Bayesian Fusion using Conditionally Independent Submaps for High
Resolution 2.5D Mapping
Liye Sun, Teresa Vidal-Calleja and Jaime Valls Miro
Abstract— Typically 2.5D maps provide a compact and efficient representation of the environment. When sensor data is
obtained from multiple sets of noisy measurements at differing
resolutions, the problem of compounding this information
together to provide an effective and efficient means of mapping
is not trivial, particularly as the size of the environment
increases. In this paper, we propose a general framework
for integrating heterogeneous sensor data to obtain largescale 2.5D probabilistic maps. Gaussian Processes are used
to generate a prior map that learns the spatial correlation
between nearby points. Bayesian data fusion is then employed
to update these prior maps with new measurements from
distinct sensor modalities. In order to deal with large scale
data, a novel submapping strategy is introduced to perform
the fusion step efficiently in dealing with large covariance
matrices. Submaps are first marginalised from the learned
correlated prior and then updated based on the property of
conditional independence. Most notably, the technique lends
itself to generate accurate estimates at arbitrary resolutions and
is able to handle varying noise from disparate sensor sources.
The framework is applied to pipeline thickness mapping, with
experimental results in fusing a high-resolution sensor and
a low-resolution sensor showing the ability of the proposed
technique to capture spatial correlations to come up with more
accurate results when compared with a naı̈ve fusion approach.
I. INTRODUCTION
Two-and-a-half dimensional (2.5D) mapping has been
widely used in robotics to represent the environment in a
compact manner. A 2.5D map is a two-dimensional grid
where each cell stores a value related to the environment. For
instance, digital elevation maps (also called terrain maps [1])
store the height of the terrain. In a recent application,
thickness maps store the remaining material in pipe walls [2].
Independently of the variable of interest, the challenge remains on how to combine multiple sets of noisy sensor data,
obtained at different measurement locations, into a single
map estimate while at the same time consider correlations
form nearby points. And doing so in an effective manner for
large scale data sets.
This paper proposes a probabilistic framework to integrate
multiple sources of information (sensors’ measurements) into
a single high-resolution map. As in our previous work [2],
Gaussian Processes (GPs) are used to incorporate and handle
uncertainty in a statistically sound manner to represent spatially correlated data. The GP model learned from one data
source can thus be used as a prior for Bayesian fusion via the
maximum a posteriori estimator. For Gaussian distributions
All authors are with the Centre for Autonomous Systems at
Faculty of Engineering and IT, University of Technology, Sydney
NSW 2007, Australia. [email protected],
[email protected] and
[email protected].
the estimator requires the inversion of the covariance matrix
that represents the correlation between points in the map,
therefore high resolution grid maps tend to be intractable. In
order to achieve high-resolution maps, a submapping strategy
is hereby proposed. The key idea is to split the map into
multiple Conditional Independent (CI) submaps [3]. Fusion
is then performed at submap level with further updates of
the state accomplished by back-propagating cross-correlation
information from the neighbouring submap.
The proposed framework is constrained by two assumptions that make the solution approximate rather than optimal.
Firstly, it is assumed that each pair of submaps are exactly CI,
i.e. there is no correlation between three submaps. Although
this is an approximation, in practice, submaps are chosen in
such a way that the correlation between every second submap
is negligible. Secondly, the last submap is estimated to be
optimal in the backpropagation step. In reality, this submap
is not necessarily optimal as the cross-correlation from the
previous submap has not been incorporated.
We apply the proposed general framework in a highresolution pipe thickness mapping scenario. Two data-sets
from two different sensors used to assess the condition of
the pipe are employed in our experiment. We evaluate the
approach by comparing with the optimal fully-correlated
method and the uncorrelated fusion approach. Experimental
results show the validity of the proposed framework in
attaining higher accuracy maps.
II. RELATED WORK
Several approaches have been proposed in the literature to
probabilistically estimate 2.5D maps [4]. Examples in terrain
mapping through elevation maps include [5], which uses the
concept of a “scatter matrix” to represent the local geometric
uncertainty in a grid, the approach in [6] accommodates sensor resolution dependency on range, and in [7] overhanging
objects are tackled where multiple sensors’ measurements
are fused by using the sum of weighted variances of the
uncorrelated data. Our method is related to the latter in the
sense that we also make use of Bayesian methods. However,
to fuse new measurements we consider a spatially correlated
prior instead of a single variance, i.e. using the covariance
matrix with cross-correlation terms learned with GPs, as
proposed in our former work [2].
Gaussian Processes [8] are powerful non-parametric learning techniques and have been recently applied for mapping [9], [10]. GPs have the advantage of being able to
yield a continuous representation of the environment and to
produce maps of a desirable resolution, effectively dealing
with both data uncertainty and incompleteness. Similarly
to our proposed approach, GPs are used in [11] to model
the spatial correlation of observed terrain data to infer
topography from the new observations. A recent approach for
occupancy grids mapping with GPs has also been presented
in [12].
Sensor data fusion using GPs has also received some attention. The work in [13] integrates heterogeneous information
within a classifier based on GPs for a protein fold recognition
application. Each feature is represented by a separate GP. In a
similar manner [14] fuses multiple heterogeneous sources of
correlated information, exploiting the use of dependent GPs
to perform the fusion. They learn a different independent GP
model for each sensor data, and then correlate them. On the
other hand, our proposed approach in this work, as well as
in [2], only learns spatial correlation for a single sensor with
a GP model; all the other data sources can then be fused
probabilistically (using standard Bayesian methods) without
the need to learn more GP models. This allows us to fuse
effectively even very sparse data. The main drawback of both
approaches however is that for large maps, the need for the
inversion of a large covariance matrix when fusing new data
is unavoidable.
Multiple techniques have been proposed in the GPs literature to deal with large-scale data. Some sparse approximation
methods are described in [8]. Choosing a subset of samples
or reduce-rank approximations do not apply to our problem,
as we are trying to capture neighbouring correlations that
results in a substantially correlated covariance matrix.
Large-scale mapping has also been tackled extensively in
the Simultaneous Localisation and Map Building (SLAM)
field [15], [16]. The main difference between the SLAM
problem and the mapping-only problem is that the localisation of the sensor is assumed in the latter. However, some of
the strategies proposed in SLAM for handling large maps can
be utilised in the context of 2.5D mapping, in particular those
based on the covariance form [16]. Most of these approaches
rely on statistical independence between submaps. There is
one work, however, that proposes conditionally independent
submaps [3]. The key idea is that local maps shared information amongst them, therefore the correlation between
submaps is used to refine the estimate of the state. As we
want to exploit correlations in large-scale mapping, building
and fusing conditionally independent submaps appears the
appropriate strategyto solve the present problem.
III. APPROACH OVERVIEW
We consider Bayesian fusion for 2.5D mapping where
prior maps are updated with measurements from other sensors, as per [2]. First, to generate prior maps the spatial
correlation of the measurements from a single source is
learned through GPs. This prior map is represented by a
mean value µ for each cell in a grid and a covariance matrix
P that depicts all the cross-correlation terms amongst all
cells. In order to achieve the targeted high-resolution maps,
we split the large correlated map into submaps that are
conditionally independent, i.e., an overlap region between
consecutive submaps exists. The fusion step then occurs
at the submap level. Finally, cross-correlation information
needs to be propagated back to the mean and variance.
In summary, the general methodology is as follow:
•
•
•
Prior map generation
submapping and fusion, and
back-propagation.
Given a sensor dataset Ψ1 = {(x1 , y1 ) , (x2 , y2 ) , ..., (xn , yn )}
with n referenced sensor readings, where xi ∈ X is the
position from which the sensor readings yi ∈ Y was taken.
The data set Ψ1 is drawn from a noisy process yi = f (xi ) +
εi = ξi + εi , where ε = {εi }ni=1 follows independent, not
necessarily identically, distributed zero-mean Gaussian with
variance Q and ξi = f (xi ) ∈ ξ . The density p (ξξ |X) must then
be estimated from Ψ1 . GPs are used to learn the distribution
p (ξξ |X, Ψ1 ) and infer p (ξξ |X ∗ , Ψ1 ) for arbitrary points X ∗ .
Given this conditional distribution, it is straightforward to
use it as the prior p(ξξ |X) for Bayesian fusion. In the same
way as in our previous work [2], a single measurement can
update the full map because the correlations in position are
already considered. This GPs prior gives us the ability to
increase or decrease map resolution (inferring more or less
points) as required by the new sensor data.
With this model as a prior, a new data set of sensor measurements Ψ2 = {(x1 , z1 ) , (x2 , z2 ) , ..., (xk , zk )} with k sensor
readings z taken from the xi ∈ X ∗ positions, can be fused
(given a sensor model and the location) using a Bayesian
framework. Let us consider a Gaussian sensor modelled as
µ z , R), where R is the uncorrelated covarip(z|ξξ , X) = N (µ
ance matrix of the new measurements. In this approach, the
sensor noise is not restricted to be constant noise, therefore
the elements in R and/or Q might be different.
The maximum a posteriori estimator is used to fuse the
prior map p(ξξ |X) with the new measurements p(z|ξξ , X)
to get the posterior1 p(ξξ |z, X) ∝ p(z|ξξ , X) × p(ξξ |X). The
µ + , P+ ) with an
fusion output is a probabilistic map N (µ
updated mean and covariance correlated based on location.
This fusion framework allows us to integrate multiple data
sets from same-type or heterogeneous sensors. Note that
the higher the resolution or the larger the area covers, the
larger the covariance matrix P is. Fusing Gaussian sensors
optimally requires the inversion of the P, which is O(n3 ).
Conditional Independent (CI) submaps {p(ξξ s1 |xs1 ),
p(ξξ s2 |xs2 ), . . . , p(ξξ sr |xsr )} are marginalised from the prior
p (ξξ |X) and then fused with Ψ2 to obtain a set of up+
+
+
µ+
µ+
µ+
dated submaps {{µ
s1 , Ps1 }, {µ
s2 , Ps2 } . . . , {µ
sr , Psr }}, where
xs1 :sr ∈ X and r is the number of submaps that cover the full
map. As the submaps are CI by construction, p(ξξ s1 |xs1 ) ∩
p(ξξ s2 |xs2 ) 6= 0, i.e. consecutive submaps have an overlapping
part. Practically, the size of the submaps is chosen as to
minimise the correlation between every second submap.
The final step is to propagate back the information between
consecutive CI submaps using the approach proposed in [3].
The influence of the neighbouring submap is propagated back
+
to the means µ +
si and covariances Psi where i = 1, 2, . . . , r −1,
as will be shown in Section VI.
1 The sensor location from where all measurements were taken it is
assumed known in this work.
IV. PRIOR MAPS
As described above, GPs are used to capture the spatial
correlation. GPs can be regarded as a Gaussian probability
distribution over functions and are fully characterised by
the mean function m and the kernel K. The mean function
can be explicitly chosen or, more commonly, set to be zero
by normalising the data appropriately. There are numerous
types of kernels which model the spatial correlation between
points. Its parameters θ can be obtained by maximising the
log marginal likelihood logp(y|X) as,
1
1
n
− (y − m (X))> Ky−1 (y − m (X)) − log|Ky | − log2π, (1)
2
2
2
where Ky = K(X, X) + Q denotes the joint prior distribution
covariance of the function at positions X. If Q represents a
constant sensor noise variance, it equals to σ 2 I, where σ is
another parameter to be learned together with θ . Standard
gradient descent can be used in this case [8] to learn all
the parameters. On the other hand, Q = σi2 I denotes a nonconstant sensor noise, where in general σi 6= σi+1 . Multiple
algorithms have been proposed to tackle this problem [14].
In general, for most of the 2.5D mapping applications, the
sensor noise variance is known or characterisable by other
means rather than GPs. Thus it is only required to learn the
kernel’s hyperparameters θ . This means that ∂∂θ logp(y|X) is
the only derivative that needs to be calculated to minimise (1)
by passing directly the known Q, which can be either
constant or variable.
Once the mean and covariance functions have been specified, GPs could infer the function values at a finite set
of query locations X ∗ . And the predicted mean µ and
covariance P are given by,
µ = m (X ∗ ) + K(X ∗ , X)Ky−1 (y − m (X))
∗
∗
P = K(X , X ) − K(X
∗
, X)Ky−1 K(X ∗ , X)> .
(2)
(3)
The matrix K(X ∗ , X), obtained from the kernel K, denotes
the cross-correlation between the function at the prediction
points X ∗ and the training inputs X.
V. BAYESIAN FUSION
Standard multivariate Bayesian fusion is applied to update
the prior map generated from a set of sensor measurements
with a new dataset of (different) sensor measurements.
µ , P) is
A spatially correlated prior map p(ξξ |X) ∼ N (µ
now available as the output of the GP inference (2) and
(3). The new independent measurements z modelled as a
µ z , R), with connormally distributed sensor p(z|ξξ , x) = N (µ
stant or non-constant noise variance are to be used to update
the prior map. Therefore, the new set of measurements are
integrated into the prior map using the maximum a posteriori
estimator. The posterior density p(ξξ |z, X) is computed as,
µ)
µ + = µ + PH > (HPH > + R)−1 (z − Hµ
+
>
>
−1
P = P − PH (HPH + R) HP ,
(4)
(5)
where H is the observation matrix, which maps the state
space to the observation space. Note that as the crosscorrelation is included in this update step, only a single
measurement can have an effect into the neighbouring points.
𝒔𝟏
𝒔𝟐
part not used
𝑏1
𝝃𝒃
𝝃𝒂
𝒛𝒂
𝒔𝒓−𝟏
𝝃𝒄
𝒛𝒃
Submap 1
𝒔𝒓
𝒛𝒄
𝑏𝑟−1
Submap 2
(a)
(b)
Fig. 1: Schematic representation of CI submapping. (a) Bayesian
network showing the probabilistic dependencies between state ξ
and measurements z. (b) Schematic illustration of the covariance
matrix when split into submaps, where bi represents the common
elements shared among submaps.
VI. CONDITIONALLY INDEPENDENT
SUBMAPPING
Considering the spatial correlation among all data points
during fusion is relatively straightforward for small scale
maps. However, for large scale data, this is usually intractable
because of the O(n3 ) covariance matrix inversion in (5).
In the SLAM literature this is a well-studied problem,
where several approaches have been proposed to reduce the
computational complexity when mapping large areas. One of
the common approaches is based on submapping algorithms,
where in the majority of the cases independent submaps
are considered. Note that independent submaps ignore the
correlations between each other producing an approximate
solution. There is one interesting work [3] based on CI
submaps, which allows the use of submapping algorithms
avoiding the limitations imposed by the requirement of
statistical independence between submaps. The approach
described in this section borrows ideas from [3] by: 1)
splitting the prior into CI submaps, and 2) propagating back
the cross-correlation between submaps after fusion in order
to obtain a more accurate solution. This approach allows us
to fuse information at submap level in O(u3 ) (u is the size
of the submap) with a backpropagation step also in O(u3 ).
A. CI Submapping and fusion
Fig. 1(a) shows an example of a Bayesian network that
represents the probabilistic dependencies between stochastic
variables involved in fusion. Node ξ j represents a set of
components j of the state at some positions in the 2.5D
map, and z j is the set of sensor measurements related to
those positions. Without loss of generality, we will use this
example to illustrate the approach. Let p(ξξ a , ξ b , ξ c ) be the
full prior map2 p(ξξ ), where the subindex represents a set of
components a, b, c. Then, to create CI submaps we simply
marginalised blocks of two sets of components that share a
common part. ξ a and ξ b are jointly marginalised to get the
first submap s1 , and ξ b and ξ c are marginalised together
to get the second submap s2 . Therefore, the state vector for
each submap is ξ s1 = [ξξ a ; ξ b ] and ξ s2 = [ξξ b ; ξ c ].
2 For the sake of notation simplicity, from now on, we remove the known
positions X, therefore p(ξξ a , ξ b , ξ c |X) = p(ξξ a , ξ b , ξ c ).
The information fusion is performed at submap level, once
all the submaps have been marginalised. The measurements
z j are fused using the maximum a posteriori estimator as
described in Section V.
To avoid double counting information, the only requirement is that the non-common part (e.g. za and zc ) of each
set of measurements is the one used to update each submap,
except for last submap where both are incorporated. In our
example, the result of the fusion produces the submaps
p(ξξ s1 |zs1 ) = p(ξξ a , ξ b |za ) and p(ξξ s2 |zs2 ) = p(ξξ b , ξ c |zb , zc ).
Note that ξ a and ξ c are D-separated, as the path between
them is blocked by ξ b . In other words, given ξ b and zb ,
submaps s1 and s2 do not carry any additional information
about each other. This shows that the submaps are conditionally independent. After the fusion
a a
a
Pa Pab
µa
a
a
µ s1 , Ps1 ) = N
p(ξξ s1 |zs1 ) ∼ N (µ
, a
Pba Pba
µ ab
(6)
bc bc
bc
µb
Pb
Pbc
bc bc
µ s2 , Ps2 ) = N
p(ξξ s2 |zs2 ) ∼ N (µ
, bc
µ bc
Pcb Pcbc
c
(7)
where the superindex indicates the set of measurements that
have been incorporated into the fusion.
B. Back-propagation of correlation
We are interested in recovering the full joint map p(ξξ |z) =
p(ξξ a , ξ b , ξ c |za , zb , zc ) as,

 abc   abc
abc Pabc
µa
Pa
Pab
ac
 , Pabc Pabc Pabc  .
(8)
N  µ abc
b
ba
b
bc
abc Pabc Pabc
abc
Pca
µc
c
cb
bc
abc
µ bc
µ abc
We need to assume that N (µ
s2 , Ps2 ) = N (µ
s2 , Ps2 ) to
apply the methods proposed in [3]. Note that this assumption
makes our solution approximated because s2 is considered as
optimal, while in reality information from za has not been
incorporated. Taking this assumption into account, the only
terms that need to be updated are the ones related to ξ a . In
other words, information propagates from the last submap to
all the previous submaps. This is done sequentially using the
backpropagation method proposed by [3] as,
a (Pa )−1 ;
1: K , Pab
b
abc
2: Pab = KPbabc ;
abc − Pa );
3: Paabc = Paa + K(Pba
ba
abc
a
a
µ abc
4: µ a = µ a + K(µ
b − µ b );
abc
abc
5: (optionally) Pac = KPbc ;
This algorithm is obtained exploiting the marginalisation
and conditioning operations for Gaussian distributions and
the CI property p(ξξ a |za , zb , zc , ξ b ) = p(ξξ a |za , zb , ξ b ) =
µ a|b , Pa|b ) (for more details please refer to [3]).
N (µ
Intuitively, the spatial correlation between submaps is lost
when they are marginalised before fusion, the backpropagation algorithm uses the estimation error in common components to correct the state, enforcing the cross-correlation back
again. Note that only the mean and covariance of the common elements among consecutive submaps are required for
the information propagation. There is no need to recover the
missing cross-correlation terms among submaps, although
could potentially be recovered to get the full map in case
of needed. Figure 1(b) shows schematically the elements of
the covariance matrix split into submaps and the information
flow for the backpropagation process.
In summary, the general algorithm for fusion and backpropagation is done sequentially from the last submap to the
first submap. Therefore, for the last submap sr the information from common and non-common measurements is fused
to the prior. Then for each si−1 , i = r, r − 1, . . . , 2, only the
non-common measurements are fused and the information
from si is backpropagated to update the non-common part.
VII. APPLICATION TO THICKNESS MAPPING
This section presents how to apply the approach described
in the above sections to pipe thickness mapping for assessing
the condition on metal water pipes.
A. Sensor information
Typical non-intrusive sensors used in assessing the condition of metal pipes are electromagnetic sensors that measure
variations in the electromagnetic field or Eddy-currents produced by the field, acoustic sensors that measure the timeof-flight of sound wave propagation, amongst others. 3D
range sensors such as laser scanners can be also employed to
measure thickness by processing the 3D profile of the pipe
sections. The use of these sensors depends on the material
and whether the corrosion or thinning can be exposed (e.g.
grit-blasting the pipe). Accurate, high-resolution, estimate of
the thickness can be obtained from these intrusive sensors.
High-resolution 2.5D thickness maps generated from 3D
lasers are less common as they required intrusion and
elaborated data processing, while electromagnetic sensors
information can be readily obtained. In our experiments,
data from a 3D-laser scanner is employed to generate highresolution pipe thickness measurements [17]. We also have
available thickness measurements (with varying uncertainty)
from a pulsed-Eddy current sensor [18]. Thus we aim to
integrate high resolution information from a 3D-laser scanner
(abbr. HR sensor) and the low resolution information from
Eddy currents sensors (abbr. LR sensor) to obtain highresolution thickness maps. These two heterogeneous sensors’
measurements correspond to data-sets Ψ1 and Ψ2 .
All sensors’ measurements were taken from a real castiron pipe section of 1000 mm length and 2087.5 mm
circumference, the HR sensor and LR sensor cover the full
area with a 2D grid of 258 × 125 and 42 × 20 respectively.
Note that the Cylindrical coordinates of the pipe are mapped
into Cartesian coordinates to represent the 2.5D plots that
will be presented in the results, where longitudinal axis is
on the horizontal axis, the circumferential is shown on the
vertical axis and thickness is shown in colour.
B. Prior thickness map
In theory, either sensor data could be used to train the GP
model that learns the spatial correlation. In practice, once
the model has been learned at any resolution, GPs are also
useful to increase or decrease the resolution of a given dataset just by inferring more or less X ∗ points, as they provide
a continuous representation. Therefore, for the results we
(a) HR sensor
(b) LR sensor inference
(c) CI-fusion be- (d) CI-fusion after
fore bp
bp
(e) naı̈ve fusion
Fig. 2: Thickness maps (mean) at high resolution.
present here at high-resolution fusion, the prior is learned
from the LR sensor data-set and fuse at high resolution with
the HR sensor data-set and vice-versa.
As stated in Section IV, the types of mean and covariance
function should be specified first to train the GPs. We use
a zero-mean mean function after normalising the data. And
we chose the Matérn covariance function [8],
!ν
!
√
√
1−ν
2νd
2νd
∗
22
KMatern (X, X ) = σ f
Kν
, (9)
Γ(ν)
l
l
which depends on data only through the distance d between
input locations. d is the shortest path on the cylinder surface
between points, Γ is the Gamma function and Kν (.) is a
modified Bessel function with the order being the differentiability parameter ν > 0. We set ν = 3/2. The length-scale l
and the signal variance σ f are the hyper-parameters θ , which
is first initialised and then optimised by maximising (1). Note
that other covariance functions, including non-stationary
functions [19], could be also used for this application.
As to begin with we want to target high-resolution maps,
the LR sensor (i.e. 42 × 20 grid) is used to train the GPs
µ and
model and inference is used to obtain the prior map (µ
P) at 258 × 125 resolution. As stated above, the LR sensor
noise is variable, i.e. the elements in Q are different (see
Figure 5(a)). Following the methodology in Section IV, Q is
passed directly to (1) to get θ .
After training, inference takes place to obtain p(ξξ |X ∗ ),
where X ∗ are locations at the desired resolution. µ and P
are calculated following (2) and (3). Figures 2(b) and 3(b)
show the mean and variances for prior map p(ξξ |X ∗ ). As
shown in Figure 2(b), this prior is very smooth as the GP
inference passes from low resolution to high resolution.
(a) HR sensor
(b) LR sensor inference
(c) CI-fusion be- (d) CI-fusion after
fore bp
bp
(e) naı̈ve fusion
Fig. 3: 2σ uncertainty maps at high resolution.
C. CI Submapping and Bayesian fusion
Once the prior map is obtained at the desired resolution
for fusion, we use the other source of data Ψ2 from the HR
sensor, which contains measurements from a 258 × 125 grid
with some patches of missing data, to update the prior. The
HR sensor noise variance R is set to a constant value of σ = 3
mm except for the missing data parts (see Figure 3(d)). Note
that the missing data variance is set to be very high as should
not be trusted during the fusion process. Another solution is
to ignore the missing data during fusion. These two methods
give similar fusion results.
Each prior submap is fused with the corresponding measurements from Ψ2 as described in Section V. The size of all
submaps is chosen to be equal and it is decided by looking
at the values of the cross-correlations. The main idea is to
minimise the cross-correlation that is left out by counting the
number of columns until it is close to zero. In this case, each
submap contains 20 columns (5160 elements per submap).
Figures 2(c) and 3(c) show the resulting estimate right after
the fusion, before the backpropagation. From these figures
the boundaries between submaps are clear.
D. Back-propagation results
We use the algorithm described in Section VI, to update
mean and variances of the estimate for all the submaps,
except for the last one as explained before.
The results of the fusion after backpropagation are shown
in Figures 2(d) and 3(d) for mean and variances respectively.
As figures show there is a clear improvement compared with
the fusion before backpropagation, where information from
common part has not been taken into account.
(a) LR sensor
(b) HR sensor inference
(d) CI-fusion be- (e) CI-fusion affore bp
ter bp
(c) correlated fusion
(f) naı̈ve fusion
Fig. 4: Thickness maps (mean) at low resolution.
(a) LR sensor
(b) HR sensor inference
(d) CI-fusion be- (e) CI-fusion after
fore bp
bp
(f) naı̈ve fusion
Fig. 5: 2σ uncertainty maps at low resolution.
𝒔𝟏
E. Evaluation results
The proposed method is compared with the “naı̈ve fusion”
method, which applies the maximum a posteriori estimator
to the uncorrelated prior (diagonal covariance matrix) [2].
Results for mean and variance using naı̈ve fusion are shown
in Figures 2(e) and 3(e) respectively. The missing data in
HR sensor’s data (the blank area in Figure 2(a) and dark
blue area in Figure 3(a)) is used to highlight the effectiveness
of the proposed method to handle data incompleteness. Our
approach (CI-fusion) shows a clear improvement with respect
to naı̈ve fusion, particularly noticeable, in these areas of
missing data. The resultant posterior mean using CI-fusion
(Figure 2(d)), shows also an improvement from the fusion
using the naı̈ve approach when looking at the areas with
lower thickness, where the detail is not lost. This is crucial
when assessing pipe condition, where low thickness areas are
the main focus. CI-fusion also produces lower uncertainties
than naı̈ve fusion therefore is more accurate, without being
optimistic, as we will see in the following experiment.
In order to show the performance of the proposed approach
with respect to the optimal solution, we would need to
show results compared with the optimal correlated fusion, i.e.
Bayesian fusion without submapping, which integrates all the
measurements into the global prior in one step as proposed
in [2]. However, the fully correlated fusion approach could
not be implemented at high-resolution (258 × 125) because
the covariance matrix is too large (32250 × 32250) and its
inversion is too costly. Therefore, to do a proper quantitative
evaluation we will show results from the same data-sets but,
this time, fusing at a low resolution (42 × 20), where the
covariance matrix is only 840 × 840.
(c) correlated fusion
all
correlation
used
𝒔𝟐
(a)
(b)
𝒔𝟏
correlation
not used
𝒔𝟐
(c)
(d)
Fig. 6: Covariance matrices: prior (a) and posterior (b) covariance
of full correlated fusion, (c) prior covariance of CI submaps, (d)
posterior covariance of CI-fusion after backpropagation.
For this experiment we use the HR sensor to learn the
spatial correlation and to produce the prior map. Then, LR
sensor data is fused into this prior. Results comparing the
three methods are shown in Figures 4 and 5.
Image plots of the CI-fusion covariance matrix compared
with the optimal correlated fusion covariance matrix are
shown in Figure 6. The correlated prior P inferred from the
GPs is shown in Figure 6(a). As expected is almost block
diagonal, showing that the each position is only correlated
with its close neighbours. The equivalent P once the prior is
split into CI submaps is shown in Figure 6(c). Figure 6(b)
and 6(d) show the updated covariance after the optimal correlated fusion and after CI-fusion including backpropagation
learning and inference have been suitably altered to handle
variable characterised sensor noises.
The proposed framework has been applied to the novel
application of 2.5D mapping of pipe thickness, with experimental results demonstrating the validity of the proposed
methodology in generating more accurate thickness maps
than comparable strategies. Current work is considering
building an optimal submap for the initialisation update step
without the need to resort to approximations.
(a)
(b)
Fig. 7: Fusion errors with respect to the fully correlated fusion:
(a) variance and (b) mean values derived from the naı̈ve (top) and
CI-fusion (below).
respectively. Notice that the approximation considered by CIfusion seems negligible after the backpropagation.
Figure 5 and the variance differences depicted in Figure 7
show that the uncertainty of the CI-fusion method is slightly
higher than the optimal estimate, thus proving that the
approximation of the proposed method is conservative.
The root mean squared error (rmse in mm) for the lowresolution experiment, using the optimal solution as the baseline, is 1.6411, 0.2811, 0.2767, 1.1133 for LR sensor, HR
sensor, CI-fusion and naivı̈ve fusion respectively. Please note
that missing data is not being considered in this computation.
It is shown how the CI-fusion is on average closest to
the optimal estimate. Figure 7 plots how close the actual
thickness values derived from the CI-fusion approximation
solution are to their fully correlated fusion counterparts.
The computational complexity of the correlated fusion
is O(n3 ), caused by the inversion of the whole covariance
matrix. For CI-fusion as the size of each submap is constant
the process of updating them is O(1) per step and O(n) for
the complete update, where n is the chosen resolution.
VIII. CONCLUSION
This paper presents a generic framework to integrate heterogeneous sources of information into a single probabilistic
2.5D map. The technique is notable in being able to deal with
large-scale data and producing (arbitrary) high resolution
maps. The uncertainty in the final estimate is also provided
to show the increased reliability of the proposed fusion
mechanism. Crucially, the technique can handle noisy data
from multiple differing sources, and is able to fuse it within
a robust common probabilistic representation.
Gaussian Processes are first used to incorporate correlations between neighbouring points in the map. These
dependencies are commonly disregarded by standard 2.5D
mapping techniques. The GP model learned from one data
source is used as prior for standard Bayesian fusion. For
large-scale data, a submapping technique borrowed from the
SLAM literature is introduced in order to avoid the high
computational complexity encountered by the inversion of
a large covariance matrix at the fusion step. Moreover, GP
ACKNOWLEDGMENT
This work is an outcome from the Critical Pipes Project
funded by Sydney Water, Water Research Foundation USA,
Melbourne Water, Water Corporation(WA), UK Water Industry Research Ltd, South Australia Water, South East Water,
Hunter Water, City West Water, Monash University, University of Technology Sydney and University of Newcastle.
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