arXiv:1704.07910v1  [cs.RO]  25 Apr 2017
Adaptive Cost Function for Pointcloud
Registration
Johan Ekekrantz∗, John Folkesson and Patric Jensfelt
Abstract—In this paper we introduce an adaptive cost-function for
pointcloud registration. The algorithm automatically estimates the sensor
noise, which is important for generalization across different sensors and
environments. Through experiments on real and synthetic data, we show
signiﬁcant improvements in accuracy and robustness over state-of-the-art
solutions.
Index Terms—pointcloud registration, robust estimation, localization,
mapping
I. INTRODUCTION
T
HE task of aligning sensor data is called registration and is an
important ﬁeld in robotics, with applications such as simultane-
ous localization and mapping, object tracking and 3D reconstruction.
For two point-clouds, registration is equivalent to ﬁnding a relative
sensor pose transformation. The modelling of sensor noise can make
registration more robust and accurate. Our approach adaptively learns
the sensor noise model using the measured point-cloud data while
performing the registration.
Registration problems are usually solved by ﬁnding corresponding
points in different point-clouds and from these correspondences
estimate a transformation between the point-clouds. Unfortunately,
there is no universal solution, which results in only correct corre-
spondences, for the point matching problem. Many popular solutions
require sensor-speciﬁc tuning of parameters to reduce the inﬂuence
of incorrect correspondences, such parameters make using the regis-
tration algorithm very complex for non-expert users. It is therefore
desirable to ﬁnd solutions which easily port between different sensors
and platforms. In practice, many factors such as illumination strength
or environment temperature may affect the sensor noise model. Such
environment variables are hard to account for beforehand, making
precise a priori noise models impractical in many scenarios. A
solution to these problems is to estimate the sensor noise model on
the ﬂy, directly from measurement data.
Often when point-cloud registration is required, a coarse initial
guess of the alignment is already available. Such guesses can come
from robot odometry or from the fact that the sensor only can
move so far in a certain amount of time. This class of problems
is called constrained registration. If no initial guess is available,
it is called unconstrained registration. In the case of constrained
registration, solutions usually perform iterative reﬁnement of the
correspondences and the estimated transform between the point-
clouds. For constrained registration, the range of convergence on the
accuracy of the initial guess is an important factor.
An additional issue can be that the sensors discretely sample
the environment, further complicating the registration. This is not
addressed by our method. In this paper we limit the problem to con-
strained registration in which the separation of inliers from outliers
is the primary source of error. In particular, we assume that the two
point clouds that are to be matched contain some common points
∗Corresponding author.
The
authors
are
with
the
Centre
for
Autonomous
System
at
KTH Royal
Institute
of Technology,
SE-100 44 Stockholm,
Sweden
{ekz,johnf,patric}@csc.kth.se
The work presented in this papers has been funded by the European Union
Seventh Framework Programme (FP7/2007- 2013) under grant agreement
No 600623 (STRANDS), the Swedish Foundation for Strategic Research
(SSF) through its Centre for Autonomous Systems and the Swedish Research
Council (VR) under grant C0475401.
that are sampled from nearly the same 3D point in the environment
and some that are outliers, ie. points in one point cloud that have
no corresponding points in the other point cloud. This is the same
assumption made by most registration algorithms.
The main contribution of this paper is a method which we call
Statistical Inlier Estimation (SIE). SIE can be used to analyze a
set of pairwise matched measurements and from that compute a
probability for every match to be correct. The basic idea for SIE
is to model the distribution of measurements using histograms of
residual errors, and from these histograms estimate the distribution of
inliers. The residuals are computed using an estimated transformation
between the point-clouds. The estimation of the inlier distribution and
the transformation are done alternatively and iteratively. Having an
estimate of the inlier probability allows us to deﬁne a cost function
that does soft matching rather than binary. That is, rather than
completely removing points classiﬁed as outliers from the cost, all
closest matched point pairs are included in the cost but with a weight
that depends on the inlier probability.
The registration can be performed without any requirement on hand
tuned parameters for speciﬁc sensors or applications. From a practical
perspective, SIE is relatively simple to implement. The same idea
has previously [1] been used for ﬁtting surface primitives such as
planes and spheres to point-cloud data. In this paper we show that
it can be used as a general purpose inlier estimation framework. We
extend the framework to allow for arbitrary noise distributions and
enable 3D point-to-point inlier estimation as opposed to only point-
to-surface in [1]. We also adapt the method by which the distribution
of inliers is estimated. We show that SIE can be used to extend
the standard ICP [2] algorithm. By adding an adaptively decreasing
regularization to the variance of our estimated noise model we
improve convergence. The regularization approximately correspond
to the current uncertainty of the estimated parameters in each iteration
of the ICP algorithm. We compare ICP registration using SIE to other
methods on both simulated and real world data.
II. RELATED WORKS
By far the most popular solution to the constrained point-cloud
registration problem is the Iterative Closest Point ICP algorithm [2].
ICP works directly on the point-clouds by alternating between
performing point matching using point-to-point nearest neighbour
matching and recomputing the relative transform between the two
frames such that it minimizes the L2-norm of the matches. If the
sensor measurement noise is Gaussian, the L2-norm correspond to
the maximum likelihood estimate of the transform.
Using the assumption that points are sampled from continuous
surfaces, several reﬁnements have been proposed to remove the
discretization noise generated by the point sampling from a surface.
In [3] point-to-plane ICP is introduced, meaning that the point
distances in one cloud are minimized as the projection distance along
the normals of the matched points. In [4] the GICP algorithm is
introduced. GICP can be seen as plane-to-plane ICP. In [5], the NICP
algorithm was introduced. During data association, NICP accounts for
the relative surface orientations of the points and the local curvature,
improving the matching and therefore the accuracy and robustness
of the estimation. NICP also change the optimization criteria to
minimize the total Mahalanobis distance of the found correspondence
point pairs and their normals. In [6] the NICP algorithm was extended
to use color information to improve the point matching.
In [7] the trimmed ICP algorithm is proposed. To cope with
outliers, correspondences with residuals outside of some chosen
quantile of the residuals for all correspondences are rejected. This can
be a good option if the number of outliers is approximately known
a priori.
Absolute residual error (m)
0
0.001
0.002
0.003
0
0.2
0.4
0.6
0.8
1
Histogram of residuals
Noise estimate
P(Inlier)
Fig. 1: Left: Two point-clouds registered using our method. Right: Resulting histogram of residuals error (inliers and outliers), noise estimate
(inliers) and probability of inliers as function of residual error. The non-overlapping outlier regions of the point-clouds appear at the tail end
of the histogram of residuals.
In [8], the authors propose to use Levenberg-Marquardt algorithm
to perform the estimation of the transformation given point correspon-
dences. The Levenberg-Marquardt algorithm is a general-purpose
non-linear optimization method, allowing for arbitrary cost-functions
to be used.
In [9] a multi-view Levenberg-Marquardt ICP algorithm is in-
troduced. The algorithm estimates the scale of the sensor noise
using the median residual. Using the median residual to estimate
the approximate size of the sensor noise removes the need for a
hand-tuned rejection criteria, but assumes that atleast half of the
residuals come from inlier correspondences and has a built-in bias
which increases as the number of outliers grow. The median residual
value has some similarity to our use of an adaptively decreasing
regularization in that the estimated noise often is larger at the start
of the ICP optimization than at the end.
In [10] trust region optimization was used to perform non-linear
optimization of robust cost-functions for ICP. Similarly to [9] the
approximate size of the sensor noise is estimated using the median
residual.
In GO-ICP [11], a branch-and-bound and ICP hybrid algorithm, is
introduced. The algorithm is proven to ﬁnd the optimal solution up to
a chosen level of accuracy for the point-to-point L2 error metric and
a trimmed ICP [7]. Unfortunately the increased range of convergence
comes at a price of signiﬁcantly increased computational complexity.
In [12], the authors propose the usage of Lp-norms where p < 2 in
order to better cope with false correspondences. This work is similar
to ours in that it does not require complicated tuning for use on a
new sensor.
When using a kd-tree (or a set of trees), the nearest neighbour
matching of ICP has the computational complexity of O(Nlog(K)),
where N is the number of points being matched and K is the number
of points matched against. Another popular alternative for range
scans is to use back projection in order to ﬁnd correspondences.
While back projection does not guarantee to ﬁnd the true nearest
neighbour match, the matching step can be done in O(N), providing
a signiﬁcant reduction in computational complexity. The registration
component of popular 3D-SLAM systems such [13] and
[14] that
require 30 frames per second performance use back projection as
a matching algorithm.
[13] uses an automatically estimated T-
distribution for estimation.
Image registration has traditionally been done using keypoints,
a subset of the pixels in the image. Common keypoint extractors
such as FAST [15] or Harris corners [16] can ﬁnd a stable subset
of points in images quickly. A large body of work has been put
into ﬁnding distinctive descriptors for visual information, such as
SIFT [17], SURF[18], BRIEF [19], ORB [20] and GRIEF [21].
Keypoint extractors and descriptors have also been created for point-
cloud data, such as 3D-sift [22], NARF [23] and SHOT [24].
In addition to descriptor similarity, geometrical constraints can be
used to ﬁlter out false correspondences. Probably the most popular
technique used for this is the RANSAC [25] algorithm. In [26]
and [27] the AICK algorithm was introduced. AICK is an ICP-
like algorithm which quickly registers two sets of 3D points with
associated descriptors. AICK use a weighted distance metric, taking
into account both feature similarity of keypoints and geometrical ﬁt.
AICK incrementally change the importance of the feature similarity
between the keypoints and the geometrical ﬁt as registration is
performed. This relaxes the requirement on geometrical ﬁt at the
beginning of the registration, gradually increasing the importance of
the geometrical ﬁt for both the matching and outlier rejection as the
registration improves from iteration.
In [28] a keypoint-based registration technique which uses a scaled
Geman-McClure estimator, where the λ parameter is sequentially
decreased as the registration is improved. This gradually increases
the importance of the geometrical ﬁt for determining the inﬂuence of
a match based on the geometrical ﬁt as the registration improves.
[29] presents a pointcloud registration method which forms his-
tograms over surface orientations. The surface orientation histograms
are then directly aligned to register the pointclouds. This technique
efﬁciently takes advantage of the high amount of co-planar surface
patches in indoor environments, leading to improved robustness to
poor initialization estimates.
The normal distribution transform (NDT) was developed to take
sensor noise into account, ﬁrst in 2D [30] and later extended to 3D
in [31], [32]. These methods represent one of the point-clouds using
normal distributions to which the points in the other point-clouds are
aligned. In [33] the technique is modiﬁed to perform registration from
a set of normal distributions to another set of normal distributions.
In [34] 3D-NDT was augmented with matched keypoints. The weight
between the matched keypoints and the NDT results was controlled
by a weight that changed over iterations based on the ﬁt of the NDT.
In [35] 3D-NDT was extended to use color information.
In [1] an algorithm for ﬁtting geometric primitives to point-
cloud data was introduced. At the core of the algorithm is an
inlier estimation system which is speciﬁcally created for Gaussian
distributed point-to-surface distances.
[1] and SIE share the idea
of ﬁtting a noise distribution to a distance-histogram. Compared
to [1], SIE allows for any distribution to be used. The application
domain also differs, as well as many of the practical details as to
how the distributions are ﬁtted. SIE also allows for multi-dimensional
residuals, which [1] does not.
III. POINT-CLOUD REGISTRATION
Point-cloud registration is done by maximum likelihood esti-
mation over the relative positions of the point-clouds. This ﬁnds
the transformation that minimizes some cost function over a set
of residual errors. In this paper we will assume that the sensor
measurement noise distribution is Gaussian. Using other distributions
require minimal changes, see appendix for details and experiments
using non-Gaussian noise.
Given two point-clouds A
=
{a0, a1, . . . , an} and B
=
{b0, b1, . . . , bm} and assuming some correspondence map between
them, the probability of residual ri = T ∗ai −bj is modelled
P(ai|bj, T ) = N(||ri||, 0, σi)
(1)
The registration problem can therefore be formulated as the com-
putation of arg maxb,T P(A|b, T ), where we indicate by b the
correspondences from B that are matched to A. We can reformulate
the optimization problem as
arg min
b,T
−log

P(A|b, T )

= arg min
b,T
n
X
i=0
(||ri||
σi )2
(2)
The Iterative Closest Point, ICP, algorithm assumes that T is
approximately known beforehand. The maximum likelihood estimate
of bi is the closest point to T ∗ai. We will refer to the ﬁrst step
of ICP, where nearest neighbour matching between the point-clouds
is performed, as the matching step. The second step of the ICP
algorithm where eq.(2) is minimized by reﬁning the estimate of
T given the previously found point-to-point correspondences, will
be referred to as the reﬁnement step. Given the correspondences b,
minimization of eq.(2) for the T variable can be performed using a
standard weighted least squares minimization.
Tk+1 = arg min
T
n
X
i=0
w(rk
i )||ri||2
(3)
Using the Iteratively Re-weighted Least Squares (IRLS) algorithm,
robust cost functions can be used to reduce the inﬂuence of outliers
by iteratively solving a set weighted least squares minimizations.
IRLS is performed using eq.(3) where k indicates the iteration
of the IRLS and the superscripts indicate the T that the residuals
were computed with. We will compare various standard choices for
w(i). Some of these are designed to give lower or even zero weight
to residuals that are less likely to be inliers. We will show in the
next section how we can approximate P(I|ri), the probability that a
residual is an inlier given its value. In our method, the cost function
is given a weight proportional to P(I|ri):
w(ri) = σ−2
i
∗P(I|ri)
(4)
IV. STATISTICAL INLIER ESTIMATION (SIE)
The ability to detect or mitigate the effect of outliers is critical
for the accuracy of point-cloud registration. Even a very small
fraction of unaccounted for outliers generally results in extremely
poor estimations.
SIE builds a histogram over the residual errors, ri, and uses
that histogram to estimate the parameters of the measurement noise
model. The residuals are the differences between the matched point
pairs using the current best estimate of the transformation between
point clouds. For estimating the measurement noise model, we exploit
the insight that, at or near the peak in the histogram, the contribution
to the histogram of outliers is small as compared to that of the inliers.
Fitting a parametric noise model to the histogram shape near the
peak therefore models the inlier distribution, with negligible effect
from the outliers. Since the histogram is the sum of the outliers
and the inliers, the ratio of the inlier distribution to the histogram
tells us about the outlier versus inlier relative probability. Figure 1
shows the estimated distributions and probabilities of SIE for a
registered pointcloud. For the registration, we use this to weight all
correspondence pairs in the cost function. This cost function is then
minimized to give a new transformation (the weights are held constant
during this minimization). We can then iterate this process with new
residuals.
A. Adaptive model ﬁtting
The adaptive model ﬁtting step takes as input a set of residuals,
in the form of a matrix R with n rows and m columns, n being the
number of correspondences and m the number of dimensions for the
residuals caused by the correspondences.
For each column j of R, we compute a histogram Hj. Each
histogram is then the sum of the distributions of the inliers and the
outliers. The distribution of outliers is in general unknown and often
of non-parametric form and is as such hard to compute directly from
the data or deﬁne a priori. The inlier measurement distribution on the
other hand can be approximated by some a-priori known parametric
function, making it well suited for estimation. In practice, normal
(Gaussian) distributions are very popular approximations of sensor
noise for a wide variety of sensor types.
In section III we made the assumption that the sensor noise,
and therefore the inliers, were distributed according to Gj = αj ∗
e
−0.5|
x−µj
σj
|2
. We make the assumption that all residuals within one
column of R are independent and identically distributed. In many
applications the sensor noise model parameters (here {αj, µj, σj})
are unknown and impossible to deﬁne a priori, due to among other
things environmental conditions. Cameras for example are susceptible
to environmental conditions such as illumination, unknown camera
exposure time or temperature changes. We believe that if the pa-
rameters are unknown or at least inaccurately known, the parameters
should be estimated from data.
The µj value is deﬁned by a clear peak in Hj, see ﬁg.(1)
for example. Assuming that the vast majority of measurements at
the peak correspond to inliers, we can draw the conclusion that
αj ≈Hj(µj). Once αj and µj are approximately known, we
estimate σj by ﬁtting the noise estimate Gj to the histogram Hj. Gj
is ﬁtted by solving eq.(5), which corresponds to maximizing the data
likelihood under the soft constraint that Gj ≤Hj. As such, we seek
a sensor noise model estimate Gj that explain as many measurements
as possible without violating probability constraints.
arg min
σj
n
X
i=0
F

Hj(i) −G(i, αj, µj, σj)

F(x) =
(
−k ∗x
if x ≤0
x
if x > 0
(5)
where k is used as a penalty term that push the minimization to
choose solutions where Hj(i) ≥G(i, αj, µj, σj).
We found that eq.(5) can be effectively minimized using bisection,
but any other suitable optimization technique would also be ﬁne.
Computing the inlier distribution parameters in the histogram space
has the advantage that, other than constructing the histogram, the
computational cost is not related to the number of samples but rather
the number of bins in the histogram. This keeps the computational
complexity of the minimization low even if there are vast quantities
of ﬁtting data and a complex inlier distribution.
B. Prediction step
Once the inlier noise distributions and the histograms are known,
one can compute the inlier probability for a single dimension j
of a measurement i. We start with the component of the residual
distribution that is due to the inliers,
P(I|Ri,j)P(Ri,j) = P(Ri,j|I)P(I) ≈Gj(Ri,j)
(6)
where P(I) is the prior probability of inliers, and P(Ri,j) is given
by the histogram. One way to estimate P(I) is by summing over the
data:
P(I) ≈
X
i,j
G(Ri,j)
(7)
We will throughout implicitly assume that we have normalized Hj
and Gj(Ri,j) so that P
i,j Hj(Ri,j) = 1, even if in many formulas
this normalization factor cancels and need not be computed. For one
dimension the inlier probability is the ratio of expected inliers to the
ratio of total residuals for a speciﬁc bin in the histogram.
P(I|Ri,j) ≈G(Ri,j, αj, µj, σj)
Hj(Ri,j)
(8)
We know that a single correspondence, a row i of R, is either an
inlier or outlier. We have computed the probability of it being an
inlier given one column of that row. We now have to compute the
probability given the evidence from all the columns of the row. The
residuals across a row are not independent unless we know whether
or not the row is an inlier. We need to ﬁnd the joint probability across
all dimensions j.
P(Ri) = P(I)
Y
j
P(Ri,j|I) + P(O)
Y
j
P(Ri,j|O)
(9)
where O indicates outlier, P(O) = 1 −P(I).
P(Ri,j|O) ≈P(Ri,j) −P(Ri,j|I)P(I)
P(O)
(10)
In terms of our estimated inlier distribution and the histogram:
P(Ri,j|O) ≈Hj(Ri,j) −G(Ri,j)
P(O)
(11)
And ﬁnally we can infer the inlier probability
P(I|Ri) ≈
Q
j G(Ri,j)
Q
j G(Ri,j) + (P(I))m−1P(O) Q
j P(Ri,j|O)
(12)
This can be rewritten as
P(I|Ri) ≈
Q
j P(I|Ri,j)
Q
j P(I|Ri,j) + γ Q
j(1 −P(I|Ri,j))
(13)
where γ = (P(I)/P(O))m−1. This formula is the inlier proba-
bility given the m dimensional residual. To avoid degenerate cases
leading to division by zero, we truncate the value of P(I|Ri,j) to
some value less then 1 (in this paper we pick 0.99). This means that,
regardless of residual value, no correspondence is completely certain
to be correct.
C. Regularizer for use in parameter estimation
Many estimation problems, including point-cloud registration, re-
quire ﬁtting some model with a set of variable parameters T to a
set of measurements. Since the parameters T are unknown/poorly
known at the start of the estimation, the residuals in R are affected
by systematic bias, we account for this by adding a secondary
regularization term βj to the estimated noise σj of the system,
resulting in Eq.(14).
P(I|Ri,j) = G(Ri,j, αj, µj, σj + βj)
Hj(Ri,j)
(14)
Once the precision of the estimated parameters T improve, the
size of βj can be reduced. For the point-cloud registration problem
we run the registration algorithm until convergence and then reduce
βj by 50 percent. Once βj has been decreased, the optimization can
be run again. We continue this until βj << σj, which means the
solution has converged.
The initial value of βj can be set using prior knowledge about the
uncertainty of T or approximated as the standard deviation of the
initial set of residuals.
D. Noise normalization
It is a well known fact that the measurement noise of many sensor
types is correlated to some a priori known factors for the sensor
type. For example, the noise of a measurement by structured light
RGBD cameras is known to increases approximately as the square
of the distance to the sensor. We formalize this by stating that each
residual value Ri,j is either an outlier or sampled with an individual
σi,j = σjF(i, j). Re-scaling Ri,j by the inverse of F(i, j) makes
the scale of the measurement noise identical for all measurements
and the techniques presented in sections IV-A and IV-B performs as
expected. For a residual computed from the difference between two
measurements, the variance of the resulting residual is the sum of the
variances of the two measurements.
E. Adaptive parameters
Two parameters are required to compute a histogram: the interval
in which the histogram is deﬁned and the number of bins in the
histogram. Through iterative updating, these values can be computed
adaptively to the data.
Given that our choice of Gj is monotonically decreasing with the
distance to the mean value µj, one can safely limit Hj to the interval
where Gj > ǫ, where epsilon is a suitably small number.
If an initial guess of Gj is not available, one can safely initialize Gj
to the maximum likelihood estimate assuming that all measurements
in R are inliers.
As a means of accounting for sample variance, the number of bins
in the histogram can preferably be computed linearly to the number
of data that falls within the range of the histogram. This keeps the
0
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1000 Inliers 100 Outliers
1000 Inliers 1000 Outliers
1000 Inliers 10000 Outliers
Translation
Rotation
SIE
Truncated L2-norm
L1-norm
L0.1-norm
T-distribution
Fig. 2: Failure ratios for compared solutions with different number of outliers and different numbers of initial transformation estimates. The
top row contain experiments of initial transformation translated along the x-axis by 0 to 1 units. The bottom row contain experiments of
initial transformation rotated around the x-axis by 0 to π radians. In some experiments, some solutions never had any complete failures.
This means that the curves overlap of along the x-axis of the ﬁgure. Therefore, if a solution is not visible in the ﬁgure, no complete failures
were recorded.
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1000 Inliers 100 Outliers
1000 Inliers 1000 Outliers
1000 Inliers 10000 Outliers
Translation
Rotation
SIE
Truncated L2-norm
L1-norm
L0.1-norm
T-distribution
Fig. 3: Mean errors of non-failure cases for compared solutions with different number of outliers and different numbers of initial transformation
estimates. If the failure rate is greater than 0.5, no mean error is displayed. The accuracy of estimation vary by orders of magnitudes between
the compared solutions, the mean error is therefore drawn on a logarithmic axis. The top row contain experiments of initial transformation
translated along the x-axis by 0 to 1 units. The bottom row contain experiments of initial transformation rotated around the x-axis by 0 to
π radians.
sample variance of the histogram approximately constant regardless
of the amount of data in Hj.
In hopes of further reducing sample variance, the histogram Hj is
also smoothed by convolution with a zero mean Gaussian kernel,
where the standard deviation is proportional to the width of the
histogram. If σ is expected to be the same for some set of the
dimensions, a joint histogram if all the residuals for those dimensions
can be used to reduce the sample variance. Similarly, the absolute
values of the residuals can be used to reduce the sample variance of
the histograms.
In eq.(5) we introduced a penalty term k, that bias the minimization
to choose solutions where Hj(i) ≥G(i, αj, µj, σj). In the ideal case
with no sample variance in Hj, one would set k = ∞. In practice,
because of sample variance, we fund that 1 ≤k ≤10 provided good
results, with K = 1 providing more precise estimations and k = 10
providing more robustness to outliers and poor initial value of T .
We therefore run the registration with k = 10 until convergence and
then adapt the value of k to the data at hand, using the heuristic
k = P(I|Hj)−3 where P(I|Hj) is the average probability for the
residuals inside the interval of Hj to be an inlier.
During iterative estimation, such as ICP minimization, the param-
eters for the histogram width, number of bins and value of k can
be carried between iterations. This reduces the computational load of
SIE.
V. EXPERIMENTAL SETUP
We compare our method to other cost functions used for point-
cloud registration in order to determine both the accuracy and
robustness of the registration relative to other popular solutions.
Below we list the cost functions used during experimentation, and
the motivation behind using them. Many cost functions have names
that indicate which norm is used in the minimization problem. The
L2-norm is equivalent to assuming that the measurement noise is
Gaussian. All tested solutions are minimized using the iteratively re-
weighted least squares (IRLS) minimization algorithm.
Truncated L2-norm
The truncated L2-norm is a very popular
alternative for point-cloud-alignment. The idea is to reject
all correspondences with an error greater than some thresh-
old. The threshold value is usually set to some value in
the range of three to four σ. Finding the optimal rejection
threshold can be hard if one does not know σ for the sensor.
Lp-norm
The use of Lp-norms, where one artiﬁcially set p to
a much smaller number than what is actually present in
the sensor, is another popular alternative. The idea is that
the effect of correspondences with large residuals can be
decreased if p is small. A useful property of the algorithm
is that knowledge of the sensor noise model is not required,
making it a practical solution in many applications where
the threshold of the truncated L2-norm is not easily tuned.
We will evaluate the result of the L1-norm and L0.1-norm
as proposed by [12].
T-distribution
In [36] and [13], the T-distribution is proposed
as a self-tuning alternative for registration of RGBD images.
Estimation of the scale parameter σ is performed using the
method from [36].
Statistical Inlier Estimation (ours)
SIE models both inlier
and outlier distributions. It therefore has the ability to adapt
to a wide range of applications. A useful property of the
algorithm is that an estimate of the sensor noise is not
required, making it a practical solution in many applications
where the threshold of the truncated L2-norm is not easily
tuned.
VI. SIMULATION
As a matter of isolating the effects of the SIE based estimation,
we deﬁne a simulated experiment where the task is to determine
the relative poses of two point-clouds, given a set of point-to-point
correspondences, some of which are correct and some which are
outliers 1.
This is equivalent to performing registration using a keypoint
matching scheme. If the points in the point-clouds are independently
sampled with zero mean Gaussian noise and transformed by some
rigid body transform T , the maximum likelihood estimate TMLE of
T is given the least squares ﬁt of the correct correspondences.
Our system aims to compute an estimate of T , which is as close
to TMLE as possible. In this paper, we measure the Accuracy of
estimation, robustness to outliers and robustness to initial guess of
T . Since this data is simulated, we can vary the number of outliers
and change the initial guess for the transformation.
The test error of an estimated transform is deﬁned as the difference
between the root-mean-square-error (RMS) of the correct correspon-
dences and the corresponding value for TMLE. The advantage of
this error metric is that we are directly measuring how much less
probable the estimated solution is to be correct, than the best possible
solution. We consider a tested transform with an error greater than
the measurement noise a complete failure.
To acquire reliable statistics we sample a set of 100 different test
instances on which we apply our test initial guesses. A total score
for an initial transformation and a set of outliers is then computed
as the mean error for all the non-failure tests. This score is useful
to determine the precision of a registration algorithm. Looking at the
ratio of failures is a good method for determining the robustness of
the estimator used.
A. Simulation Results
We test three different cases, an easy case with 1000 inliers and
100 outliers, a medium case with 1000 inliers and 1000 outliers,
and a hard case with 1000 inliers and 10000 outliers. To test the
range of convergence, two sets of test transformations are created,
one where translation on the X-axis in the range of [0, 1] is tested
and the other where rotation around the X-axis is tested in the range
of [0, π] radians. From ﬁgures 2 and 3, we observe that SIE reliably
and signiﬁcantly outperform the other cost functions with regards to
robustness to poor initialization for T0, robustness to outliers and
precision of estimation.
VII. PRIMESENSE DATA
In [37], a benchmark for RGBD-image slam and visual odometry
was presented. [37] deﬁnes two measurement errors: relative-pose-
error (RPE) and absolute-trajectory-error (ATE). The translation RPE
is used to test visual odometry and registration algorithms, making
it suitable for our purposes.
As previously stated, the noise of a measurement i in the prime-
sense cameras increase approximately by the distance to the sensor
squared. We therefore try both σi = z2
i and σi = 1 and scale the
residuals accordingly for all tested methods. From experience, we
1We deﬁne a function F (k, n, σ) = {A, B} that samples a set of test
instance point correspondences A, B, where k is the number of inlier matches
and n is the number of outlier matches. The k inliers in A are uniformly
sampled in [0, 1]3 and the inliers in B are identical to the inliers in A but
with added Gaussian noise, with a standard deviation of 0.01. The n outliers
in A are uniformly sampled in [0, 1]3 and each corresponding outlier in B
is uniformly sampled around the outlier in A in the interval of [−1, 1]3.
For an initial guess T0, the inlier points are transformed by T0 and the
outlier points are kept as is. This ensures that the outlier points do not
beneﬁcially change the estimation of T.
TABLE I: Translational RMSE Relative Pose Error(RPE) in m/s for data from [37].
Sequence
Range to noise increase
SIE
Threshold 0.007 m
L1-norm
L0.1-norm
T-distribution
freiburg1 xyz
quadratic
0.023
0.022
0.038
0.022
0.022
freiburg1 rpy
quadratic
0.044
0.045
0.057
0.045
0.049
freiburg1 desk
quadratic
0.048
0.052
0.093
0.057
0.052
freiburg1 desk2
quadratic
0.054
0.065
0.088
0.062
0.058
freiburg1 room
quadratic
0.062
0.069
0.095
0.065
0.065
freiburg1 360
quadratic
0.100
0.117
0.148
0.106
0.112
freiburg1 xyz
constant
0.031
0.037
0. 035
0.031
0.034
freiburg1 rpy
constant
0.091
0.153
0.132
0.152
0.114
freiburg1 desk
constant
0.064
0. 093
0.100
0.077
0.068
freiburg1 desk2
constant
0.065
0. 091
0.094
0.078
0.071
freiburg1 room
constant
0.081
0. 140
0.129
0.168
0.086
freiburg1 360
constant
0.135
0.296
0.200
0.246
0.142
know that the noise of the primesense cameras is in the range of 0.001
to 0.002 m. We set the threshold for the truncated L2-norm is set to
0.007 m. For the other algorithms all parameters are kept identical
to the simulation experiments, despite there being a difference of an
order of magnitude in the size of the noise.
For computational reasons, the registration algorithm is limited
to 2000 randomly sampled nearest neighbour matches for all cost
functions. We replace the point-to-point distance used in section VI
with the point-to-plane distance metric.
From the results in table I, we can clearly see that SIE consistently
outperforms the other cost functions, SIE is has the lowest score or
shared lowest score in 11 out of 12 tests. SIE does well especially
on the hard cases where all the costfunctions have a high error
score. It is also clear that knowing the connection between range
and measurement noise has a clear positive effect on the estimation
for the tested solutions.
VIII. CONCLUSIONS
In this paper we introduced a statistical inlier estimation (SIE)
system. SIE is used to determine the probability of pairwise matches
to be correct without the need hand tuned parameters. This makes
using the system simple and ﬂexible. SIE is intrinsically portable and
we have tried it in two different scenarios for sensors with vastly dif-
ferent amounts of noise, without changing any parameters. We extend
the ICP algorithm to utilize the SIE system for outlier rejection and
show that the improved ICP algorithm outperforms comparable cost
functions on the dataset from [37]. Similarly, we show on synthetic
data that SIE improves both accuracy and robustness when estimating
transforms from point-to-point correspondences, especially in the
case of many outliers and poor initialization values.
APPENDIX
In the main body of this paper, we made the assumption that the
sensor measurement noise was Gaussian. This is not required by the
SIE system presented in section IV. If we simulate a set of inliers
similarly to section V, but with Laplacian noise instead of Gaussian,
we can investigate if SIE can be used to estimate the parameters of
the Laplacian noise. Both the Laplacian and Gaussian distributions
are subsets of the Generalized Gaussian Distribution (GGD)
G(x, µ, σ, p) =
p
2σΓ(p−1)e−( |x−µ|
σ
)p
(15)
where Γ denotes the gamma function. The Laplacian distribition is
the special case of a GGD when p = 1 and the Gaussian distribution
the special case of a GGD when p = 2.
The equivalent of eq.(2, with Gaussian noise becomes
arg min
b,T
−log

P(A|b, T )

= arg min
b,T
n
X
i=0
||ri||
σi
pi
(16)
which is equivalent to minimizing over the Lp-norm instead of the
L2-norm. The minimization using the probabilities from SIE can be
performed using the IRLS algorithm where
w(ri) = σ−pi
i
max(||ri||, δ)(pi−2)P(I|ri)
(17)
where δ is a very small number which makes sure that division by
zero is avoided.
For the Laplacian distribution, the test error of an estimated
transform is deﬁned as the difference between the mean-absolute-
error of the correct correspondences and the corresponding value for
TMLE. Similarly to the RMS error for the Gaussian distribution, the
advantage of the mean-absolute-error metric for Laplacian data is that
we directly measure the probability of the estimated solution.
If p is unknown, SIE can be used to ﬁnd an estimate of p. Naturally
this will result in reduced precision, as compared to knowing the
exact value of p apriori. In ﬁgures 4 and 5 we perform experiments
using laplacian noise, we observe that using p = 1 results in the best
estimation, with SIE estimating p outperforming the estimation when
p = 2 when there are few to moderate amounts of outliers relative
to inliers.
Both the Laplacian distribution and the estimated p norm allow for
fat-tail distributions, resulting in the estimated model over-ﬁtting to
the outliers if the outliers signiﬁcantly outnumber the inliers.
In ﬁgures 6 and 7 we observe that p = 2 is the best solution when
the measurement noise is Gaussian. The second best solution is using
SIE to estimate p. Assuming that p = 1 providing the worst estimates.
We draw the conclusion that estimating p is a good alternative when
the exact shape of the measurement noise distribution is not precisely
known.
0
0.2
0.4
0.6
0.8
1
1e-8
1e-7
1e-6
1e-5
1e-4
1e-3
translation start offset
mean registration error
0
0.2
0.4
0.6
0.8
1
1e-7
1e-6
1e-5
1e-4
1e-3
translation start offset
mean registration error
0
0.2
0.4
0.6
0.8
1
1e-6
1e-5
1e-4
1e-3
1e-2
translation start offset
mean registration error
0
0.5
1
1.5
2
2.5
3
1e-8
1e-7
1e-6
1e-5
1e-4
1e-3
Angle start offset (rad)
mean registration error
0
0.5
1
1.5
2
2.5
3
1e-8
1e-7
1e-6
1e-5
1e-4
1e-3
Angle start offset (rad)
mean registration error
0
0.2
0.4
0.6
0.8
1
1e-6
1e-5
1e-4
1e-3
1e-2
Angle start offset (rad)
mean registration error
Laplacian noise
1000 Inliers 100 Outliers
1000 Inliers 1000 Outliers
1000 Inliers 10000 Outliers
Translation
Rotation
SIE L2-norm
SIE L1-norm
SIE estimated norm
Fig. 4: Mean errors of non-failure cases for compared solutions with different number of outliers and different numbers of initial transformation
estimates. The measurements are sampled with Laplacian noise. If the failure rate is greater than 0.5, no mean error is displayed. The accuracy
of estimation vary by orders of magnitudes between the compared solutions, the mean error is therefore drawn on a logarithmic axis. The
top row contain experiments of initial transformation translated along the x-axis by 0 to 1 units. The bottom row contain experiments of
initial transformation rotated around the x-axis by 0 to π radians.
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
translation start offset
Failiure ratio
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
translation start offset
Failiure ratio
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
translation start offset
Failiure ratio
0
0.5
1
1.5
2
2.5
3
0
0.2
0.4
0.6
0.8
1
Angle start offset (rad)
Failiure ratio
0
0.5
1
1.5
2
2.5
3
0
0.2
0.4
0.6
0.8
1
Angle start offset (rad)
Failiure ratio
0
0.5
1
1.5
2
2.5
3
0
0.2
0.4
0.6
0.8
1
Angle start offset (rad)
Failiure ratio
1000 Inliers 100 Outliers
1000 Inliers 1000 Outliers
1000 Inliers 10000 Outliers
Translation
Rotation
SIE L2-norm
SIE L1-norm
SIE estimated norm
Fig. 5: Failure ratios for compared solutions with different number of outliers and different numbers of initial transformation estimates. The
measurements are sampled with Laplacian noise. The top row contain experiments of initial transformation translated along the x-axis by 0
to 1 units. The bottom row contain experiments of initial transformation rotated around the x-axis by 0 to π radians. In some experiments,
some solutions never had any complete failures. This means that the curves overlap of along the x-axis of the ﬁgure. Therefore, if a solution
is not visible in the ﬁgure, no complete failures were recorded.
0
0.2
0.4
0.6
0.8
1
1e-8
1e-7
1e-6
1e-5
translation start offset
mean registration error
0
0.2
0.4
0.6
0.8
1
1e-9
1e-8
1e-7
1e-6
1e-5
translation start offset
mean registration error
0
0.2
0.4
0.6
0.8
1
1e-8
1e-7
1e-6
1e-5
1e-4
1e-3
1e-2
translation start offset
mean registration error
0
0.5
1
1.5
2
2.5
3
1e-8
1e-7
1e-6
1e-5
Angle start offset (rad)
mean registration error
0
0.5
1
1.5
2
2.5
3
1e-9
1e-8
1e-7
1e-6
1e-5
Angle start offset (rad)
mean registration error
0
0.2
0.4
0.6
0.8
1
1e-8
1e-7
1e-6
1e-5
1e-4
1e-3
1e-2
Angle start offset (rad)
mean registration error
Gaussian noise
1000 Inliers 100 Outliers
1000 Inliers 1000 Outliers
1000 Inliers 10000 Outliers
Translation
Rotation
SIE L2-norm
SIE L1-norm
SIE estimated norm
Fig. 6: Mean errors of non-failure cases for compared solutions with different number of outliers and different numbers of initial transformation
estimates. The measurements are sampled with Gaussian noise. If the failure rate is greater than 0.5, no mean error is displayed. The accuracy
of estimation vary by orders of magnitudes between the compared solutions, the mean error is therefore drawn on a logarithmic axis. The
top row contain experiments of initial transformation translated along the x-axis by 0 to 1 units. The bottom row contain experiments of
initial transformation rotated around the x-axis by 0 to π radians.
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
translation start offset
Failiure ratio
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
translation start offset
Failiure ratio
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
translation start offset
Failiure ratio
0
0.5
1
1.5
2
2.5
3
0
0.2
0.4
0.6
0.8
1
Angle start offset (rad)
Failiure ratio
0
0.5
1
1.5
2
2.5
3
0
0.2
0.4
0.6
0.8
1
Angle start offset (rad)
Failiure ratio
0
0.5
1
1.5
2
2.5
3
0
0.2
0.4
0.6
0.8
1
Angle start offset (rad)
Failiure ratio
1000 Inliers 100 Outliers
1000 Inliers 1000 Outliers
1000 Inliers 10000 Outliers
Translation
Rotation
SIE L2-norm
SIE L1-norm
SIE estimated norm
Fig. 7: Failure ratios for compared solutions with different number of outliers and different numbers of initial transformation estimates. The
measurements are sampled with Gaussian noise. The top row contain experiments of initial transformation translated along the x-axis by 0
to 1 units. The bottom row contain experiments of initial transformation rotated around the x-axis by 0 to π radians. In some experiments,
some solutions never had any complete failures. This means that the curves overlap of along the x-axis of the ﬁgure. Therefore, if a solution
is not visible in the ﬁgure, no complete failures were recorded.
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