from a PCD using local analysis
• covariance analysis for local frame
• quadric fitting for principal curvatures
Registration of PCD
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
0 otherwise.
222
21 ))(())(())((),( pxpxpxxF nee 21P
Registration of PCD
Iterated Closest Point (ICP)
Iterate
1. Find correspondence between P and Q.
• closest point (point-to-point).
• tangent plane of closest point (point-to-plane).
2. Solve for the best rigid transform given the correspondence.
Registration of PCD
ICP in Our Framework
0))((),( 2 j nF pxx P
1)(),( 2 j F pxx P
• Point-to-plane ICP (good for small d)
• Point-to-point ICP (good for large d)
Registration of PCD
Convergence Properties
Gradient decent over the error landscapeGauss -Newton Iteration
Zero residue problem (model and data PCD-s match)Quadratic Convergence
For fractional steps, Armijo condition usedDamped Gauss-Newton IterationLinear convergence
can be improved by quadratic motion approximation (not currently used)
Registration of PCD
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
Registration of PCD
Convergence Funnel
Translation in x-z plane. Rotation about y-axis.
Converges
Does not converge
Registration of PCD
Convergence Funnel
Our algorithmPlane-to-plane ICP
Registration of PCD
Convergence Rate I
Bad Initial Alignment
Registration of PCD
Convergence Rate II
Good Initial Alignment
Registration of PCD
Partial Alignment
Starting Position
Registration of PCD
Partial Alignment
After 6 iterations
Registration of PCD
Partial Alignment
After 6 iterationsDifferent sampling density
Registration of PCD
Future Work
Partial matching
Global registration
Non-rigid transforms
Registration of PCD
Questions?
