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Evaluation of the Robustness of Planar-Patches based 3D-Registration using
Marker-based Ground-Truth in an Outdoor Urban Scenario
Kaustubh Pathak, Dorit Borrmann, Jan Elseberg, Narunas Vaskevicius, Andreas Birk, and Andreas N¨
uchter
Abstract—The recently introduced Minimum Uncertainty Maximum
Consensus (MUMC) algorithm for 3D scene registration using planar-
patches is tested in alarge outdoor urban setting without any prior
motion estimate whatsoever.With the aid of anew overlap metric based
on unmatched patches, the algorithm is shown to work successfully in
most cases. The absolute accuracy of its computed result is corroborated
for the first time by ground-truth obtained using reflectivemarkers.
Therewereacouple of unsuccessful scan-pairs. These areanalyzed for
the reason of failureby formulating twokinds of overlap metrics: one
based on the actual overlapping surface-area and another based on the
extent of agreement of range-image pixels. Weconclude that neither
metric in isolation is able to predict all failures,but that both taken
together areable to predict the difficulty level of ascan-pair vis-`
a-vis
registration by MUMC.
I.INTROD UCTIO N
A3D registration algorithm [1] based on matching large planar-
patches extracted from “point-clouds” sampled from 3D sensors
was recently introduced by some of the authors. The algorithm
is termed Minimum Uncertainty Maximum Consensus (MUMC)
and as the name suggests, it tries to find aset of correspondences
between planar-patches from the twoscans being matched which
minimizes the uncertainty-volume of the registration result as
measured by the determinant of the 6×6covariance matrix of
the computed pose. Aclosed-form least-squares solution of the
registration was also presented along with explicit expressions for
its uncertainty.
The MUMC algorithm was tested on avariety of sensors and
compared to point based methods likeIterativeClosest Point (ICP),
in both its point-to-point [2] and point-to-plane [3] incarnations, and
3D Normal Distribution Transform (3D NDT). It was found that
the algorithm had abigger convergence radius than its competitors
because it does aglobal search and does not depend on alocal
attraction to the nearest locally optimum solution. The goodness-
of-fit was measured by the quality of alignment of scans, which is
arelativemeasure. The algorithm was also used— embedded in a
pose-graph for Simultaneous Localization and Mapping (SLAM)—
to generate 3D maps of disaster scenarios [4]. In that work, clearly
visible ground-truth structures were used for aqualitative evalua-
tion. The main reason for the use of these somewhat subjective
criteria for evaluation was that ground-truth in 3D is hard to
come by.In the field of mobile robot navigation, ground-truth
comparison either requires asimulation-study or precise GPS [5].
In [6] surveying data of buildings, stored in vector format, and
available from government land registration offices was used as the
source of ground-truth. Satellite images can also be used to obtain
arough ground-truth.
In this paper,we present acomparison of the MUMC algorithm
results to the ground-truth for the first time. The ground-truth
This work was supported by the Deutsche Forschungsgemeinschaft (Ger-
man Research Foundation).
The authors are with the Dept. of EECS, Jacobs University
Bremen, 28751 Bremen, Germany.kaust@ieee.org,
a.birk@jacobs-university.de
Oℓ
Or
ℓpc
rpc
ℓCpp
rCpp
Fℓ
Fr
ℓ
rR,ℓ
rt
ℓPi↔rPj
Fig. 1. The sensor in twocoordinate-frames observes the same physical
plane. Only apatch of the plane is visible from anypose; the polygonized
patches are shown in color,with the sampled points shown as dots and
crosses from the left and the right frame respectively.
was obtained based on the commercial reflectivemarker-based
solution available on the high-end RIEGL VZ-400 3D laser scanner.
The dataset consists of afairly big outdoor urban scenario, viz.
the old city-center of Bremen, Germany.The data was originally
obtained as point-clouds of about 22.5 million points per scan,
which were subsequently sub-sampled to range-images of half a
million points per scan, from which planar-patches were extracted
using aregion-growing method described in [7]. This method also
involves computation of the uncertainties of the plane-parameters
using asensor range error-model [8].
A. Notation Overview
The nomenclature used is briefly reviewed in this section. An
infinite plane P(ˆ
m,ρ)is given by the equation ˆ
m·p=ρ,where
ρis the signed distance from the origin in the direction of the unit
plane normal ˆ
m.Wesee that P(ˆ
m,ρ)≡P(−ˆ
m,−ρ).Toachieve
aconsistent sign convention, we define planes as P(ˆ
n,d),where,
d,|ρ|≥0,and ˆ
n,σ(ρ)ˆ
m,where, σ(ρ)=−1if ρ<0and
+1 otherwise. If ρ=0,then we choose the maximum component
of ˆ
nto be positive.
For registration, we consider tworobot-frames as shown in Fig. 1:
aleft one denoted as Fℓwith origin Oℓfrom which the indexed
plane-set ℓPis observed, and aright one Frwith origin Orfrom
which the indexed plane-set rPis observed. The equations of the
planes are
ℓˆ
ni·ℓp=ℓdi,rˆ
ni·rp=rdi.(1)
An indexed set kPof planar-patches is extracted [7] from
apoint-cloud associated with the k-th robot-frame Fkby seg-
mentation of the range-image using region-growing followed by
polygonization. As shown in Fig. 1, apatch has aset of points
The 2010 IEEE/RSJ International Conference on
Intelligent Robots and Systems
October 18-22, 2010, Taipei, Taiwan
978-1-4244-6676-4/10/$25.00 ©2010 IEEE
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pjassociated with it: the weighted scatter matrix associated with
these points is denoted as Cpp and is depicted as the dotted
ellipsoid in the figure. The weighted centroid is denoted as pc;
the weights being taken to be inversely proportional to the trace of
the covariance of the individual points pj.The normal of the plane
turns out to be the eigenvector of Cpp corresponding to its least
eigenvalue. Apart from the planar patch’sˆ
nand dparameters, the
extraction procedure also gives [8] their 4×4covariance matrix
C.Thus, the plane kPis an ordered set of triplets given by
kP,{kPihkˆ
ni,kdi,kCii,i=1. . . Nk}.(2)
If the robot moves from Fℓto Fr,and observes the coordinates
of the same physical point as ℓpand rprespectively,these
coordinates are related by [9]
ℓp=ℓ
rRrp+ℓ
rt,(3)
where, the translation ℓ
rt,−−−→
OℓOr,resolved in Fℓ.
ℓPi↔rPjmeans that the ith patch in the scan taken at Fℓ
corresponds to the jth patch in Fr.If we assume that the planes
in the twoframes havebeen renumbered so that planes with the
same index physically correspond, then substituting (3) in (1) and
comparing coefficients gives
ℓˆ
ni=ℓ
rRrˆ
ni,ℓˆ
ni·ℓ
rt=ℓdi−rdi.(4a)
The registration problem nowconsists of estimating ℓ
rRand ℓ
rt
by solving the abovein aleast-squares sense. This procedure
also yields the covariance of the registration solution. Tofind the
actual correspondences between the planar-patches is the task of
the MUMC algorithm [1].
II.ROBUSTNESS INTHECASEOFACOMPL ET E ABSENCEOF
AN INITIALMOTIO N ESTIMATE
Algorithms likeICP which match scans by iterativeattraction
to alocal minimum of the registration-cost-function rely on a
good initial guess of inter-scan pose change— e.g. that provided
by vehicle odometry– for finding the right registration. Matching
twoscans in the absence of anyinitial guess is aformidable task,
especially if the inter-scan pose change is considerable compared to
the field of view(FOV) and the maximum range of the 3D range-
sensor involved. In case the sensor has alarge FOV,the main culprit
for matching failures is occlusion.
MUMC searches the global correspondence-space of large
planar-patches for the consensus which maximizes geometrical
consistencyand hence minimizes pose-uncertainty.The globality
of the search implies that MUMC does not necessarily need to
havean initial guess for the pose difference. However,if such a
guess is available along with its uncertainty,it can still be utilized
by MUMC to do χ2-tests and prune the global search space—
hence speeding-up its execution. This paper focuses on making the
performance of MUMC more robust in the case of large, totally
unknown inter-scan movements, assuming arelatively big FOVof
the sensor.In other words, we focus on improving and evaluating
the robustness of MUMC primarily w.r.t. occlusion.
A. Improving Robustness: Unmatched Planes Overlap Metric µu
The central dilemma facing ascan-matching algorithm is to
weigh the size of scan-overlap againstthe quality of overlap. If
only the size is given priority,poorly overlapping scans cannot
be matched or are wrongly matched. In case only the quality of
match is considered, avery small area may be accurately matched
without there being aglobal agreement. MUMC proposes to solve
this issue by employing the uncertainty-volume (determinant of the
covariance matrix) of the computed pose-registration as ametric
to minimize: if consistent patch-correspondences are added to the
matched set, this metric reduces; this avoids the temptation to
greedily collect all patch-correspondences which merely satisfy
some threshold.
In the original formulation of MUMC [1], the agreement of
unmatched planes was not explicitly considered. In case of complete
absence of initial guesses and the presence of occlusion, robustness
can be improved by also evaluating planes in the twoscans for
which no correspondences were found. Weintroduce here ametric
µuto compute the extent of overlap of unmatched planes. It is
computed after [1, step 11 of Algorithm 2], when aset of potential
plane correspondences Γhas been found, and the least-squares
registration ℓ
rR,ℓ
rtit implies, is computed. Nowthe set Γneeds
to be evaluated w.r.t. the unmatched planes.
The basic idea is that the overlap can be measured as aχ2
distance in terms of the weightedscatter matrix and the weighted
centroid of the patches as depicted in Fig. 1. Assume that we want
to evaluate whether the previously unmatched planes ℓPiand rPj
overlap. The matched set Γand its associated registration ℓ
rR,ℓ
rt
are considered fixed and certain.
ℓqj,ℓ
rRrpc,j +ℓ
rt,(5)
Σ,ℓCpp,i +ℓ
rRrCpp,j
ℓ
rRT,(6)
χ2
v=(ℓqj−ℓpc,i)TΣ−1(ℓqj−ℓpc,i ).(7)
For each unmatched plane rPj,we find the ℓPihaving the
minimum χ2
vto it, such that, additionally,rPjand ℓPiare also
translationally consistent [1, Sec. III-A2] with ℓ
rtand rotationally
consistent, i.e. ℓˆ
ni·(ℓ
rRrˆ
nj)≈1.Tobe considered feasible, this
minimum χ2
vshould also be less than χ2
3,t%,which is the χ2value
for 3d.o.f. at the significance level of t%.Weselected t=1%.If a
specific indexiin the ℓ-set is found to pair with more than one index
in the r-set, the pairing with the lesser value of the minimum χ2
vis
taken and the other is rejected. Wenowhavean additional set of
surmised correspondences denoted Γsfrom among the unmatched
set of planes in addition to the previously fixed set Γof matched
planes. Hence,
µu=#Γs+#Γ
Nr
,Γ←Γ∪Γs.(8)
where, #denotes the set’ssize and Nris the number of r-patches
being matched. In [1, step 12 of Algorithm 2], we nowconsider
the appended set Γanyfurther only if µu≥¯µu,i.e. the overlap is
at least as big as agiven threshold.
III.EVALUATION OFSCA NS F ORMATCHABILITY G IVENTHE
GROU ND -TRUTHREGISTRATION
A. The Spatial Surface-Area Overlap Metric µs
The spatial overlapping surface area mutually visible from the
twoscans to be matched is one of the factors affecting matching
success of MUMC. Wepropose to evaluate it using an Octree to
discretize the space and represent the point-clouds of the scans. It
uses the original point-cloud directly and not the extracted planes.
First, the twoscans are aligned using the given ground-truth. The
volume covered by the points is recursively split into cubic octets
until the size of the voxels fallsbelowathreshold ∆t.Each voxel
that does not contain anypoints is deleted. From each remaining
voxel, the center point is taken for the evaluation. Avoxel in one
scan is considered to correspond to the closest voxel in the other
scan if their mutual distanceis within athreshold ¯
ds.The number
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0 1 2 3 4 5 6 7 8 9 10 11 12
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Scan number
Metric
µe
µp
µn
Fig. 7. Fractions of various kinds of points in the range-image. µeis the
fraction of empty or maximum-range beams, µpis the fraction of points
lying on planar-patches, and µnis the fraction of the remaining points.
0−1 1−2 2−3 3−4 4−5 5−6 6−7 7−8 8−9 9−10 10−1111−12
0
0.5
1
1.5
2
2.5
3
3.5 x 106
Scan 1
Spatial Overlap
Scan 2
Fig. 8. Showing the absolute pairwise overlap µsof scans. The overlap is
computed by finding corresponding voxel in scan 1for all voxels in scan
2. Certain points in scan 2may havethe same corresponding point in scan
1, as is the case in pair 9-10. The chart clearly shows that scan-pair 5−6
is the worst according to this metric.
REFERENCES
[1] K. Pathak, A. Birk, N. Vaskevicius, and J. Poppinga, “Fast Registra-
tion Based on Noisy Planes with Unknown Correspondences for 3D
Mapping,”IEEE Transactions on Robotics,vol. (in press), 2010.
[2] P.J. Besl and N. D. McKay,“Amethod for registration of 3-d shapes,”
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no. 2, pp. 239–256, Feb 1992.
[3] Y.Chen and G. Medioni, “Object Modeling by Registration of Multiple
Range Images,”Imageand Vision Computing,vol. 10, no. 3, pp. 145–
155, 1992.
[4] K. Pathak, A. Birk, N. Vaskevicius, M. Pfingsthorn, S. Schwertfeger,
and J. Poppinga, “Online 3D SLAM by Registration of Large Planar
Surface Segments and Closed Form Pose-Graph Relaxation,”Journal
of Field Robotics, Special Issue on 3D Mapping,vol. 27, no. 1, pp.
52–84, 2010.
[5] S. Schuhmacher and J. B¨
ohm, “Georeferencing of terrestrial laser
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XXXVI/5/W17, 2005. [Online]. Available: http://www.ifp.uni-stuttgart.
de/publications/2005/schuhmacher05 venedig.pdf
0−1 1−2 2−3 3−4 4−5 5−6 6−7 7−8 8−9 9−10 10−1111−12
−12
−10
−8
−6
−4
−2
0
Log RangeImage Overlap Metric
Fig. 9. Showing the range-image pairwise overlap µrof scans in log-scale.
The chart clearly shows that scan-pair 0−1is the worst according to this
metric.
[6] O. Wulf, A. N¨
uchter,J. Hertzberg, and B. Wagner,“Benchmarking
urban six-degree-of-freedom simultaneous localization and mapping,”
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tion and Polygonalization in noisy 3D Range Images,”in International
Conference on Intelligent Robots and Systems (IROS).Nice, France:
IEEE Press, 2008, pp. 3378 –3383.
[8] K. Pathak, N. Vaskevicius, and A. Birk, “Uncertainty analysis for
optimum plane extraction from noisy 3D range-sensor point-clouds.”
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[9] J. J. Craig, Introduction to robotics –Mechanics and control.Prentice
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