www.lr.tudelft.nl

2010 International Conference on Indoor Positioning and Indoor Navigation (IPIN), 15-17 September 2010, Zürich, Switzerland
Automated Localization of a Laser Scanner in
Indoor Environments Using Planar Objects
Kourosh Khoshelham
Optical and Laser Remote Sensing, Dept. of Remote Sensing, Delft University of Technology, Delft, The Netherlands.
Email: [email protected]
Abstract— A method is presented for automated localization
of a laser scanner in indoor environments by matching
planar features extracted from range data. Plane
correspondences are used in a linear least-squares
adjustment model to estimate the relative scanner positions
in consecutive scans. The performance of the method is
demonstrated using two datasets of building interiors.
Accuracy assessment of the computed positions shows
localization errors of a few centimeters.
Keywords—Navigation; plane matching; search; registration.
I.
INTRODUCTION
A new trend in terrestrial laser scanning is the
development of an autonomous system that is able to scan
an indoor environment from a number of predefined
positions, register the scans, and provide an accurate and
complete point cloud of the scene. Such a system would
require the automation of two main processes: scanner
localization and the registration of the scans. In theory,
automated registration precedes the localization problem,
because a correct registration of consecutive scans
provides the relative position and orientation of the
scanner in every pair of scans. In practice, however,
existing registration methods are slow and iterative, and
require an approximate estimate of the transformation
between the two scans. Alternatively, if the motion of the
scanner from one scan position to another can be
estimated from the data, it can serve as an initial
approximation to perform an iterative fine registration of
the scans, which will in turn lead to an improved accuracy
for the localization of the scanner.
The registration of laser scanner data has been a topic
of intensive research in the past. Salvi et al., [1] provide a
review of the more recent methods. Perhaps the most
common approach is the so-called iterative closest point
(ICP) algorithm [2]. The ICP algorithm uses an iterative
estimation model to find a transformation that minimizes
the sum of the squared distances between a set of
corresponding points in the two scans. To establish
correspondence between the points the ICP algorithm
requires a good approximation of the transformation
between the two scans. The concept of point-to-point
correspondences was employed in a number of other
methods [3-6], which all rely on the availability of an
initial approximation. Another type of correspondence
was proposed by Chen and Medioni [7], who developed
an estimation model based on minimizing point-to-plane
distances. The point-to-plane correspondences lead to a
more robust estimation model, which converges in fewer
978-1-4244-5864-6/10/$26.00 ©2010 IEEE
iterations [1]. However, this method also requires an
initial approximation of the transformation.
The problem of initial approximation can be dealt with
in different ways. A straightforward solution is to directly
measure the scanner’s position and orientation using GPS
and navigation sensors [8]. This solution is applicable
only in outdoor scenes where GPS signals can be
received. A more general approach would be to estimate
the motion between a pair of consecutive scans using the
properties of the scanned surface itself. Chung et al., [9]
used principal component analysis to find the relative
orientation between two overlapping scans. This coarseregistration approach is likely to fail when complete 360degree scans are to be registered. Brenner and Dold [10]
proposed the use of plane correspondences for coarse
registration, and investigated the combinatorial
complexity of the search for corresponding planes. In a
previous work, we proposed a method to estimate the
relative position of the scanner by intersecting
corresponding planes extracted from two overlapping
range images [11]. The relative positions make up the
trajectory of the scanner, but are insufficient for a full
registration of the scans.
This paper extends the plane-based localization method
to enable a complete pair-wise registration of overlapping
scans. This is done by inferring a transformation from a
set of plane correspondences within a linear least-squares
plane matching model. The benefit of a linear estimation
model is that it requires no initial approximations, and
leads to a more efficient search for correspondences. The
correspondence problem is approached with an initiateand-extend search strategy, which begins with initial
correspondence
hypotheses
and
extends
the
correspondences to all plane pairs that can contribute to an
accurate estimation of the transformation.
The paper proceeds with a description of the plane
intersection and the extended plane matching model in
Section 2. Section 3 presents the initiate-and-extend
strategy for finding the largest set of corresponding
planes. Experiments and results are presented in Section 4.
The paper concludes with some remarks in Section 5.
II.
LOCALIZATION AND REGISTRATION USING PLANE
CORRESPONDENCES
The basic principle of plane-based localization and
registration is that a minimum of three plane
correspondences sufficiently constrain the motion in the
3D space, subject to the condition that the planes have an
intersection point. In the following, first the method for
motion estimation by intersecting the planes is described.
Then the method is extended to a more general plane
2010 International Conference on Indoor Positioning and Indoor Navigation (IPIN), 15-17 September 2010, Zürich, Switzerland
matching model, which forms the basis for inferring a
complete transformation from two sets of planes.
A. The Plane Intersection Method
Suppose a minimum of three planes that intersect in a
point appear in two consecutive scans. The motion
between the two scans can be computed from the
differences in the measured ranges to the planes. Figure 1
illustrates this concept. Distances to three planes are
measured from the scanner positions S1 and S2. If planes
P1, P2, and P3 in the coordinate system of S1 are shifted
by -ρ12, -ρ22, and –ρ32 respectively in the direction of their
normals, they intersect at point S2. Let a plane be defined
with a normal vector n = (n1, n2, n3)T and a perpendicular
distance ρ to the origin of the coordinate system. The
condition that the plane contains a point x is described by
ρ = nTx. Now, the intersection of the three shifted planes
can be expressed as:
⎧ ρ 1 − ρ 2 = n 1T s 1
1
1
2
⎪ 1
⎪ 1
2
1T 1
−
=
ρ
ρ
n
s
⎨ 2
2
2
2
⎪ 1
2
1T 1
ρ
ρ
−
=
n
s
⎪⎩ 3
3
3
2
where x = (x1, x2, x3, 1)T is the homogenous representation
of a point. The transformation of a set of m points from
the coordinate system of a second scan to that of a first
scan is expressed as:
2
x14 xm = H 4x 4 x 4x
m
where superscripts denote the coordinate system, and
subscripts show the size of the matrices. In principle, H
can be a similarity transformation:
⎡ sR
H = ⎢ T3x 3
⎣ 01x 3
t 3x1 ⎤
1 ⎥⎦
π1j = H − T π 2j
π1 x1 = ( H − T π 2 ) Hx 2 = π 2 H −1Hx 2 = π 2 x 2 = 0
T
T
T
B. Extension to Plane Matching
To estimate a complete transformation from a set of
corresponding planes all the plane parameters should be
used in the adjustment. Let π = (nT, –ρ)T denote a plane,
and the point-on-plane condition be expressed as:
πTx = 0
(2)
T
(6)
Equation (5) can be written as:
T
This plane intersection solution does not provide any
information about the relative rotation between the two
scans. The rotation parameters can be derived from the
directions of the plane normals, which are absent in the
system of equations (1).
(5)
because the points-on-plane condition is invariant to the
transformation, i.e. :
π1j H = π 2j
Figure 1. Scanner localization by plane intersection.
(4)
consisting of a scale s (normally equal to 1), a 3D rotation
R and a translation vector t. The corresponding
transformation for transforming the plane j that contains
the points from the second scan to the first is:
(1)
where n and ρ are plane parameters, subscripts denote
plane numbers, and superscripts denote the coordinate
system in which the plane parameters are defined. Solving
the system of equations given in (1) for s21 gives the
location of s2 in the coordinate system of s1, which is the
motion between the two scans.
(3)
T
(7)
Equation (7) is the basic observation equation in the
plane matching estimation model. Given a set of
corresponding planes a similarity transformation can be
T
T
estimated such that the norm || π2 – π1 H || is minimized.
To obtain an estimation of the transformation without
iteration we may ignore the special orthogonality of R and
assume that H is an affine transformation. In effect, we
over-parameterize the transformation to make the
estimation model linear. The result is the familiar direct
linear transform (DLT), which is commonly used in
photogrammetry and computer vision [12, 13]. The
application of this model has been shown also for the
registration of aerial laser strips [14].
To solve for the transformation, Equation (7) is
rearranged so as to have the entries of H stacked in a
vector:
⎡⎣ n11 j I 4
n12 j I 4
⎡ n12j ⎤
⎡ h1 ⎤
⎢
⎥
2
⎢h ⎥ = ⎢ n2 j ⎥
n31 j I 4 ⎤⎦
2
⎥
4 x12 ⎢
⎢ n2
⎥
⎢⎣ h 3 ⎥⎦12 x1 ⎢ 2 3 j 1 ⎥
⎢⎣ − d j + d j ⎥⎦ 4 x1
(8)
where h1T, h2T, and h3T are the first three rows of the
transformation matrix H. For k planes we will have a
system of equations of the form given in (8), for which the
least-squares solution is:
2010 International Conference on Indoor Positioning and Indoor Navigation (IPIN), 15-17 September 2010, Zürich, Switzerland
A 4k x12 ⋅ X12 x1 = B4k x1
→
T
−1
T
X = ( A WA) A WB
(9)
where W is a weight matrix to assign appropriate weights
to the two observation types in B.
It can be seen that the plane matching equation given in
(7) is a general form that contains the plane intersection
model as a special case in which only distances are used
as observables. Let us decompose the plane parameters in
(7) such that:
⎡n1T
⎣⎢ j
⎡ R t ⎤ ⎡ 2T
− ρ 1j ⎤⎥ ⎢
= n
⎦ ⎣01x 3 1⎥⎦ ⎣⎢ j
− ρ 2j ⎤⎥
⎦
(10)
where R is the rotation and t = s21 is the translation from
the second scan to the first scan. We have:
⎡n1T R n1T s1 − ρ 1 ⎤ = ⎡n 2T
j 2
j⎥
⎢⎣ j
⎦ ⎢⎣ j
− ρ 2j ⎤⎥
⎦
(11)
which gives rise to two equations:
⎧ n1T R = n 2T
j
⎪⎪ j
⎨
⎪ 1
2
1T 1
⎪⎩ ρ j − ρ j = n j s 2
(12a)
by imposing tight constraints one can substantially reduce
the search space. Such tight constraints would then
increase the risk of eliminating a correct combination due
to small errors in plane parameters. In addition, the
constrained search does not result in a single combination
containing correct correspondences, but a set of
combinations. That is, it may end up with a set of correct
yet slightly different transformations.
To deal with the combinatorial complexity of finding
plane correspondences we use a variant of the hillclimbing algorithm [15], which we call the initiate-andextend search. It works in two steps. In the first step,
initial combinations of three (or four) plane pairs are
created using a small subset of the planes in each scan. At
this stage, loose constraints are imposed to further reduce
the number of initial combinations while maintaining the
correct correspondences. A transformation is estimated for
each remaining combination that now we call an initial
match set. In the second step, each initial match set is
extended with new planes in the two scans that fit to the
estimated transformation. Figure 2 illustrates the initiateand-extend search. A main aspect of this method is that a
large number of planes in an extended match set
contribute to the estimation of the transformation. In
addition, extending the initial match sets provides a
straightforward method for finding the correct
transformation by picking the largest extended match set
(the winning match set). The following sections describe
the initiate-and-extend search method in more details.
(12b)
As it can be seen, Equation (12b) is the plane intersection
equation as in (1). Note that in the intersection model only
the distance parameter of a plane is used as observable,
whereas in the general plane matching model all the plane
parameters take part. As a result, the solution to the
general model still requires a minimum of three
corresponding planes even though the number of
unknown variables is raised to 12.
III.
INITIATE-AND-EXTEND SEARCH FOR
CORRESPONDENCES
Using the plane matching equation a transformation can
be estimated between two sets of corresponding planes in
two consecutive scans. A major issue now is how to
establish correspondence between the planes. The
correspondence problem is usually approached with a
search algorithm. Since a minimum of three planes is
sufficient for estimating the transformation, a simple
search strategy could be to create all combinations of three
planes out of all planes in the two scans, and then estimate
a transformation between every pair of these plane triplets.
The correct transformation can then be identified using an
objective score function. Such an exhaustive search
strategy will be enormously expensive as the number of
planes increases. For example, with 50 planes per scan
more than one billion combinations can be created, which
should all undergo the transformation estimation and the
scoring process. To reduce the computational cost of the
search, Brenner and Dold [10] suggested a hierarchy of
constraints to progressively prune the search tree. A
drawback of such a constrained search strategy is that only
Figure 2.
The initiate-and-extend search for plane correspondences.
A.
Creating Initial Match Sets
In principle, the total number of combinations of k
plane pairs in two sets of m and n planes is expressed as:
nc = Ckm ⋅ Pkn =
m!
n!
×
k !(m − k )! (n − k )!
(13)
where C and P denote combination (unordered selection)
and permutation (ordered selection) respectively. It can be
readily seen that by starting with a small subset of the
planes (small m and n) the number of initial match sets
can be reduced dramatically. For example, with 10 planes
per scan the number of combinations of three plane pairs
will be in the order of 105. Imposing a loose relative
orientation constraint [16] further reduces the number of
initial combinations, typically by a factor of 10~102. The
computational cost of processing this number of initial
2010 International Conference on Indoor Positioning and Indoor Navigation (IPIN), 15-17 September 2010, Zürich, Switzerland
combinations is far more affordable than that of, for
instance, 109 combinations when starting with 50 planes.
An important issue in initiating with a small number of
planes is that the initial match sets should contain at least
one combination that includes correct correspondences,
and can, therefore, lead to a correct transformation.
B. Extending Initial Match Sets
If an initial match set contains correct correspondences,
then the estimated transformation can be used to find other
correct correspondences in the two scans. Extending the
plane matching estimation model with these new
correspondences will result in a more precise estimation
of the transformation. In contrast, if a transformation is
estimated from wrong correspondences in an initial match
set, the likelihood of other planes randomly fitting in the
transformation is very small. Therefore, the number of
planes correspondences in an extended match set is an
indication of the correctness of the transformation.
The extension of the initial match sets follows the
concept of the hill climbing algorithm. Each initial match
set is extended by taking a new plane from the first scan
and transforming it to the second scan (using the
transformation estimated by the initial match set). In the
second scan, a nearest neighbor operation is performed to
find the nearest plane (having the most similar
parameters) to the transformed plane. This plane and the
plane taken from the first scan are added to the match set,
and a new transformation is estimated. If the estimation
yields residuals smaller than a threshold, then the
extension is accepted; otherwise, the two new planes are
removed from the match set. The extension is tried with
all the planes in the two scans, which are not already in
the match set; the process terminates when no new planes
can be added to the match set.
Once the extension process is performed for all the
initial match sets, a winning match set is identified as one
that contains the largest number of plane correspondences.
The number of planes in the winning match set is an
indication of the reliability of the estimated
transformation. A winning match set with too few planes
might imply that none of the initial match sets contained a
correct set of correspondences. In such case, the matching
should be initiated with a larger number of planes.
IV. EXPERIMENTS
The performance of the plane matching method was
tested using two indoor datasets. The first dataset included
a set of seven scans acquired in a corridor of
approximately 2x5x3 meters dimension. The second
dataset was obtained in an interior of the Aula building of
TU Delft with a volume of about 8x25x3 meters, and
Figure 3.
included a set of six panoramic scans. Figure 3 shows
reflectance images pertaining to one scan in each dataset.
Both datasets were acquired by a FARO LS 880 laser
scanner. The position of the scanner at each scan was
manually measured to serve as reference in the evaluation
of the localization results. In the corridor dataset, 2D
measurements were made by a tape, while in the Aula
interior a total-station was used to make accurate
measurements of the scanner positions in 3D. Planar
segments were extracted from all scans using a range
image segmentation algorithm as described in [11].
The plane matching process was performed with the
planes extracted from the corridor scans in a pair-wise
fashion. From the estimated transformation the position of
the scanner at each scan was computed. Figure 4 depicts
the computed trajectory of the scanner plotted together
with the manual measurements of the scanner positions.
As can be seen, discrepancies are within a few
centimeters. Table I summarizes the results of the corridor
test. The values δx and δy are discrepancies between the
computed and measured coordinates of the position of the
scanner at each scan. The root-mean-squared distance
residuals and of normal residuals are derived from the
estimation model (of the winning match set) in each
registration, and indicate the precision of the registration.
The precision measures given in Table I as the RMS
residual of the distance parameters of planes range
between 1.7 cm and 2.4 cm. Considering the nominal
range precision of the scanner (1.1 to 4.2 mm at 25 m)
these residuals are noticeably large. The low precision of
the registration process represented by large residuals is
possibly due to the low precision of the plane parameters
obtained by the segmentation algorithm. In the regiongrowing segmentation, the plane parameters are computed
within a 5x5 neighborhood around every point, and once a
region is fully grown the average of the parameters over
the region is associated to the plane. Such a procedure is
local and sensitive to noise, and may not provide sufficient
accuracy for the plane parameters.
Figure 5 shows the number of planes involved in the
matching process for the registration of each pair of
consecutive scans in the corridor test. A total of 20 largest
planes in each scan were used for the registration, out of
which 10 were used for the creation of the initial match
sets. Each initial match set contained 4 plane pairs. It can
be seen from Figure 5 that the winning match sets contain
between 10 to 15 planes, which is an indication that a
correct set of correspondences was present in the initial
match sets, and that a large number of plane pairs
contributed in the estimation of the transformation in all
registrations.
Reflectance images of a scan in the corridor (left) and a scan in the Aula interior (right).
2010 International Conference on Indoor Positioning and Indoor Navigation (IPIN), 15-17 September 2010, Zürich, Switzerland
Figure 6 shows in a magnified box the closing error of the
entire set of estimated registrations as the distance
between the initial position of point p1 (which was
measured manually) and its computed position obtained
from all six pair-wise registrations. Table II summarizes
the discrepancies between the measured coordinates of the
scanner positions and the computed ones obtained from
the estimated registration parameters. As it can be seen,
the discrepancies are slightly smaller than those of the
corridor test, except for the points p4 and p5, which have
larger discrepancies. These points were the farthest to the
total-station, and it is likely that their reference measured
coordinates were less accurate. The discrepancies at P1
represent the closing error, which is 2.7 cm as the distance
between the reference and the computed position of the
point.
s9
440
420
s8
400
s7
380
s6
Y
360
340
320
s5
s4
300
Total nr planes
Nr planes winning matchSet
Nr planes initial matchSets
280
24
22
260
s3
40
60
80
20
100
X
120
140
160
Figure 4. The computed trajectory of the laser scanner in the
corridor test. The manually measured positions are shown with red
marks. The coordinates are in centimeters.
TABLE I.
ACCURACY OF REGISTRATION AND LOCALIZATION
IN THE CORRIDOR TEST.
18
16
Nr planes
240
20
14
12
10
8
6
Scanner
position
Localization
δx
(cm)
δy
(cm)
s3
0
0
s4
3.6
2.8
s5
2.2
1.0
s6
1.0
-2.6
s7
-1.8
-0.3
s8
-1.5
1.8
s9
-3.4
-0.5
Mean
0.0
0.3
RMS
2.3
1.7
Registration
4
RMS residual
of normals
RMS residual
of distances
(cm)
0.010
2.4
0.029
1.7
0.011
2.3
0.024
2.1
0.021
2.3
0.024
2.0
To verify the influence of the accuracy of plane
parameters on the precision of the registration, in the
Aula-interior dataset a least-squares plane fitting was
applied to the points within the regions of the segmented
range images. The plane matching process was carried out
with the plane parameters from this fitting procedure to
yield the registration parameters as well as the motion of
the scanner. Figure 6 shows the positions of the scanner
derived from the estimated transformation parameters in
the Aula-interior test. As it can be seen, the scanner
trajectory forms a polygon so that both sides of a wall in
the middle of the area can be scanned. This setting made
possible a registration of the last scan and the first scan.
2
0
s3_s4
s4_s5
s5_s6 s6_s7
Registration
s7_s8
s8_s9
Figure 5. The number of planes involved in the plane matching
process in the corridor test.
The results of the Aula-interior test shown in Table II
exhibit a noticeable improvement over those of the
corridor test in terms of the internal precision of the
registrations. Here the RMS residuals of the distance
parameters of the planes range from 0.4 to 1.0cm. Also,
the RMS residuals of the normal vector parameters are
significantly smaller than those of the corridor test. These
results indicate that a precise fine registration can be
performed using planar segments if the parameters of the
planes are sufficiently accurate. Figure 7 shows the
registered point cloud of the Aula interior obtained by
applying the registration parameters estimated in the pairwise application of the plane matching process.
The processing time needed for the registration of all
scans was measured in both tests. The algorithms were run
on a desktop PC with 3.2 GHz CPU speed and 2.00 GB
memory. In the corridor test, where initial match sets were
created with four plane pairs out of 10 planes in each scan,
the entire registration process took 38 seconds. In the
Aula-interior test, where initial match sets contained three
plane pairs selected out of 7 planes in each scan the
process took 155 seconds. The faster processing of the
corridor dataset is due to the fact that initial match sets
contained four plane pairs. This allowed for a quick
identification of wrong correspondences (using the
residual values), which in turn speeds up the extension
process.
2010 International Conference on Indoor Positioning and Indoor Navigation (IPIN), 15-17 September 2010, Zürich, Switzerland
11
p4
10
p5
9
8
p3
7
Y
6
5
4
p2
1.32
3
p6
1.31
Y
2
1.3
p1
1
1.29
-4
-3
-2
-1
0
1
p1
2
1.28
1.29
X
1.3
1.31 1.32 1.33
X
Figure 6. The computed trajectory of the laser scanner in the Aulainterior test. The manually measured positions are shown with red
marks. The coordinates are in meters.
TABLE II.
ACCURACY OF REGISTRATION AND LOCALIZATION IN
THE AULA-INTERIOR TEST.
Localization
Scanner
position
correspondences are effectively eliminated from the
estimation model by rejecting plane pairs whose
parameters have large residuals. This results in a high
level of robustness against noisy and outlier planes as
these normally cannot survive the extension process.
The plane matching approach to pointcloud registration
and scanner localization requires that a sufficient number
of planes are available in each scan, and that at least three
of these planes intersect in a point so as to have a unique
solution for the estimation model. A degenerate
configuration of the planes can be thought of as a set of
only vertical walls in two scans, which can not constrain
the motion of the scanner in the direction of Z axis. Such a
constraint can be provided by including the planes of the
floor or ceiling in the transformation estimation model.
Another requirement for a fine registration using planar
objects is the high accuracy of the input plane parameters.
Generally, a segmentation algorithm does not provide
highly accurate plane parameters, and a least-squares
fitting process is necessary. It was shown that plane
parameters from the segmentation lead to a rather coarse
registration, which may only be sufficient as an initial
approximation for an iterative fine registration method
such as ICP. However, with accurate plane parameters
derived from a least-squares fitting process, a fine
registration can be performed in a single step, with a
precision better than 1 cm, which obviates the need for an
iterative second registration.
Registration
δx
(cm)
δy
(cm)
δz
(cm)
p1
0
0
0
p2
-0.7
1.0
-0.6
p3
-0.6
1.5
-0.7
p4
1.0
-13.8
0.5
p5
0.6
-4.8
-4.7
p6
-0.7
-1.3
0.0
p1
-2.3
-0.7
1.2
Mean
-0.4
-2.6
-0.6
RMS
1.1
5.6
1.9
RMS
residual of
normals
RMS
residual of
distances
(cm)
0.002
0.4
0.003
0.7
0.002
0.7
0.001
0.0
0.003
1.0
0.003
0.9
V. CONCLUDING REMARKS
The paper introduced a plane matching method for
pointcloud registration and scanner localization. The
transformation between two overlapping scans is
estimated in a linear least-squares adjustment model,
which is independent of an initial approximation. Plane
correspondences are found using an initiate-and-extend
search strategy that begins with matching initial
combinations of three or four plane pairs, and extends
when more planes fit into the estimated transformation.
This way, a correct transformation can be found by
looking for the largest extended match set, while incorrect
Figure 7. The registered point cloud of Aula interior. The ceilings have
been removed to provide a better view of the interior. The circles
represent scanner positions.
ACKNOWLEDGMENT
The author would like to thank Ben Gorte for
segmenting the range images, and Jochem Lesparre and
Giorgi Mikadze for their assistance during the scanning of
the Aula interior.
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2010 International Conference on Indoor Positioning and Indoor Navigation (IPIN), 15-17 September 2010, Zürich, Switzerland
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