This article proposes an approach for determination of radiation dose pro le in a radiation-susceptible

environment, aiming to guide an autonomous robot in acting on those environments, reducing the human

exposure to dangerous amount of dose. The approach consists of an active learning method based on

information entropy reduction, using log-normally warped Gaussian Process (GP) as surrogate model,

resulting in non-linear online regression with sequential measurements. Experiments with simulated

radiation dose  elds of varying complexity were made, and results showed that the approach was e ective

in reconstruct the  eld with high accuracy, through relatively few measurements. The technique was

also shown some robustness in presence measurement noise, present in real measurements, by assuming

Gaussian noise

Pereira, Claudio Márcio Nacimento Abreu

Carpe dIEN

2017 International Nuclear Atlantic Conference - INAC 2017
Belo Horizonte, MG, Brazil, October 22-27, 2017
ASSOCIAC¸A˜O BRASILEIRA DE ENERGIA NUCLEAR - ABEN
FIELD DOSE RADIATION DETERMINATION BY ACTIVE LEARNING
WITH GAUSSIAN PROCESS FOR AUTONOMOUS ROBOT GUIDING
de Freitas Naiff, Danilo1, Paulo R. Silveira2 and Pereira, Claudio M. N. A.3
1Universidade Federal do Rio de Janeiro - PEN/COPPE/UFRJ
Ilha do Fundo, s/n
21975-970, Rio de Janeiro, RJ
danilonaiff1992@poli.ufrj.br
2Universidade Federal do Rio de Janeiro - PEN/COPPE/UFRJ
Ilha do Fundo, s/n
21975-970, Rio de Janeiro, RJ
paulo@lmp.ufrj.br
3Instituto de Engenharia Nuclear (IEN/CNEN)
R. Hlio de Almeida, 75 - Ilha do Fundo
21941-614, Rio de Janeiro, RJ
cmnap@ien.gov.br
ABSTRACT
This article proposes an approach for determination of radiation dose profile in a radiation-susceptible
environment, aiming to guide an autonomous robot in acting on those environments, reducing the human
exposure to dangerous amount of dose. The approach consists of an active learning method based on
information entropy reduction, using log-normally warped Gaussian Process (GP) as surrogate model,
resulting in non-linear online regression with sequential measurements. Experiments with simulated
radiation dose fields of varying complexity were made, and results showed that the approach was effective
in reconstruct the field with high accuracy, through relatively few measurements. The technique was
also shown some robustness in presence measurement noise, present in real measurements, by assuming
Gaussian noise.
1. INTRODUCTION
The reconstruction of radiation dose profile in a spatial environment is a important task
given the effects that ionizing radiation has on living organisms. Due to it’s importance,
an efficient method to determine the measurement points in a environment, and to es-
timate the full profile given those measurements is of interest, in particular for guiding
an autonomous, robotic agent through the environment. The approach used in this work
uses a Gaussian Process for both estimation and determination of point measurements by
the agent. Previous works using Gaussian Process for active learning and spatial function
reconstruction includes [1, 2, 3].
2. METHODOLOGY
2.1. Mathematical Formulation
Let X ⊂ R2 be the map containing the dose radiation field, and g : X → R+ the
unknown dose radiation field function. The objective is to sequentially design a set of
n measurement points of X, getting the data Dn = {(xi, zi)}i=1:n, such as, with those
measures, we can estimate a regression function function g∗ : X → R+ such as g∗ is as
close as possible from g. We assume further that for each xi, zi is a noisy measure of xi,
given by:
zi = g(xi)(1 + ) (1)
Here  ∼ N (0, σ2e) is a normally distributed random variable.
The regression function g∗ is constructed with the use of a Gaussian Process (GP) over
the logarithm of g. The Gaussian Process is a regression method that considers a function
f : X → R to be a random function from the space of functions from X to R, where this
random function has the property that, for each finite subset of points {xi}i=1:m ⊂ X,
{f(xi)}i=1:m follows a multivariate Gaussian distribution [4] Thus, the GP is completely
specified by it’s mean function m(x) := E[f(x)], usually and in this work assumed to be
zero, and it’s covariance function k(x,x′) := cov(f(x), f(x
′
)), which defines the structure
of the GP. In this work, the covariance function used is the isotropic Matrn covariance
function with parameter ν = 3/2:
k(x,x′) = C3/2(x,x′) = θ20 exp
(
−1
l
√
3r
)(
1 +
1
l
√
3r
)
(2)
Where r = ||x−x′||. The hyperparameters θ20 and l are called respectively the amplitude
parameter and the shape parameter.
With f distributed according to a GP with mean m(x) = 0, and Cn = {(xi, yi)}i=1:n
being a previously acquired data, with yi = f(xi) + ,  ∼ N (0, σ2e), we have that f(x)
is a normally distributed random variable such as f(x) ∼ p(f(x)|Cn) = N (µn(x), σ2n(x)),
where:
µn(x) = k(x)
T (K + σ2eI)
−1y
σ2n(x) = k(x,x) + k(x)
T (K + σ2eI)
−1k(x)
(3)
Where k(x) := (k(x1,x), . . . , k(xn,x))
T is the covariance vector between x e x1, . . . ,xn,
y := (y1, . . . , yn)
T is the observed values vector, and K = (Kij) ∈ Rn×n, Kij := k(xi,xj)
is the covariance matrix between x1, . . . ,xn.
By letting f = log(g), and yi = log(zi) = log(g(xi)(1 + )) ≈ log(g(xi)) + , we can
apply the GP over the logarithm of g. This way, given Dn as defined above, we have
that g(x) is a log-normally distributed random variable such as g(x) ∼ p(g(x)|Dn) =
Lognormal(µn(x), σ
2
n(x)), where µn and σ
2
n are as defined above.
The hyperparameters θ0 and l, collectively called Θ, can be optimized by maximizing the
INAC 2017, Belo Horizonte, MG, Brazil.
log-likelihood of Dn [5]:
L = log(p(Dn|Θ)) =− 1
2
log det(K + σ2eI)−
1
2
yT (K + σ2eI)
−1y
− n
2
log(2pi)−
n∑
i=1
log zi
(4)
The regression function g∗(x) of g(x) is given by a point measure of p(g(x)|Dn). In
this work we use the median, minimizing the expected absolute error, so we have that
g∗(x) = exp(µn(x)).
The GP can also be used to sequentially determine the set of points to be measure. This
practice of autonomous determination of learning points is also known as active learning
[6]. Given Dn, we want to choose a point xn+1 so the expected data Dn+1 = Dn∪{(x, y)}
can improve our knowledge of g as much as possible. One criteria that can be used
is based on a information entropy criterion, in which xn+1 is chosen so that g(xn+1)
has the maximal differential entropy [1]. Since g(x) is log-normally distributed, this
means that we want to maximize h(g(x)) = µn(x)+
1
2
log(2pieσ2n(x)), then getting xn+1 as
arg maxx αn(x) := h(g(x)). Thus function h(g(x)) has to be optimized for getting xn+1.
The function αn is also called the acquisition function.
2.2. Algorithm
Using the above method, it is designed algorithm for autonomous determination of point
measurements of dose radiation in a known map, with the objective of reconstructing the
field of dose radiation. In this algorithm we manually set the hyperparameters optimiza-
tion frequency Nopt, the initial hyperparameters θ0 and l for the first Nopt measurements,
the initial position of the agent x0, and a stopping criterion, in this work being a total
number N of measurements. The pseudocode of the algorithm is shown in figure 1.
GP← initialize GP
x← x0
z ← MEASURE RADIATION()
GP← ADD DATA(GP,{x,log z})
counter← 1
while counter ≤ N − 1 do
αn ← GET AQUISITION FUNCTION(GP,x)
x← OPTIMIZE(αn)
y ← MEASURE RADIATION(x)
GP.ADD DATA({x,y})
counter← counter + 1
if i mod Nopt = 0 then GP← UPDATE PARAMETERS(GP)
g∗ ← GET MEDIAN(GP)
return g∗
Figure 1: Algorithm for autonomous determination of point measurements
INAC 2017, Belo Horizonte, MG, Brazil.
2.3. Simulations
The active regression method was studied in two hypothetical dose functions ga and
gb in a 20m × 20m room, which were constructed as sums of basis functions of form
φ(x, y;x0, y0, h,K) = K/((x−x0)2 +(y−y0)2 +h2), so that gp(x) = g(x, y) = ∑i φip(x, y),
with p denoting either the index a or b. The function ga is composed of 4 roughly equally
spaced and equally shaped basis functions, while gb is composed of 12 unevenly shaped
and spaced basis functions, giving it a much higher complexity than ga. The plot of both
functions are shown in figure 2.
(a) 4 basis functions (b) 12 basis functions
Figure 2: Dose functions used in simulations
For each of the dose radiation functions, two scenarios are considered, one where there is
no measurement error, (σe equals 0), and a 1% measurement error (σe equals 0.01). The
agent initial position x0 is set to 10m × 10m, so the first measurement is made in the
middle of the map, θ0 and l to 1 and 1m respectively, Nopt to 10, and N to 100. The
acquisition function was optimized using a differential evolution algorithm. Given the
stochastic nature of measurements (where σe is not 0), and of the optimization algorithm,
three trials were made for each of the cases above.
3. RESULTS
3.1. Four Basis Functions Field Case
3.1.1. Without measurement error
In scenario assuming no measurement error, the dose field reconstruction of ga was highly
accurate, with the three trials finding the four peaks within 75 evaluations, as shown in
the the error figure 3. The final mean relative error of the three trials after 100 evaluations
were around 1%.
The final field estimation for one of the trials is shown in figure 4, where is seen that the
estimation was done closely resembling the original function. The evaluation points are
shown in figure 5, where clusters of evaluation points around the peaks are seen.
INAC 2017, Belo Horizonte, MG, Brazil.
Figure 3: Evolution of the estimation mean relative error of ga in the no
measurement error case.
Figure 4: Final estimation of ga in the no measurement error case.
INAC 2017, Belo Horizonte, MG, Brazil.
Figure 5: Evaluation points for ga in the no measurement error case.
Yellower points are early evaluations, while redder points are later
evaluations.
3.1.2. With measurement error
In the scenario assuming a 1% measurement error, the reconstruction showed problems,
with only one of the three trials the agent finding all of the four peaks after 100 evaluations,
with the other two trials finding only three of the peaks, as shown in figure 6.
Figure 6: Evolution of the estimation error of ga in the 1% measurement
error case.
INAC 2017, Belo Horizonte, MG, Brazil.
The final field found in one of the failed trials is shown in figure 7, The evaluation points
order in figure 8 shows that there was indeed a region unexplored in the algorithm, where
the remaining peak was located.
Figure 7: Final estimation of ga in the 1% measurement error case.
Figure 8: Evaluation points for ga in the 1% measurement error case
INAC 2017, Belo Horizonte, MG, Brazil.
3.2. Twelve Basis Functions Field Case
3.2.1. Without measurement error
In scenario assuming no measurement error, even thought the gb function is more complex
than the previous one, the reconstruction was still accurate, as shown in figure 9, with a
final mean relative error of the three trials after 100 evaluations were around 2.5%.
Figure 9: Evolution of the estimation error for gb in the no measurement
error case
The final field estimation for one of the trials shown in figure 10 resembles closely the
original field. The evaluation points showed in figure 11 shows disperse evaluation points,
which may be due the higher complexity of the measured function, preventing formation
of clusters of measurements.
INAC 2017, Belo Horizonte, MG, Brazil.
Figure 10: Final estimation of gb in the no measurement error case
Figure 11: Evaluation points for the gb function measurement, showing more
disperse evaluation points
3.2.2. With measurement error
In the scenario assuming a 1% measurement error, although the reconstruction was flawed,
it was more accurate than in the ga function case, as shown in figure 12, resulting in a
final estimation error around 5%. The final plot of one of the trials is shown in figure
INAC 2017, Belo Horizonte, MG, Brazil.
13. This higher accuracy, compared to the ga case, be due to the scattered points of
measurements, which makes it harder to miss larger peaks.
Figure 12: Evolution of the estimation error for gb in the 1% measurement
error case
Figure 13: Final estimation for gb in the 1% measurement error case
4. CONCLUSIONS
The method for autonomous determination of point measurements and regression outlined
above fared well in both simulated situations, specially with the no measurement error
INAC 2017, Belo Horizonte, MG, Brazil.
assumption. Further tests with actual physical agents remains to be done to see how the
method fares in a real setting.
ACKNOWLEDGMENTS
This work has been possible with the support of Laborato´rio de Monitoramento de Process
(LMP) - COPPE/UFRJ.
REFERENCES
1. A. Krause, A. Singh, C. Guestrin, “Near-optimal sensor placements in Gaussian
processes: Theory, efficient algorithms and empirical studies”, J. Machine Learning
Research, 9, pp. 235-284, (2008).
2. A. Krause and C. Guestrin, “Nonmyopic active learning of Gaussian processes: an
exploration-exploitation approach”, ICML ’07: Proc. of the 24th int. conf. on Ma-
chine learning, pp. 449-456, (2007).
3. Dongbing Gu and Huosheng Hu, “Active learning of Gaussian processes for spatial
functions in mobile sensor networks”, Preprints of the 18th IFAC World Congress,
Milano (Italy), August 28 - September 2, (2011).
4. C. E. Rasmussen and C. K. I. Williams, Gaussian Processes for Machine Learning,
The MIT Press, (2006).
5. E. Snelson, C. E. Rasmussen, Z. Ghahramani, “Warped Gaussian Processes”, Ad-
vances in Neural Information Processing Systems, (2006).
6. B. Settles, “Active Learning Literature Survey”,Computer Sciences Technical Report,
1648, University of Winconsin-Madison, (2010).
INAC 2017, Belo Horizonte, MG, Brazil.


Field dose radiation determination by active learning with gaussian process for autonomous robot guiding

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Field dose radiation determination by active learning with gaussian process for autonomous robot guiding

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