CC BY
59
UDC 519.7: 004.8
Bodyanskiy Ye. V.1, Kulishova N. E.2
1Doctor of Science, Professor, Kharkiv National University of Radio Electronics, Ukraine
2Ph.D., Professor, Kharkiv National University of Radio Electronics, E-mail: kunonna@mail.ru, Ukraine
EXTENDED NEO-FUZZY NEURON IN THE TASK OF IMAGES _FILTERING_
The paper describes a modification of the neo-fuzzy neuron called as «extended neo-fuzzy neuron» (ENFN) that characterized by improved approximating properties. The adaptive learning algorithm for ENFN is proposed, that has both following and smoothing properties and allows to solve problems of prediction, filtering and smoothing of non-stationary disturbed stochastic and chaotic signals. A distinctive feature of ENFN is its implementation computational simplicity compared with artificial neural networks and neuro-fuzzy systems. These properties of the proposed neo-fuzzy neuron make it very effective in suppressing noise in image filtering.
Keywords: color images, disturbance, contours, filtering, neo-fuzzy neuron.
INTRODUCTION
Digital images are often exposed to noise when they are created and transmitted over communication channels. For reasons of noise and distortion can be attributed atmospheric phenomena (for images obtained on TV), originals surface defects (scanning), and low-light during shooting (for digital cameras). The main problem in this case is the need for effective compensation of distortion and noise while preserving image features such as edges, textures, and small details. Existing image smoothing filters, suppressing noise, greatly blur contours and reduce image sharpness.
The aim of this work is to develop an adaptive filter, which can compensate the noise on digital images without significant reducing their quality.
Artificial neural networks (ANN) and fuzzy inference system (FIS) in recent years have proliferated to address a large class of data mining tasks of various natures under a priori and the current uncertainty. Hybrid neuro-fuzzy system (NFS), that have appeared at the junction of the two main areas of computational intelligence [1-4], and absorbed their best features. Thus, the neuro-fuzzy system is capable of learning like ANN and provide linguistic interpretability and «transparency» of the results like FIS. However, NFS's calculation bulkiness and low speed training limit their applicability to image processing problems.
To overcome some of the noted problems, neuro-fuzzy system,called by the authors as «neo-fuzzy neuron (NFN)», was introduced and studied in [5-7]. Fig. 1 shows the architecture of the neo-fuzzy neuron.
Neo-fuzzy neuron is a nonlinear learning system with multiple inputs and one output, that realizes the mapping
n
y = Z f (x)
i=1
where xi - i-th component of the n-dimensional input signals vector, x = (x1,...,xi,...,xn) e Rn , y- scalar
© Bodyanskiy Ye. V, Kulishova N. E., 2014 112 DOI 10.15588/1607-3274-2014-1-16
Fig. 1. Neo-fuzzy neuron
NFN's output. Neo-fuzzy neuron structural blocks are nonlinear synapses NSi, performing a nonlinear
transformation of the i-th component xi in the form
h
f, (x ) = Z wi№ii (x )
i=1
where wii - i-th synaptic weight of i-th nonlinear synapse, i = 1,2,...,h , i = 1,2,...,n; v-i, (x, ) - i-th membership function in the i-th nonlinear synapse, producing a fuzzification of crisp component xi.
Thus, transformation, realized by NFN, can be written as
n h
y = ^Lwiivi, (x, ). i=1 i=1
A fuzzy inference implemented by this same NFN, has the form
IF xi IS xh THEN THE OUTPUT IS wli,l = 1,2,...,h, i.e. actually nonlinear synapse implements zero order Takagi-Sugeno fuzzy inference [8, 9].
Neo-fuzzy neuron authors [5-7] as membership functions were used traditional triangular construction that meet the conditions of unity partition
Pi, =
xi - cl- 1,i
cli - cl- -1,i
cl+1,i - xi
cl+1, i - cli
0,
if X e[c-1,i ,cu ],
if xi e[cli,cl+1,i], otherwise,
where cu - rather arbitrarily selected (usually uniformly distributed) centers of membership functions on the interval [0, 1] , thus, naturally 0 < xl < 1.
Such choice of the membership functions leads to that ith component of the input signal activates only the two adjacent functions, thus their sum is equal to unity, i.e.
and
Pi, (x, ) + Pl+1,i (X ) = ! f (x, ) = wi, Pli (x, ) + wl+1,i Pl+1,i (x, ).
It is this circumstance allowed to synthesize simple and effective adaptive controllers for nonlinear control objects [10, 11].
Of course, besides triangular as membership functions can be used and other forms and, first of all, the B-splines [12], proved to be effective in the composition of neo-fuzzy neuron [13]. The general form of membership functions based on q-th degree B-spline can be presented in the form:
pf (x , q ) =
xi- cli
[l, if X £ [cü , cl+1,i ]]
cl+q-1,i - cli
-pf (x , q -1)+-
0, otherwise
cl+q,i — Xi cl+q,i - cl+1,i
for q = 1,
pf+1,i (x , q -1)for q >1, l =1,2,..., h - q.
In the case when q = 2 we obtain the traditional triangular functions. It should be noted also that the B-splines also provide a single partition in the form of
h
Zpf (x, , q ) =1
l=1
are non-negative, i.e.
pf (x, , q)0
and have local domain
pf (xi, q) = 0 for xi i [cli, cl+q,i ].
Thus, when applied to NFN's input the vector signal x(k) = (x1 (k),...,xl (k),...,xn (k)) (k = 1,2,... here - the current discrete time) at its output appears scalar value
n h
y (k ) = ££wlI (k - 1)Pli (xi (k)),
i=1l=1
(1)
where wil (k -1) - the current value of the adjusting synaptic weights resulting from learning on previous k-1 observations.
Introducing the (nh x 1) vector of membership functions
P( x(k)) = (P1 (x1 (k))'..' Ph1 (x1 (k))' P1 ( x1 ( k )),...,
P12 (x2 (k))Pli (xi (k))Phn (xn (k))) and the corresponding vector of synaptic weights
w(k - 1) = (wn (k - 1),..., wh1 (k - 1), w2 (k - 1)wli(k - ^^,
T
. ., Whn (k -1)) ,we can rewrite the transformation (1), implemented by NFN, in a compact form
y(k) = wT (k- 1)p(x(k)).
(2)
To adjust the neo-fuzzy neuron parameters, the authors used the gradient procedure that minimizes the learning criterion
* (k )=2 (y (k)-y (k ))2=2 e2 (k )=
i n h \2
y (k)-ZZ^üPli (x (k))
i=1l=1
and having the form
wl, (k) = wl, (k -!) + ne (k) Vi, (x, (k)) = = w (k -1) + n (y (k)- y (k)) Vi (x, (k)) =
( n h \
= wl, (k -1) + n y (k)- ZZ wl,Vii (x, (k)) Vii (x, (k)) ,
^ i=1l=1 )
where y (k) - external training signal, e (k) - learning error, n - learning rate parameter.
To accelerate the NFN learning process in [14] special algorithm was introduced, having both following (for non-stationary signal processing) and filtering (for «noisy» data processing) properties
w (k ) = w (k -1) + r (k )e (k )p(x (k )), r (k) = ar (k -1) + ||p(x(k))) ,0 <a< 1.
(3)
Wherein when a = 0, algorithm (3) is identical in structure to the Kaczmarz-Widrow- Hoff one-step learning algorithm [15], and when a = 1 - to Goodwin-Ramage-Caines stochastic approximation algorithm [16].
Note also, that the neo-fuzzy neuron synaptic weights learning can be used by many other algorithms for learning and identification, including the traditional method of least squares with all its modifications.
EXTENDED NEO-FUZZY NEURON
As was noted above, the neo-fuzzy neuron is nonlinear synapse NSi implements zero-order Takagi-Sugeno inference, thus being the Wang-Mendel elementary neuro-fuzzy system [17-19]. It is possible to improve approximating properties of such system using a structural unit, which we called «extended nonlinear synapse» (ENSi) (see Fig. 2) and synthesized on its basis the «extended neo-fuzzy neuron» (ENFN), containing as elements ENSi instead of the usual nonlinear synapses NSi.
By introducing the additional variables
y¡, (x ) = I (x ) (w0 + w/,x + wu
h , f ( xi ) = Z Iii (xi ) ( w° + w1xi + wii
xf +... + wf*f
2 2 -,ixi ■
-w<y I-
= w0Ili (xi ) + w1ixiIli (xi )+ ■■■ + w1iXiPM-1i (xi ) + w°il2i (xi )+ - + +w21XPI2i (x, ) +... + KX,PIhi (xi )>
Í 0 1 p 0 p py : (w1i, w1i,..., <, w2i,..., w2i,..., whi ) ,
1 (x) = (l1i (x),xil1i (x),...,xpI1i (x),hi (x),...,
\T
...,xpP2i (x),...,xpPhi (x)) ,
Xi(k)
r" 1 ENSi ¡
1 Wii + wliXi + WÍiX2+ p p 0 ii(Xi(k)) 1
\ I
1 ^ v.'
■ p 1 2 2 W2i + W2iXi+W2iXi + p p
1
1 1 ^ / i
1 Whi + WhiXi + WhiX2+ p p / !
^hi 0hi(Xi(k))
1
fi(Xi(k))
Fig. 2. Extended non-linear synapse
we can write
f (x ) = wT p i (x ),
y = Z f (x ) = Z wT p (x ) = w T P (x ), i-1 i-1
where
-T Í T T w - I W1 ,..., wi ,
, w,
<n
V (X) = (Vf (x1 vT (Xi ),..., vT (xn )) .
It's easy to see that ENFN contains (p + 1)h
adjusting synaptic weights and fuzzy output, implemented by each ENSi, has the form
IF x, IS x, THEN THE OUTPUT IS
w0 + w^x, +... + wpxp,l = 1,2,...,h,
i.e. essentially coincides with p-order Takagi-Sugeno inference.
Let's note also that ENFN has a much simpler architecture than the traditional neuro-fuzzy system that simplifies its numerical implementation.
When the ENFN's input is vector signal x (k), at the output scalar value appears
y(k) = wT (k - 1)V(x(k)),
whereby this expression differs from (2) only in that it comprises in a (p +1) times more number of tuning parameters than conventional NFN. It is clear that learning parameters ENFN algorithm may be used such as (3), obtaining in this case the form
w (k) - w (k -1) + r~X (k) e (k) p (x(k)), r (k) - ar (k -1) + ||p (x(k)))2 ,0 <a< 1.
Fig. 3 shows the architecture of an extended neo-fuzzy neuron.
EXPERIMENT
The effectiveness of the proposed architecture has been investigated on a set of test images (Fig. 4).
Images were damaged by different types of noise: the Poisson, Gaussian, impulse, multiplicative. Neo-fuzzy neuron performance for noise compensation was estimated by two objective measures (MSE, PSNR) and subjective visual evaluation. Mean square error MSE is calculated by the formula:
M N
-72,)
MSE =
i=1 ,=1
MN
where I. 12.. - the original and filtered images, respectively,
i = 1,2,..., M, j = 1,2,..., N - numbers ofimage pixels, M, N -image sizes.
To calculate the signal/noise ratio PSNR used:
PSNR = 10 • lg
f R2_ ^
MSE
where R is a coefficient depending on the encoding of images (for 8-bit encoding R = 255, for floating-point R = 1). For comparison were also used standard filters - averaging, median, and Wiener filter. Neo-fuzzy neuron learning was carried out in two versions: for pure signal and the Wiener preliminary filtered, that can be used in cases there is no clear signal.
Image quality estimations after filtering are given in Table 1, some examples of images before and after the noise compensation are shown in Fig. 5.
Fig. 3. Extended neo-fuzzy neuron
It is obvious that, although numerical estimates demonstrate higher quality filtering for some standard filters, visual evaluation certainly suggests a high efficiency of the neo-fuzzy neuron. This is appeared in the keeping contours, fine details and textures. If a clean signal for training is available, can use the Wiener filter for learning signal for the neo-fuzzy neuron.
CONCLUSION
The paper proposes an extended architecture of neo-fuzzy neuron, which is a generalization of the standard neo-fuzzy neuron in case of fuzzy inference order above zero. The learning algorithm is introduced having both following and filtering properties. Considered NFN has improved approximating properties, characterized by a high learning rate, has simple numerical implementation.
SPISOK LITERATURY
1. Rutkowski, L. Computational Intelligence. Methods and Techniques / Rutkowski L. - Berlin : Springer-Verlag, 2008. -514 p.
2. Mumford, C. L. Computational Intelligence / C. L. Mumford, L. C. Jain. - Berlin : Springer-Verlag, 2009. - 725 p.
3. Kruse, R. Computational Intelligence. A Methodological Introduction / [Kruse R., Borgelt C., Klawonn F., Moewes C., Steinbrecher M., Held P.]. - Berlin : Springer-Verlag, 2013. -488 p.
4. Du, K.-L. Neural Networks and Statistical Learning / K.-L. Du, M. N. S. Swamy. - London : Springer-Verlag, 2014. -815 p.
5. Yamakawa, J. A neo-fuzzy neuron and its application to system identification and prediction of the system behavior / J. Yamakawa, E. Uchino, J. Miki, H. Kusanagi // Proc. 2-nd Int. Conf. on Fuzzy Logic and Neural Networks «IIZUKA-92», Iizuka, Japan, 1992. - P. 477-483.
6. Uchino, E. Soft computing based signal prediction, restoration and filtering / E. Uchino, J. Yamakawa ; Ed. Da Ruan «Intelligent Hybrid Systems: Fuzzy Logic, Neural Networks and Genetic Algoritms». - Boston : Kluwer Academic Publishers, 1997. - P. 331-349.
7. Miki, J. Analog implementation of neo-fuzzy neuron and its on-board learning / J. Miki, J. Yamakawa ; Ed. by N. E. Mastorakis «Computational Intelligence and Applications». - Piraeus : WSES Press, 1999. - P. 144-149.
8. Takagi T. Fuzzy identification of systems and its application to modeling and control / T. Takagi, M. Sugeno // IEEE Trans. on System, Man and Cybernetics. - 1985. - vol. 15, no. 1. -P. 116-132.
original image
original image
original image
Fig. 4. Test images
Table 1. Image quality estimation after filtration
Averaging Filter Median Filter Wiener Filter Neo-Fuzzy Neuron with clear learning signal Neo-Fuzzy Neuron with Wiener filtered learning signal
Image «Car»
Poisson 0,0209 16,8 0,0138 18,6 0,0078 21,1 0,0068 21,7 0,0093 20,3
Gaussian, m=0, a=0,0005 0,0296 15,3 0,0231 16,4 0,0154 18,1 0,0159 17,9 0,0172 17,6
Gaussian, m=0, a=0,001 0,0418 13,8 0,0365 14,4 0,0263 15,8 0,0292 15,3 0,0284 15,5
Gaussian, m=0, a=0,05 0,0423 13,7 0,0366 14,4 0,0265 15,8 0,0370 14,3 0,0290 15,4
Gaussian, m=0, a=0,01 0,0418 13,8 0,0364 14,4 0,0263 15,8 0,0292 15,3 0,0285 15,5
Gaussian, m=0, a=0,1 0,1968 7,1 0,2075 6,8 0,1519 8,2 0,2083 6,8 0,1591 7,9
Impulse, a=0,005 0,0203 16,9 0,0132 18,8 0,052 22,9 0,0154 18,1 0,118 9,3
Impulse, a=0,01 0,0248 16,1 0,0178 17,5 0,0069 21,6 0,0118 19,3 0,0307 15,1
Multiplicative, a=0,005 0,0198 17,0 0,0128 18,9 0,0068 21,7 0,0057 22,5 0,0084 20,8
Multiplicative, a=0,001 0,0168 17,8 0,0094 20,2 0,0042 23,7 0,0021 26,8 0,0085 20,7
Multiplicative, a=0,01 0,0234 16,3 0,0168 17,7 0,0098 20,1 0,0095 20,2 0,0117 19,3
Multiplicative, a=0,1 0,0739 11,3 0,0752 11,2 0,0393 14,1 0,0673 11,7 0,0426 13,7
Image «Lena»
Poisson 0,0126 18,9 0,0093 20,3 0,0065 21,8 0,0070 21,5 0,0083 20,8
Gaussian, m=0, a=0,0005 0,0366 14,4 0,0336 14,7 0,0260 15,8 0,301 15,2 0,0285 15,4
Gaussian, m=0, a=0,001 0,0365 14,4 0,0336 14,7 0,0260 15,8 0,0300 15,2 0,0285 15,5
Gaussian, m=0, a=0,05 0,0366 14,4 0,0333 14,8 0,0256 15,9 0,0373 14,3 0,0289 15,4
Gaussian, m=0, a=0,01 0,0365 14,4 0,0335 14,8 0,0259 15,9 0,0304 15,2 0,0286 15,4
Gaussian, m=0, a=0,1 0,0361 14,4 0,0324 14,9 0,0248 16,1 0,0583 12,3 0,0293 15,3
Impulse, a=0,005 0,0133 18,8 0,0086 20,6 0,0034 24,7 0,0046 23,3 0,0045 23,5
Impulse, a=0,01 0,0177 17,5 0,0132 18,8 0,0049 23,1 0,0096 20,2 0,0062 22,1
Multiplicative, a=0,005 0,0136 18,7 0,0091 20,4 0,0060 22,2 0,0059 20,9 0,0081 20,9
Multiplicative, a=0,001 0,0101 19,9 0,0054 22,7 0,0030 25,2 0,0018 27,4 0,0045 23,5
Multiplicative, a=0,01 0,0175 17,6 0,0136 18,7 0,0097 20,1 0,0105 19,8 0,0120 19,1
Multiplicative, a=0,1 0,0782 11,1 0,0812 10,9 0,0540 12,7 0,0788 11,1 0,0582 12,3
Image «Parrot»
Poisson 0,0109 19,6 0,0083 20,8 0,0044 23,6 0,0031 25,1 0,0049 23,1
Gaussian, m=0, a=0,0005 0,0318 14,9 0,0308 15,1 0,0222 16,5 0,0244 16,1 0,0224 16,5
Gaussian, m=0, a=0,001 0,0318 14,9 0,0308 15,1 0,0223 16,5 0,0244 16,1 0,0224 16,5
Gaussian, m=0, a=0,05 0,0343 14,7 0,0330 14,8 0,0244 16,1 0,0349 14,6 0,0249 16,0
Gaussian, m=0, a=0,01 0,0325 14,9 0,0314 15,0 0,0228 16,4 0,0255 15,9 0,0230 16,4
Gaussian, m=0, a=0,1 0,0355 14,5 0,0340 14,7 0,0257 15,9 0,0584 12,3 0,0266 15,7
Impulse, a=0,005 0,0134 18,7 0,0108 19,7 0,0032 24,9 0,0060 22,2 0,0037 24,3
noised image
Averaging Filtering
Median Filtering
Wiener filtering
noised image
Wiener filtering
Neo Fuzzy Neuron Filtering (pure signal)
Averagng Fltering
Neo Fuzzy Neuron Filtering (pure signal)
Averaging Filtering
Neo Fuzzy Neuron Filtering (pure signal)
Neo Fuzzy Neuron Filtering (Wiener learning signal)
f t W'-f;*
I I it*
fvwl V-'W } il. «
Fig. 5. Images after filtering (multiplicative noise, a=0,1)
НЕЙРО1НФОРМАТИКА ТА ШТЕЛЕКТУАЛЬШ СИСТЕМИ
9. Jang, J.-S. Neuro-Fuzzy and Soft Computing: A Computational Approach to Learning and Machine Intalligence / J.-S. Jang, C.-T. Sun, E. Mizutani. - Upper Saddle River : Prentice Hall, 1997. - 640 p.
10. Bodyanskiy, Ye. Adaptive nonlinear control using neo-fuzzy model / Ye. Bodyanskiy, V. Kolodyazhniy ; Eds. O. Sawodny, P. Sharif «Sinergies Between Information Processing and Automation». - Aachen : Shaker Verlag, 2004. - P. 122-127.
11. Bodyanskiy, Ye. A self-training robust neo-fuzzy controller with constraints on control actions / [Bodyanskiy Ye., Kokshenev I., Kolodyazhniy V., Otto P.] // 50 Int. Wiss. Koll. Tagungsband. - TU Ilmenau, 2005. - P. 125-126.
12. Zhang, J. Constructing fuzzy-controllers with B-spline models - Principles and Applications / J. Zhang, H. Knoll // Int. J. of Intelligent Systems. - 1998. - vol. 13. - P. 257-285.
13. Kolodyazhniy, V. Cascaded multiresolution spline-based fuzzy neural network / Kolodyazhniy V., Bodyanskiy Ye. ; Eds. P. Angelov, D. Filev, N. Kasabov. // Proc. Int. Symp. On Evolving Intelligent Systems. - Leicester, UK : DeMontfort Univercity, 2010. - P. 26-29.
14. Bodyanskiy, Ye. An adaptive learning algorithm for a neo-fuzzy neuron / Ye. Bodyanskiy, I. Kokshenev, V. Kolodyazhniy // Proc. 3rd Int. Conf. of European Union Sosiety for Fuzzy Logic and Technology (EUSFLAT 2003), Zittau, Germany, 2003. - P. 375-379.
15. Haykin, S. Neural Networks. A Comprehensive Foundation / Haykin S. - Upper Saddle River : Prentice Hall, 1999. - 842 p.
16. Goodwin, G. C. Discrete time stochastic adaptive control / G. C. Goodwin, P. J. Ramage, P. E. Caines // SIAM J. Control and Optimisation. - 1981. - vol. 19. - P. 829-853.
17. Wang, L. X. Fuzzy basis functions, universal approximation and orthogonal least squares learning / L. X. Wang, J. M. Mendel // IEEE Trans. on Neural Networks. - 1993. -vol. 3. -P. 807-814.
18. Wang, L. X. Adaptive Fuzzy Systems and Control. Design and Stability Analysis / Wang L. X. - Upper Saddle River : Prentice Hall, 1994. - 256 p.
19. Osowski, S. Sieci neuronowe do przetwarzania informacij / Osowski S. - Warszawa : Oficijna Wydawnicza Politechniki Warszawskiej, 2006. - 422 p.
Стаття надшшла до редакци 14.04.2014.
Бодянский Е. В.1, Кулишова Н. Е.2
'Д-р техн. наук, профессор, Харьковский национальный университета радиоэлектроники, Украина
2Канд. техн. наук, професор, Харьковский национальный университет радиоэлектроники, Украина
РАСШИРЕННЫЙ НЕО-ФАЗЗИ НЕЙРОН В ЗАДАЧАХ ФИЛЬТРАЦИИ ИЗОБРАЖЕНИЙ
В статье предлагается модификация нео-фаззи нейрона, названная нами «расширенный нео-фаззи нейрон» (Е№ТЯ) и характеризующаяся улучшенными аппроксимирующими свойствами. Введен адаптивный алгоритм обучения Е№К, обладающий следящими и сглаживающими свойствами и позволяющий решать задачи прогнозирования, фильтрации и сглаживания нестационарных «зашумленных» стохастических и хаотических сигналов. Отличительной особенностью Е№К является вычислительная простота его реализации по сравнению с искусственными нейронными сетями и нейро-фаззи системами. Эти свойства предложенного нео-фаззи нейрона делают его очень эффективным при подавлении шумов на изображениях в ходе фильтрации.
Ключевые слова: цветные изображения, помеха, контуры, фильтрация, нео-фаззи нейрон.
Бодянський С. В.1, Кутшова Н. С.2
'Д-р техн. наук, профессор, Харювський нацюнальний ушверситет радюелектронжи, Украша
2Канд. техн. наук, профессор, Харювський нацюнальний ушверситет радюелектронжи, Украша
РОЗШИРЕНИЙ НЕО-ФАЗЗ1 НЕЙРОН В ЗАДАЧАХ ФШЬТРАЦП ЗОБРАЖЕНЬ
У статп пропонуеться модифжащя нео-фаззi нейрона, що названа нами «розширений нео-фаззi нейрон» (Е№К) i характеризуемся полшшеними апроксимуючими властивостями. Введено адаптивний алгоритм навчання ENFN, що мае слiдкуючi i згладжувальш властивост i дозволяе виршувати завдання прогнозування, фшьтрацй i згладжування нестацюнарних «зашумле-них» стохастичних i хаотичних сигналiв. Вщмшною особливютю ENFN е обчислювальна простота його реалiзацй в порiвняннi з штучними нейронними мережами i нейро-фаззi системами. Ц властивост запропонованого нео-фаззi нейрона роблять його ефективним для пригшчення шушв на зображеннях в ходi фшьтрацй.
Ключовi слова: кольоровi зображення, завада, контури, фшьтращя, нео-фаззi нейрон.
REFERENCES
1. Rutkowski L. Computational Intelligence. Methods and Techniques. Berlin, Springer-Verlag, 2008, 514 p.
2. Mumford C. L., Jain L. C. Computational Intelligence, Berlin, Springer-Verlag, 2009, 725 p.
3. Kruse R., Borgelt C., Klawonn F., Moewes C., Steinbrecher M., Held P. Computational Intelligence. A Methodological Introduction. Berlin, Springer-Verlag, 2013, 488 p.
4. Du K.-L., Swamy M. N. S. Neural Networks and Statistical Learning. London, Springer-Verlag, 2014, 815 p.
5. Yamakawa J., Uchino E., Miki J., Kusanagi H. A neo-fuzzy neuron and its application to system identification and prediction of the system behavior, Proc. 2-nd Int. Conf. on Fuzzy Logic and Neural Networks «IIZUKA-92», Iizuka, Japan, 1992, pp. 477-483.
6. Uchino E., Yamakawa J. Soft computing based signal prediction, restoration and filtering, Ed. Da Ruan «Intelligent Hybrid Systems: Fuzzy Logic, Neural Networks and Genetic Algoritms». Boston, Kluwer Academic Publishers, 1997, pp. 331-349.
7. Miki J., Yamakawa J. Analog implementation of neo-fuzzy neuron and its on-board learning, Ed. by N. E. Mastorakis «Computational Intelligence and Applications». Piraeus, WSES Press, 1999, pp. 144-149.
8. Takagi T., Sugeno M. Fuzzy identification of systems and its application to modeling and control, IEEE Trans. On System, Man and Cybernetics, 1985. vol. 15, no. 1, pp. 116-132.
9. Jang J.-S., Sun C.-T., Mizutani E. Neuro-Fuzzy and Soft Computing: AComputational Approach to Learning and Machine Intalligence, Upper Saddle River, Prentice Hall, 1997, 640 p.
10. Bodyanskiy Ye., Kolodyazhniy V. Adaptive nonlinear control using neo-fuzzy model, Eds. O. Sawodny, P. Shaff «Sinergies Between Information Processing and Automation», Aachen, Shakar Verlag, 2004, pp. 122-127.
11. Bodyanskiy Ye., Kokshenev I., Kolodyazhniy V., Otto P. A self-training robust neo-fuzzy controller with constraints on control actions, 50 Int. Wiss. Koll. Tagungsband. TU Ilmenau, 2005, pp. 125-126.
12. Zhang J., Knoll H. Constructing fuzzy-controllers with B-spline models - Principles and Applications, Int. J. of Intelligent Systems, 1998, vol. 13, pp. 257-285.
13. Kolodyazhniy V., Bodyanskiy Ye. Cascaded multiresolution spline-based fuzzy neural network, Eds. P. Angelov, D. Filev, N. Kasabov, Proc. Int. Symp. On Evolving Intelligent Systems, Leicester, UK, DeMontfort Univercity, 2010, pp. 26-29.
14. Bodyanskiy Ye., Kokshenev I., Kolodyazhniy V. An adaptive learning algorithm for a neo-fuzzy neuron, Proc. 3rd Int. Conf.
of European Union Sosiety for Fuzzy Logic and Technology (EUSFLAT 2003), Zittau, Germany, 2003, pp. 375-379.
15. Haykin S. Neural Networks. A Comprehensive Foundation. Upper Saddle River, Prentice Hall, 1999, 842 p.
16. Goodwin G. C., Ramage P. J., Caines P. E. Discrete time stochastic adaptive control, SIAM J. Control and Optimisation, 1981.vol. 19, pp. 829- 853.
17. Wang L. X., Mendel J. M. Fuzzy basis functions, universal approximation and orthogonal least squares learning, IEEE Trans. on Neural Networks, 1993, vol. 3, pp. 807-814.
18. Wang L. X. Adaptive Fuzzy Systems and Control. Design and Stability Analysis. Upper Saddle River, Prentice Hall, 1994, 256 p.
19. Osowski S. Sieci neuronowe do przetwarzania informacij. Warszawa, Oficijna Wydawnicza Politechniki Warszawskiej, 2006, 422 p.
