Outlier Robust ICP Minimizing Fractional RMSD
Jeff M. Phillips, Ran Liu, Carlo Tomasi | Department of Computer Science, Duke University
Distance! Functions
RMSD(D, M, T, µ)
=
Motivation: Registration with Outliers
p∈D
Outlier detection depends
on registration
Problem 1 [minimize rmsd]:
Align data point set D to model point set M
under a set of transformations T to minimize
min
T ∈T
µ:D→M
We register point sets and
find outliers in one algorithm
RMSD (D, M, T, µ)
=
!
"
1 " 1 $
#
||T (p) − µ(p)||2
f λ |Df |
4. Compute fraction: fi ∈ [0, 1]
λ = 1.3 for R2
λ = .95 for R3
λ
λ
1.6
f
FICP
60.1
78.8
0.6668
0.6668
1.0
time (s)
# iter.
RMSD
FRMSD
f
Stanford dragon:
16.5
17.3
0.0052
0.0124
0.750
scans at 0° and 48°
and 4. At each step FRMSD(D, M, f, T, µ) cannot increase.
d=2
1.4
1.3
1.2
1
.95
0
d=3
0.1
0.2
0.3
0.4
model
noise
0.5
0.6
0.7
0.8
0.9
ω/ωmax
● FRMSD is robust for
λ ∈ [1, 5]
● FICP has larger radius
of convergence with λ = 3
data
Theorem: Fractional ICP aligning D and M always
converges to a local minimum of FRMSD in the space of all
transformations T, matchings {D → M }, and fractions of
inliers [0, 1].
Proof Sketch: The state (f, T, µ) only changes at steps 2, 3,
2.2
2
RMSD
FRMSD
until ( µi = µi−1 and fi = fi−1 )
● minimum near true fraction of outliers
1.8
ICP
time (s)
# iter.
to minimize FRMSD(D, M, fi , Ti , µi )
T ∈T
µ:D→M
f ∈ [0, 1]
time (s)
0.142
0.069
0.059
0.061
0.062
0.063
model
Stanford bunny:
25% deformation
5° rotation
to minimize RMSD(D, M, Ti , µi )
Problem 2 [minimize frmsd]:
Align data point set D to model point set M under
a set of transformations T and fractions to
minimize
min
FRMSD (D, M, f, T, µ)
λ
1
1.3
2
3
4
5
Experiments
3. Compute matching: µi ∈ D → M
Let Df be f |D| points p ∈ D with smallest residuals
||p − µ(p)||.
0.8
● measurement error
● spurious/unrelated
data
to minimize RMSD(Df , M, Ti , µi−1 )
p∈Df
so that aligned points are more
likely inliers than outliers
New Data
● partial matches
● changes over time
● scans from different view ● comparison of similar
objects
● data set has grown
repeat
1. Compute closest f |D| points: Df
2. Compute transformation: Ti ∈ T
FRMSD(D, M, f, T, µ)
noise of model & fraction of inliers (weakly)
Deformation
Fractional ICP
● T = rotations, translations, scale, ...
● µ = matchings from D to M
● hard to optimize over both T and µ
● susceptible to outliers
Optimal value of λ depends on
Occlusion
Registration is often skewed
by outliers
1 "
||T (p) − µ(p)||2
|D|
# iter.
10.38
3.81
3.06
3.17
3.21
3.30
RMSD
FRMSD
0.158
0.170
0.170
0.170
0.171
0.172
0.225
0.248
0.303
0.404
0.538
0.717
f
0.701
0.749
0.750
0.750
0.751
0.751
1
● Fixing a distance threshold for µ may not converge.
● Fixing a fraction f (TrICP) does not find a local minimum.
Funnel of convergence is larger
than TrICP, but smaller than ICP.
ICP has smaller parameter space.
Algo.
ICP
TrICP
FICP
FICP
λ
3
3
1.3
5◦
0.999
0.875
0.952
0.857
● TrICP heuristically searches for f outside of loop.
rotation◦
10◦
0.997
0.870
0.945
0.473
25
0.994
0.853
0.909
0.141
50◦
0.962
0.816
0.875
0.060
ICP
FICP