Sensor Adaptation and Development in Robots by Entropy
Maximization of Sensory Data∗
Lars Olsson∗ , Chrystopher L. Nehaniv∗† , Daniel Polani∗†
Adaptive Systems∗ and Algorithms† Research Groups
School of Computer Science
University of Hertfordshire
College Lane, Hatfield Herts AL10 9AB
United Kingdom
{L.A.Olsson, C.L.Nehaniv, D.Polani}@herts.ac.uk
Abstract— A method is presented for adapting the sensors of a robot to the statistical structure of its current
environment. This enables the robot to compress incoming
sensory information and to find informational relationships
between sensors. The method is applied to creating sensoritopic maps of the informational relationships of the sensors
of a developing robot, where the informational distance
between sensors is computed using information theory and
adaptive binning. The adaptive binning method constantly
estimates the probability distribution of the latest inputs
to maximize the entropy in each individual sensor, while
conserving the correlations between different sensors. Results
from simulations and robotic experiments with visual sensors
show how adaptive binning of the sensory data helps the
system to discover structure not found by ordinary binning.
This enables the developing perceptual system of the robot to
be more adapted to the particular embodiment of the robot
and the environment.
Index Terms— Ontogenetic robotics, sensory systems, entropy maximization
I. I NTRODUCTION
One of the major tasks of many sensory processing system
is compression of incoming sensory signals to representations more suitable to compute the specific quantities
needed for that specific animal or robot to function in
the world. It is believed that in many animals the functionality of the sensory organs and nervous system is
almost completely innate, while in others it develops during
the lifetime of the individual [6]. This development and
adaptation is in part dependent on the structure of the
incoming sensory signals, and there are also indications
that individual neurons adapt to the statistical structure
of their incoming sensory signals. This paper presents a
robotic system that constantly adapts its visual sensors
to the statistical structure of its environment by entropy
maximization of the incoming sensory data.
The structure of the incoming sensory signals depends on
the embodiment and actions of the agent and the environment. Research in to the structure of natural signals is still
at an early and explorative phase, but there are indications
∗ The work described in this paper was partially conducted within the
EU Integrated Project RobotCub (“Robotic Open-architecture Technology
for Cognition, Understanding, and Behaviours”) and was funded by the
European Commission through the E5 Unit (Cognition) of FP6-IST under
Contract FP6-004370.
that signals of different sensory modalities such as acoustic
waveforms, odor concentrations, and visual contrast share
some statistical properties [20]. For example, in general the
local contrast in natural images has the same exponential
form of the probability distributions as sound pressure in
musical pieces [20]. Another commonality between signals
of different modalities is coherence over time and, in many
cases, spatial coherence. Coherence between signals means
that one part of the signal can be predicted by another part
of the signal. In other words, natural signals contain some
redundancy. For example, consider nearby photoreceptors,
which usually sample regions of visual space close to each
other. Thus, nearby photoreceptors often sample the same
object in natural scenes which usually is coherent in respect
to colour, orientation, and other parameters. Contrast this
with an image where each pixel is generated independently
from a random distribution. An image like this will contain
no redundancy.
Given the statistical structure and redundancy of natural
signals is it natural to ponder whether this structure is
exploited by animals to optimize their sensory systems.
Barlow suggested in 1961 [1] that the visual system of
animals “knows” about the structure of natural signals and
uses this knowledge to represent visual signals. Thus, the
sensory data can be represented in a more efficient way
than if no structure in the data is known. In 1981 Laughlin
recorded the distribution of contrasts as seen through a
lens with the aperture the size of a fly photoreceptor
while moving in a forest [12]. A single cell in the fly
encodes contrast variations with a graded voltage response.
The distribution of contrasts has some shape and Laughlin
was interested in whether the voltage response conveyed
the maximal amount of information given the specific
distribution of contrasts by maximizing the entropy of the
voltage distribution. This can be viewed as single neuron
cell version of Linsker’s Infomax principle [13]. Laughlin
compared the computed ideal conversion of contrast to voltage given his collected data from the forrest and found the
match to be very good with the measured response of the
second order neurons in the fly visual system. This result
suggests that the early visual system of the fly is adapted to
the statistical structure of natural scenes in its environment.
Since Laughlin’s work focused on global statistics the
adaptation must have taken place over evolutionary time.
Recent results indicate that the fly visual system also adapts
to the current conditions in much shorter timescales, on
the order of seconds or minutes [5]. This means that
individual neurons adapt their input/out relations depending
the structure of incoming signals. There is also evidence
[4] that the coding of the signals adapts to the input
distributions in the vertebrate retina.
In many robotic systems the processing of sensory data
is often, like in the visual system of organisms, limited
by the physical limits of sensation [3], memory capacity,
processing speed, heat generation, power consumption [8],
and limited bandwidth of data transfer (e.g [10]). Thus,
there is a need for robots to extract relevant information
[15], [19] from the incoming streams of sensory data and
also to represent this information as efficiently as possible.
One way information can be represented as efficiently
as possible given memory and processing constraints is
by maximization the entropy in each sensory channel as
described above in the case of the fly. This is a well
known statistical technique also known as adaptive binning
[23] or histogram or sampling equalization [14]. This
maximization also enables the robot to find more structure
in ensembles of sensors since the maxmization conserves
the correlations between sensors while more resolution is
achieved in the parts of the input distribution where most of
the data is located. Entropy maximization is also closely
related to Linsker’s Infomax principle [13] and Becker’s
mutual information maximization method [2] that models
cortical self-organisation.
This paper a describes a sensory system that maximizes
the information a robot with limited computational resources can have about the world. The sensory system is
constantly adapting to the structure of its current environment using entropy maximization of each sensor using a
sliding window mechanism. We show in simulation how
an agent using this method can find informational relationships in the sensory data using the sensory reconstruction
method [17] not found by a non-adapting system using
the twice the amount of memory to represent the data.
We also present results from experiments using a SONY
AIBO1 robot. The results show how the visual signals
in different natural environments have different statistics
and how the adaptive binning method helps the developing
robot to reconstruct its visual field.
The structure of the rest of this paper is as follows. The
next section describes the idea of entropy maximization
and the information theory background. In section III the
sensory reconstruction method is described and section
IV presents the performed simulations and robotic experiments. Finally, section V concludes and points out some
possible future areas of research.
II. E NTROPY M AXIMIZATION OF S ENSORY DATA
To get a better understanding of entropy maximization,
this section contains a short introduction to the general
1 AIBO
is a registered trademark of SONY Corporation.
concepts of entropy and information theory [21]. Then
entropy maximization is introduced and exemplified.
A. Information Theory
Let X be the alphabet of values of a discrete random
variable (information source, in this paper a sensor) X with
a probability mass function p(x), where x ∈ X . Then the
entropy, or uncertainty associated with X is
X
p(x) log2 p(x)
(1)
H(X) = −
x∈X
and the conditional entropy
XX
H(Y |X) = −
p(x, y) log2 p(y|x)
(2)
x∈X y∈Y
is the uncertainty associated with the discrete random
variable Y if we know the value of X. In other words,
how much more information do we need to fully predict
Y once we know X.
The mutual information is the information shared between the two random variables X and Y and is defined
as
I(X; Y ) = H(X) − H(X|Y ) = H(Y ) − H(Y |X). (3)
To measure the dissimilarity in the information in two
sources Crutchfield’s information distance [7] can be used.
The information metric is the sum of two conditional
entropies, or formally
d(X, Y ) = H(X|Y ) + H(Y |X).
(4)
Note that X and Y in our system are information sources
whose H(Y |X) and H(X|Y ) are estimated from the time
series of two sensors using (2).
B. Entropy Maximization
Due to memory and processing constraints, as well as to
simplify learning, it is often desirable to compress incoming sensory data. One common method to achieve this is
binning, whereby the range of incoming data is mapped to
a smaller number of values using a transfer function. For
example, consider the grey-scale image depicting Claude
Shannon in fig. 1(a) where each pixel can have a value
between 0 and 255. How could this image be compressed
if only 5 different pixel values were allowed? Maybe the
first method that comes to mind is to divide the range
{0, 1, . . . , 255} in to 5 bins of size 51, where all values
between 0 and 50 would be encoded as 0, 51 to 102 as
1, and so forth. This method, which does not take in to
account the statistics of the data, is called uniform binning,
and the corresponding image is shown in fig. 1(c). As seen
in fig. 1(d) the distribution of grey-scales in fig. 1(a) is not
uniform, with most pixels in the range {100, 101, . . . , 200}.
The entropy of the encoded image is ≈ 1.97, which is
less than the maximal theoretical entropy of log 2 5 ≈ 2.32.
From an information theoretical point of view this means
that this encoding is non-optimal since the entropy of the
encoded image is less than the maximal possible entropy
of the image. Now, consider fig. 1(e) which also uses 5
bins, where (at least if studied from a distance) the image
seems to convey more detail about the original image. Here
the original values have been binned in such a way that
each bin contains approximately the same number of pixels,
which means that the entropy of the image is close to the
maximum of log2 5 ≈ 2.32. This can also be considered as
adding more resolution where most of the data is located.
As discussed in the introduction, it has been found that
the contrast encoding in the visual system of the fly is
adapted to the specific statistics of it environment [12].
This basically means that, just as in the image of Claude
Shannon above, the entropy of the graded response is
maximized. More formally, given a sensor X we want
to find a partitioning of the data in to the N bins of the
alphabet X such that each bin is equally likely. That is,
P (X = c) =
1
(∀c ∈ X )
|X |
40
30
25
20
15
10
5
0
(a) No binning
III. S ENSORY R ECONSTRUCTION METHOD
In the sensory reconstruction method [18], [17] sensoritopic maps are created that show the informational
relationships between sensors, where sensors that are informationally related are close to each other in the maps.
The sensoritopic maps might also reflect the real physical
relations and positions of sensors. For example, if each
pixel of camera is considered a sensor, it is possible to
reconstruct the organization of these sensors even though
nothing about their positions is known. It is important to
note that using only the sensory reconstruction method,
only the positional relations between sensors can be found,
and not the real physical orientation of the visual layout. To
do this requires higher level feature processing and world
knowledge or knowledge about the movement of the agent
[17].
To create a sensoritopic map the value for each sensor
at each time step is saved, where in this paper each sensor
is a specific pixel in an image captured by the robot. A
number of frames of sensory data are captured from the
robot and each frame is one time step. The first step in the
method is to compute the distances between each pair of
0
50
100
150
200
250
(b) No binning - histogram
1200
line 1
1000
(5)
which means that the entropy, H(X) is equal to the
maximal capacity C of the sensor.
The experiments performed by Brenner et al. [5] also
indicates that this kind of entropy maximization constantly
is happening in the motion sensitive neurons of the fly.
This can be implemented by estimating the probability
distribution each time step of the most recent T time
steps and changing the transfer function accordingly. In the
experiments performed in this paper we have implemented
this algorithm using histogram estimators to estimate the
probability distributions. In our implementation all instances of the same value are added to the same bin, which
explains why the distribution in fig. 1(f) is not completely
uniform. The sliding window in our implementation does
not use decay, which means that more recent values do
not affect the distribution more then older ones within the
window.
line 1
35
800
600
400
200
0
(c) 5 uniform bins
0
50
100
150
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300
(d) 5 uniform bins - histogram
pixels per bin
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400
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100
0
(e) 5 adaptive bins
0
50
100
150
200
250
300
(f) 5 adaptive bins - histogram
Fig. 1. Example of adaptive binning. fig. 1(a) shows a 50 by 50 pixels
grey-scale image of the founder of information theory, Claude Shannon,
and fig. 1(b) the corresponding histogram of pixels between 0 and 255.
fig. 1(c) shows the same picture where the pixel data (0-255) is binned
in to only 5 different pixel values using uniform binning and fig. 1(d) the
frequency of pixel values. Finally, fig. 1(e) shows the same picture with
the pixel data binned in to 5 bins using adaptive binning and fig. 1(f) the
corresponding histogram. The entropy of the normal binning distribution
is ≈ 1.97 while the entropy for the adaptive binning distribution is close to
the theoretical maximum of log2 5 ≈ 2.32. The adaptive binning (entropy
maximization) increases the resolution where most of the data is located.
(Best viewed at a distance.)
sensors. This is computed by considering the time series
of sensor values from a particular sensor as an information
source X. The distance between two sensors X and Y is
then computed using the information metric, equation II-A.
From this 2-dimensional distance matrix a 2-dimensional
sensoritopic map can be created using a number of different
methods such as metric-scaling [11] which positions the
sensors in the two dimensions of the metric projection.
2500
0
1000
Frequency
3000
1500
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0
50
100
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250
0
50
Pixel value
100
150
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250
Pixel value
(b) Green histogram 6000
time steps
20
0
10
Frequency
30
(a) Red histogram 6000 time
steps
0
On a 500 x 350 pixel environment (see fig. 2) a 8 x 8
pixel agent represented as square moves a maximum of one
pixel per time step in the x-direction and a maximum of 1
pixel in the y-direction. Hence dx and dy ∈ {−1, 0, 1},
but both cannot be 0 at the same time. Each time step there
is a 15% probability that either dx or dy, or both, change
value by -1 or 1. Every pixel of the agent has 4 sensors,
one for the red intensity, one for the green, one for the blue,
and one for the average intensity of that pixel. Thus, the
agent has a total of 256 sensors with Rn , Gn , Bn , In , 1 ≤
n ≤ 64, located in the same position. For each time step
the values of all the 256 sensors are used as the input to
the sensory reconstruction method.
15000
A. Simulation
Frequency
IV. E XPERIMENTS
Frequency
In our experiments we have used the relaxation algorithm
described in [18].
0
50
100
150
200
250
0
50
Pixel value
100
150
200
250
Pixel value
(d) Red histogram 10 time
steps
fig. 3 shows the histograms of all sensors of each type
accumulated over the whole simulation of 6000 time steps,
and also examples of histograms for each sensor type over
10 consecutive time steps. The red and green sensors are
quite uniformly distributed over almost the whole range
while the blue has a high peak at 0. In fig. 3(d) to 3(f)
we can see that the ranges of values in the red and green
sensors are greater than in the blue sensors during these 10
time steps, something that was true for most frames.
Given the structure of the input data it is expected
that adaptive binning with a sliding window would be
advantageous for the sensory reconstruction method. fig.
4 shows results of the simulation. In fig. 4(a) the input
to the sensory reconstruction method is sensory data from
the 256 sensors partitioned in to 16 uniform bins (4 bits
per sensor). The graph shows that some structure is found
and some sensors that are closely positioned in the agent
are close in the sensoritopic map. One exception is the
blue sensors, B1 to B64 , all located to the left. Clearly,
if all the informational structure could be found the map
should correspond to the physical order and, for example,
R1 should be close to B1 .
6
4
2
Frequency
30
10
0
Fig. 2. The environment where the agent is moving. The image depicts
autumn leaves and has higher variation in the red and green channels than
the blue channel.
0
Frequency
8
(c) Blue histogram 6000
time steps
0
50
100
150
200
250
Pixel value
(e) Green Histogram 10 time
steps
0
50
100
150
200
250
Pixel value
(f) Blue Histogram 10 time
steps
Fig. 3. Figures 3(a), 3(b), and 3(c) shows histograms of red, green, and
blue sensors from the image in fig. 2 collected from 6000 timesteps of
movement from all sensors. Figures 3(d), 3(e), and 3(f) show examples
of histograms from 10 consecutive time steps.
Now consider fig. 4(b). Here the input data was binned
into only 4 bins (2 bits per sensor) using entropy maximization with a sliding window of size 100. Here the sensoritopic map clearly shows the informational and physical
relationships between the sensors, where sensors that are
closely located in the layout of the agent are clustered in the
map. This means that the real physical layout of the sensors
has been recovered from the raw input data, something that
the same method failed to do using uniform binning and
double the amount of resolution.
−1
0
1
2
1000
0
400
Frequency
800
400
Frequency
200
250
0
50
100
Pixel value
150
200
250
Pixel value
3
V1
(a) Red histogram
(b) Adaptive binning - 4 bins
Frequency
Fig. 4. Sensoritopic maps of the sensory data where R1-64 is the red
channel sensors, G1-64 the green, B1-64 the blue, and I1-64 the intensity
sensors . In fig. 4(a) uniform binning is used with 16 bins. fig. 4(b)
shows the a sensoritopic map using the same input data using entropy
maximization of each sensor with only 4 bins. Notice that in the entropy
maximization case the physical order, e.g. B1 is close to R1, is found,
while in the uniform binning case with 16 bins it is not. Thus, the 2 bits
of information in each sensor in the entropy maximization case manages
to find correlations not found by the 4 bits per sensor in the uniform
binning case.
(b) Green histogram
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V2
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−1
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500
(a) Uniform binning - 16 bins
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Frequency
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B5 R6B6 R7G7I7B7 R8
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I32R32B32
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B26 G26
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R1 R2 R3 R4 R5 R6
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G1R17 G2 G3R11G4R12G5 I6 G6 I7G7 I8 R7 R8
R13 R14
I5
I1 G9 I2G10 I3 I4
G15I16R15
G16R16
R18 G11 G12 G13 I14G14 I15
R25I9G17
I13
I10
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B. Robotic Experiments
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B13
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(d) Intensity histogram
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Histograms of red, greeen, blue, and intensity sensors.
1.5
Fig. 5.
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Pixel value
(c) Blue histogram
0.0
The robotic experiments were performed with a SONY
AIBO robot wandering in an office environment with both
artificial lights and two windows. Images of size 88 by 72
pixels from the robot’s camera were saved over a dedicated
wireless network with an average frame rate of 15 frames
per second. The images were transformed to 8 by 8 pixel
images by either pixelation with averaging or by using only
8 by 8 pixel from the centre of each image. Either transformation produced similar results in subsequent experiments.
Each pixel has three sensors, Rn , Gn , Bn , 1 ≤ n ≤ 64,
with R1 in the upper left corner and R64 in lower right
corner. The robot performed a simple exploring behaviour
with obstacle avoidance walking around in the office.
fig. 5 shows histograms of all sensors of each type
(red, green, blue, intensity) combined over 1000 time
steps where each sensor can have a value in the range
{0, 1, . . . , 255}. The red and green sensors have most
values between 0 and 170 with two clear peaks at roughly
70 and 150. The blue sensors had a narrower range, with
most sensor values between 0 and 80, and two narrow peaks
at roughly 25 and 75. The peaks are due to the windows;
when walking towards the windows the ambient light is
brighter. Similarly to the simulation above, the histograms
in any given frame show a narrower range of the data for
the blue sensors. Thus, it is expected that the blue sensors
are more difficult to reconstruct using uniform binning.
As seen in fig. 6 this is the case. fig. 6(a) shows a
sensoritopic map of the blue sensors constructed from 16
uniform bins. Contrast this with fig. 6(b) using only 6
adaptive bins where the organisation of the visual field has
been reconstructed. In fig. 7 sensoritopic maps of all the
red, green and blue sensors combined are shown. Here we
can again see how the adaptive binning enables the sensory
reconstruction method to find the positional relations of
150
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B52
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B51
B58
B50
B42
B49
B41
2
B57
3
V1
(a) Uniform binning - 16 bins
(b) Adaptive binning - 6 bins
Fig. 6. Sensoritopic maps of blue sensors. fig. 6(a) shows the sensoritopic
map created with 16 uniform bins and fig. 6(b) with six adaptive bins using
a sliding window.
the sensors of the different types of sensors, where sensor
from the same physical position, e.g. R8 , G8 , and B8 ,
are clustered together. We can also see that the order with
R1 at the opposite corner of R64 has been found. In the
sensoritopic map in fig. 7(a) created using 16 uniform bins
we see that the blue sensors are separated from the red and
green. The structure of the red and green sensors are also
less clear compared to fig. 7(b).
V. C ONCLUSIONS
This short paper has discussed entropy maximization of
sensory data in the fly visual system and how a similar
system can be implemented in a robot. The system constantly adapts the input/output mapping of sensory data
−2
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R5
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0
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V1
(b) Adaptive binning - 6 bins
Fig. 7. Combined sensoritopic maps of red (R), green (G), and blue (B)
sensors. In fig. 7(a) the map is created with 16 uniform bins and in fig.
7(b) with six adaptive binning using a sliding window.
by estimating the distribution of input data and adapts the
output distribution by entropy maximization of the data.
This mapping of input/output data compresses the data
while maintaining correlations between sensors. Results
from simulation show how an agent using this adaptive
technique can reconstruct its visual field with a resolution
of only two bits per sensor using the sensory reconstruction
method. Using four bits per sensor and uniform binning
the sensory reconstruction method failed to reconstruct the
visual field. This result indicates that adaptive binning is
useful for compressing sensory data and to find correlations
between sensors. Results from experiments with a SONY
AIBO robot show some statistical properties of different
indoor environments and how adapting to this structure
helps the robot find structure in the sensory data.
Adaptation of sensory systems to the specific environment of a particular species is also studied in the field
of sensory ecology [9]. Here many results also seem to
indicate that sensory systems evolve to be “tuned” to
the average statistical structure of the environment. For
example, it has been found that the spectral sensitivity
of many unrelated fishes has converged to similar patterns
depending on the water colour and ambient light.
It is interesting to note how the sensory reconstruction
method manages to merge sensors of different types such
as red, blue, and green light sensors and reconstruct their
their visual layout without any knowledge of their physical
structure as seen in fig 7(b). This is an example of autonomous sensory fusion. Maybe the most studied example
of this in neuroscience is the optic tectum of the rattlesnake,
where nerves from heat-sensitive organs are combined with
nerves from the eyes [16]. This kind of multimodal sensor
integration is something that will be studied in future work.
This paper has presented initial work and there are many
possibilities for developing and extending these ideas. For
example, should the sliding window use some kind of decaying function and how long should the window be? One
more important issue is on what level entropy maximization
is the most effective. For example, is entropy maximization
more effective for the agent on sets of sensors or a more
abstract object level than the single sensor level? In this
case, how are these sets or objects selected? These are
some questions that can be addressed in future work. We
are also interested in robots that develop and learn over
time and adapt to their particular environment and tasks.
Results indicate that constraints on perception during development may improve the perceptual efficiency by reducing
the information complexity [22]. In this context adaptive
binning can be applied to a developmental system starting
with low resolution, where more resolution is added using
adaptive binning as the robot develops and learns about its
particular environment.
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