Content uploaded by Sergejs Kodors
Author content
All content in this area was uploaded by Sergejs Kodors on Jun 27, 2017
Content may be subject to copyright.
Information Technology and Management Science
2012 /15
20
doi: 10.2478/v10313-012-0003-x
The Analysis of Noise Level of RGB Image
Generated Using SOM
Sergejs Kodors1, Peter Grabusts2, 1-2Rezekne Higher Education Institution
Abstract – The article discusses how to generate RGB images
with noise using Kohonen’s self-organizing map (SOM). The
article also describes the adaptation process and structure of
SOM, which can be used to generate RGB images with noise. The
authors of the article evaluate the influence of SOM parameters
(a learning coefficient, adaptation time, effective width) on the
noise level of RGB image generated using SOM. According to
these observations, the authors formulate several
recommendations how to control the noise level by adjusting
SOM parameters.
Keywords – additive noise, image generation, self-organizing
map
I. INTRODUCTION
The cross-validation method is used to determine the neural
network ability to recognize the images [2]. According to the
cross-validation method, the total array of images is divided
into two groups: a validation set and an estimation set. The
estimation set includes the images, which are used during the
adaptation process of SOM, but the validation set comprises
the images, which are not used during the adaptation process
of SOM.
The neural network ability to recognize the images is
determined as the ratio of the recognized images from the
validation set. Therefore, the larger the validation set is, the
more accurate the result is.
However, to determine the neural network ability to
recognize the images with noise, the validation set must
contain the images with noise. If there are only the images
without noise, there is need for an automatic system to
generate the images with noise.
SOM can be used to generate RGB images with noise for
the validation process.
The aim of this research is to investigate the influence of
SOM parameters (a learning coefficient, adaptation time,
effective width) on the noise level of RGB image generated
using SOM. This analysis will enable the authors of the article
to develop the recommendations on how to control the noise
level of RGB image by adjusting SOM parameters.
II. KOHONEN’S SELF-ORGANIZING MAP
SOM is a neural network with a feature to configure the
neurons so that the close neurons begin to react to the similar
signals [2], [6].
SOM consists of the following elements:
• the adaptation time – the step count of the adaptation
process;
• the SOM lattice – the structure of SOM, where the
neurons are placed in the nodes or cells. The lattice can
be one-dimensional or multidimensional;
• the winning neuron – the neuron with a minimal distance
to an input signal;
• the learning coefficient – the SOM parameter, which
determines a learning speed in every step of the
adaptation process;
• the topological neighbourhood – a mathematical
function, which determines the influence field with a
winning neuron in its centre;
• the effective width – the parameter of topological
neighbourhood, which affects the topological
neighbourhood width.
III. THE ADAPTATION PROCESS OF SELF-ORGANIZING MAP
The adaptation process of SOM is based on the competitive
learning, when the neurons compete among themselves, which
is more similar to an input signal [2]-[3], [4], [6]-[7]:
1. Initial learning coefficient η0 is inputted.
2. Adaptation time t is inputted.
3. Initial effective width δ0 is inputted.
4. Random values to the neuron weights are generated.
5. Random input signal x is generated.
6. The winning neuron i(x) is determined by Euclidean
distance (1):
j
jwxxi minarg)( , (1)
where j – the neuron, wj – j neuron weights.
7. All neurons are adapted by the formula (2):
))1(()()()1()( )(,
nwxnhnnwnw jxijjj
(2)
where n – the adaptation step, hj,i(x)(n) – the topological
neighbourhood.
8. 5–7 steps are repeated t-1 times.
To calculate the topological neighbourhood hj,i(x)(n), any
mathematical function can be used, which satisfies these
conditions [2]:
• the function is symmetric at the winning neuron;
• the function decreases, when the distance to the winning
neuron increases, but it never achieves zero.
Gaussian function satisfies these conditions (3).
)(2
exp 2
2
)(, n
d
hxij
, (3)
where d – the Euclidean distance from neuron j to the
winning neuron, δ – the effective width.
Unauthenticated
Download Date | 6/27/17 6:43 PM
Information Technology and Management Science
2012 /15
21
The learning coefficient and effective width must decrease
for the adaptation process; to satisfy this condition, it is
recommended to use the following formulas [2]:
)/exp()( 10
nn , (4)
where τ1 – some coefficient, n – the adaptation step.
)/exp()( 20
nn , (5)
where τ2 – some coefficient, n – the adaptation step.
The learning step n is calculated beginning with zero, then
the first adaptation step contains these values: η(0) = η0 and
δ(0) = δ0.
The adaptation process of SOM consists of two phases: the
organization of neurons and the stabilization process of values.
It is recommended to use particular values of parameters for
every phase [2].
The organization process of neurons:
• the learning coefficient: η0 = 0.1 and τ1 = 1000;
• the effective width has such a value that the initial
topological neighbourhood practically influences all
neurons;
• the adaptation time: 1000 steps.
The stabilization process of values:
• the learning coefficient is static and equal to 0.01;
• the effective width has such a value that the topological
neighbourhood only influences adjacent neurons;
• the adaptation time is approximately 500 times larger
than the number of neurons.
IV. RGB IMAGE GENERATION USING SELF-ORGANIZING MAP
Every synapse (weight) of a neuron can only get one value,
therefore RGB image (an input signal) must be transformed
into the one-dimensional array to obtain the association of the
neuron weights with an input signal.
The one-dimensional array size is 3·n·m, where n – the
image width, m – the image height.
Fig. 1. The association of RGB image with neurons weights
To create the association of the neuron weights with the
input signal the following concept can be used: the first n·m
elements are associated with red colour components of the
RGB image, the next n·m elements – with green colour
components, and the remaining ones – with blue colour
components. To distinguish the image rows, for every colour
group (the group size is n·m elements, where m is the number
of rows) the following concept is used: the first n elements are
associated with the first image row, the second – with the
second row and so on for each colour group. The sketch of this
association is illustrated in Fig. 1.
Fig. 2. The hexagonal SOM to generate RGB images with noise
To generate RGB images with noise, the following feature
of SOM is used: the winning neuron weights aim to be equal
to an input signal, but adjacent neuron weights aim to be equal
to the winning neuron weights. Therefore, the input signal is
selected from the array {RGB, RG, GB, RB, R, G, B}, where
the letters indicate unturned colour components of the
generated image (other elements of the image are equal to
zero).
If there is a need to generate an equal number of the images,
which are inclined to some colour, a hexagonal type of the
lattice with the predefined winning neurons can be used: the
central neuron weights are equal to the generated image colour
values (to input signal RGB), the corner neuron weights are
equal to inputs {R, RG, G, GB, B, RB}. Other neuron weights
have the random values (see Fig. 2).
Fig. 3. The coordinates of the hexagonal lattice
Than RGB images with noise are generated among the
predefined winning neurons.
The distance between two neurons of the hexagonal SOM is
calculated by the formula (6).
babad 22 , (6)
where Δa – the distance between two neurons on the
coordinate X, Δb – on the coordinate Y, and 60o between the
coordinates (see Fig. 3).
V.
ADDITIVE NOISE OF RGB IMAGE
Unauthenticated
Download Date | 6/27/17 6:43 PM
Information Technology and Management Science
2012 /15
22
The additive noise is the signal, which is added to the
original signal [1]. If the original image pixel value is
represented as (7), then the pixel value with the additive noise
is represented as (8).
),,( bgrC , (7)
where r – the red colour component, g – green, b – blue.
),,(' bbggrrC . (8)
Then the noise level can be calculated as the mean square
error (9) [5].
1
0
1
0
222
3
n
i
m
j
ijijij
mn
bgr
MSE , (9)
where n – the image width, m – the image height.
The initial noise level of the images of the neurons with the
random weights can be calculated as the expected value of the
distance between two pixels, between the same colour
components. This regularity (10) may be seen in Fig. 4.
Fig. 4. All variations of the distance
5.10922
256
)256(
2
255
12
2
i
ii
, (10)
where µ – the expected value of the distance.
VI. SELF-ORGANIZING MAP AND NOISE LEVEL
The aim of this research is to investigate the influence of
SOM parameters on the noise level of RGB image generated
by SOM. This analysis will enable the authors of the article to
formulate recommendations on how to control the noise level
by adjusting SOM parameters.
Within the framework of the research, the following SOM
parameters have been analysed:
• the learning coefficient;
• the adaptation time;
• the effective width.
The effective width is determined through the initial
distance and the initial topological neighbourhood function
value in the analysis process (11).
)ln(2
),( 1
h
d
hd
, (11)
where d – the initial distance, h – the initial topological
neighbourhood function value.
Fig. 5. The experiment model
SI – the neuron with the weights, which are equal to the
generated image (it is the random image in the experiment);
RI – the neurons with the random weights (all 3 neurons are
equal); FI – the neuron with the weights, which are equal to
the filtered image by the colour: one group by the red colour,
other by the green-blue colour (see Fig. 5).
The first RI describes the neurons, which are closer to SI,
the second RI describes the neurons between SI and FI, and
the third RI – which are closer to FI.
The same set of SI and RI images and the same sequence of
winning neurons have been used in every observation of the
experiment; all these elements have been generated before the
experiment.
The experiment factors are displayed in Table I.
The static parameters have been taken according to [2]
recommendations.
The results of the experiments are illustrated in Fig. 6-10.
The experiment with 5 neurons of SOM has shown the
relation: the neurons before the middle neuron and after it;
therefore the SOM experiment has been extended to 7 neurons
(see Fig. 10).
The extension to 7 neurons allows observing how the
neurons behave, when they approach the middle neuron.
Fig. 10a-10b describe direction from SI to the middle neuron;
Fig. 10d-10e describe direction from FI to the middle neuron;
Fig. 10c – the middle neuron.
TABLE I
EXPERIMENT PARAMETERS
Image size: 10x10
Count of observations: 100
SOM parameters Range Step Static parameters
Learning coefficient η0 0.001-0.01 0.001
δ0(d=4, h=0.1)
t=1000
τ1=1000
τ2=1000
Learning coefficient η0 0.01-0.2 0.01
Learning coefficient η0 0.2-3 0.1
Adaptation time t 100-2000 100 η0=0.1
δ0(4, 0.1)
Effective width δ0 ]7;1[
d 1 h=0.1, η0=0.1
t=1000
Unauthenticated
Download Date | 6/27/17 6:43 PM
Information Technology and Management Science
2012 /15
23
(a)
(b)
(c)
Fig. 6. The pit-effect of the learning coefficient and the noise level. (a) – the
first RI neuron from SI, (b) – the second, (c) – the third, where X axis is the
learning coefficient, Y – the noise level, where the top line is the red colour FI,
the bottom – the green-blue colour FI
(a)
(b)
(c)
Fig. 7. The decrease in the learning coefficient and stable noise. (a) – the first
RI neuron from SI, (b) – the second, (c) – the third, where X axis is the
learning coefficient, Y – the noise level, where the top line is the red colour FI,
the bottom – the green-blue colour FI
(a)
(b)
(c)
Fig. 8. The learning coefficient, the unstable range (a) – the first RI neuron
from SI, (b) – the second, (c) – the third, where X axis is the learning
coefficient, Y – the noise level, where the top line is the red colour FI, the
bottom – the green-blue colour FI
Unauthenticated
Download Date | 6/27/17 6:43 PM
Information Technology and Management Science
2012 /15
24
(a)
(b)
(c)
Fig. 9. The adaptation time. (a) – the first RI neuron from SI, (b) – the second,
(c) – the third, where X axis is the adaptation time, Y – the noise level, where
the top line is the red colour FI, the bottom – the green-blue colour FI
(a)
(b)
(c)
(d)
(e)
Fig. 10. The effective width. (a) – the first RI neuron from SI, (b) – the
second, (c) – the third, (d) – the fourth, (e) – the fifth, where X axis is the
effective width, Y – the noise level, where the top line is the red colour FI, the
bottom – the green-blue colour FI
VII. CONCLUSIONS AND RECOMMENDATIONS
Despite initial different distances (MSE) between SI and FI
of 2 groups (filtered by the red and green-blue colour), the
result graphs change identically (see Fig. 6-10). There are the
differences in the noise level: the less the initial distance is,
the less the resulting noise level is.
The noise level goes through 3 phases while the learning
coefficient η0 changes:
• the noise level makes a small pit, when the learning
coefficient is too small (η0 (0;0.07] in the experiment);
• when the noise level gets through the rise, it begins to
decrease gradually (η0 [0.1;1] in the experiment);
• when the learning coefficient is too large, the noise level
is not stable (see Fig. 8).
There are 2 phases of the noise level in Fig. 9: the
organization process of neurons and the stabilization process
of values, mentioned previously in the article. The
stabilization process of values begins, when the adaptation
time exceeds 1000.
Unauthenticated
Download Date | 6/27/17 6:43 PM
Information Technology and Management Science
2012 /15
25
When the initial effective width δ0(d;0.1) distance d does
not achieve the middle neuron (the middle neuron is achieved
in the experiment, when d = 3), the islands of images are
produced (see Fig. 11).
Fig. 11. The parts of the generated RGB images using the hexagonal SOM,
where the numbers are MSE of the images, the first row d = 1,
the second – d = 2, the third – d = 3, the fourth – d = 4, the fifth – d = 5,
the sixth – d = 6, the seventh – d = 7
According to the obtained information, the following
conclusions are drawn:
• a better range of the learning coefficient to control the
noise level is η0 [0.1;1], when the noise level decreases
gradually;
• a better range of the adaptation time to control the noise
level is t > 1000, when there is the stabilization process
of values. However, it does not suit the middle neurons,
because their noise level slightly changes trying to obtain
the static value;
• the initial effective width must be of such a value so that
the topological neighbourhood influences all neurons,
when there is the winning neuron in the centre of the
SOM lattice.
REFERENCES
[1] S. J. Sangwine, R. E. N. Horne, The Color Image Processing Handbook,
1st ed. London: Chapman & Hall, 1998, 440p.
[2] С. Хайкин, Нейронные сети: полный курс, 2-е изд., испр. : Пер. с
англ. – Москва: ООО «И.Д. Вильямс», 2006, 1104 с.
[3] Л. Н. Ясницкий, Введение в искусственный интеллект. Москва:
Издательский центр «Академия», 2005, 176 с.
[4] B. Curry, P. H. Morgan, Evaluating Kohonen’s learning rule: An
approach through genetic algorithms, in European Journal of
Operational Research, vol. 154, 2004, pp. 191-205.
[5] J. Pęksiński, G. Mikołajczak, Removing Noise from Digital Images
Using of Mean Square Approximation, in Procedia Environmental
Sciences, vol. 10, 2011, 980-985.
[6] C. C. Yang, H. Chen, K. Hong, Visualization of large category map for
Internet browsing, in Decision Support Systems,
vol. 35, 2003, pp. 89-102.
[7] L. Yang, Z. Ouyang, Y. Shi, A Modified Clustering Method Based on
Self-Organizing Maps and Its Applications, in Procedia Computer
Science, vol. 9, 2012, pp. 1371-1379.
Sergejs Kodors was born in Rezekne, Latvia. He received his bachelor
degree in Information Technology from Rezekne Higher Education Institution
in 2011. Now he is a master student of the study programme “Computer
Systems” at Rezekne Higher Education Institution.
His research interest focuses on artificial intelligence. His current research
includes image recognition and neural networks.
E-mail: sk_7@inbox.lv
Peter Grabusts was born in Rezekne, Latvia. He received his Dr.sc.ing.
degree in Information Technology from Riga Technical university in 2006.
Since 1996 he has been working at Rezekne Higher Education Institution.
Since 2008 he is an Associate Professor at the Department of Computer
Science.
His research interests include data mining technologies, neural networks and
clustering methods. His current research focuses on techniques for clustering
and fuzzy clustering.
E-mail: peter@ru.lv
Sergejs Kodors, Pēteris Grabusts. Ar pašorganizējošās kartes palīdzību ģenerēto RGB attēlu trokšņa līmeņa analīze
Lai noteiktu mākslīgo neironu tīklu iespējas atpazīt attēlus, tiek izmantota metode ar nosaukumu „krusteniskā validācija”. Atbilstoši šai metodei neironu tīkla
spēja atpazīt attēlus tiek izteikta kā atpazīto attēlu procents no pārbaudes kopas. Tādējādi, lai turpmāk varētu noteikt neironu tīkla spēju atpazīt attēlus ar troksni,
vispirms ir jāsagatavo pārbaudes kopa ar trokšņainiem attēliem. Šī problēma kļūst aktuāla, kad pētnieka rīcībā ir tikai attēli bez trokšņa. Tad ir nepieciešams
automātisks rīks, lai ģenerētu attēlus ar troksni. RGB attēlus ar troksni var ģenerēt, izmantojot Kohonena pašorganizējošās kartes (SOM). Pētījuma mērķis bija
izpētīt RGB attēla trokšņa līmeni atkarībā no SOM parametriem: apmācības koeficienta, apmācības laika un slīpuma koeficienta. Rakstā ir aprakstīta visa
nepieciešamā teorija, kā ģenerēt RGB attēlus ar troksni, izmantojot SOM: SOM sastāvošie elementi, SOM apmācības process, SOM īpašība, kas tiek izmantota
RGB attēla ar troksni ģenerēšanai, asociācijas starp RGB attēlu un neironu svariem izveidošana, trokšņa līmeņa aprēķināšanas formula un SOM sešstūrveida
struktūras apraksts, kuru izmantojot, var ģenerēt ar troksni novirzītus attēlus katrā krāsas komponentē. Pamatojoties uz šo informāciju, tika organizēts
eksperiments. Eksperimentā tika izmantots viendimensijas SOM ar 5 neironiem, lai aprakstītu trokšņa līmeņa atkarību no SOM parametriem. Tāds neironu skaits
tika izvēlēts, lai apkopotu rezultātus uz loģiskām neironu grupām. Lai iegūtu papildinformāciju par slīpuma koeficienta ietekmi uz trokšņa līmeni, eksperimenta
modelis tika paplašināts līdz 7 neironiem. Pamatojoties uz eksperimenta rezultātiem, tika izstrādātas rekomendācijas (noteikti SOM parametru diapazoni) trokšņa
līmeņa regulēšanai.
Сергей Кодорс, Петерис Грабустс. Анализ уровня шума RGB изображений, созданных посредством самоорганизующихся карт
Чтобы выразить, насколько эффективно нейросеть распознает изображения, используется метод перекрестной проверки. Опираясь на данный метод,
эффективность нейросети определяется как процент корректно опознанных изображений. Следовательно, чтобы выразить способность распознавать
зашумленные изображения, необходимо подготовить коллекцию изображений с заданным диапазоном шума. Данное условие требует наличия
автоматической системы для генерации изображений с шумом, когда в распоряжении исследователя есть только изображения без шума. RGB
изображения с шумом можно сгенерировать посредством самоорганизующихся карт Кохонена (SOM). Целью данной работы является исследование
зависимости уровня шума от параметров самоорганизующихся карт Кохонена: количества итераций обучения, скорости обучения и эффективной
ширины. В данной работе предоставлена вся необходимая информация для генерации RGB изображений с шумом посредством использования SOM:
описание SOM элементов, процесс обучения SOM, создание ассоциации между RGB изображением и весами синапсов нейрона, расчет уровня шума и
шестиугольная SOM для генерации RGB изображений с отклонением цвета в сторону каждого цветового компонента. Для исследования зависимости
уровня шума от параметров SOM была использована одномерная самоорганизующаяся карта из 5 нейронов. Модель из 5 нейронов была выбрана с
целью объединить результаты в логические группы нейронов. Чтобы извлечь дополнительную информацию о влиянии эффективной ширины на
уровень шума, экспериментальная модель была расширена до 7 нейронов. На основании полученных результатов были составлены рекомендации:
какие диапазоны значений параметров самоорганизующийся карты использовать для контроля уровня шума RGB изображений.
Unauthenticated
Download Date | 6/27/17 6:43 PM
