=Paper=
{{Paper
|id=Vol-2864/paper41
|storemode=property
|title=Improvement of Grayscale Images in Orthogonal Basis of the Type-2 Membership Function
|pdfUrl=https://ceur-ws.org/Vol-2864/paper41.pdf
|volume=Vol-2864
|authors=Lyudmila Akhmetshina,Artyom Yegorov
|dblpUrl=https://dblp.org/rec/conf/cmis/AkhmetshinaY21
}}
==Improvement of Grayscale Images in Orthogonal Basis of the Type-2 Membership Function==
https://ceur-ws.org/Vol-2864/paper41.pdf
Improvement of Grayscale Images in Orthogonal Basis of the
Type‐2 Membership Function
Lyudmila Akhmetshinaa and Artyom Yegorova
a
Dnieper National University Named by Oles Honchar, Gagarin Avenue, house 72, Dnieper, 49010, Ukraine
Abstract
While analyzing different images, it is very important to identify similar and/or homogeneous
areas and boundaries of the objects of interest. Certain ambiguity occurred at this step can be
caused by physical characteristics of the used equipment and noise in the process of the
image formation, on the one part, and by inaccuracy and fuzziness introduced during digital
representation and by processing algorithms, on the other part. It is shown that transition of
the features to a fuzzy space, followed by the use of orthogonal transformation and
visualization of characteristics synthesized on the basis of their eigenvalues, improves
reliability of the objects of interest identification during analyzing of the grayscale images.
Informational capabilities of characteristics synthesized with the use of the method of
singular decomposition of the type-2 features in a fuzzy space are considered from the aspect
of improvement of the grayscale image quality. The obtained experimental results are shown
on the example of the real microscopic images.
Keywords 1
image processing, orthogonal transformations, singular decomposition, fuzzy logic,
Membership Function Type-2.
1. Introduction
The number of practical problems associated with digital image processing, which are obtained
with the help of standard research methods, and which are used, for example, in materials science, or
medicine, or flaw detection, is constantly growing. Images formed by various information, tracking or
diagnostic systems very often have a quality that is not sufficient for performing a reliable analysis.
As a rule, these images contain distortions caused by physical system of their creation and process of
their formation (heterogeneity of detectors, lighting, background, dynamic distortions), on the one
hand, and by the methods of their representation and displaying in the processing system
(discretization and quantization errors, changes in color reproduction, gray ambiguity, etc.), on the
other hand.
In order to improve reliability of image analysis performed either by unaided eye or with the help
of automated systems, it is desirable, first, to improve brightness characteristics of the images. Their
characteristics such as contrast, brightness and resolution are of especial importance for visual
perception, for example, in medical applications. For the cases when automated processing systems
are used for solving a specific problem and for obtaining quantitative indicators, it is recommended to
begin with determining and identifying the necessary value parameters on the basis of initial data for
further identifying of the objects of interest [1].
The inherent inconsistency of the image transformation process lies in the fact that it is essential to
ensure, on the one hand, maximum sensitivity of the used methods to insignificant local variations in
brightness, and, on the other hand, resistance to the effects of the structure- and measurement-
CMIS-2021: The Fourth International Workshop on Computer Modeling and Intelligent Systems, April 27, 2021, Zaporizhzhia, Ukraine
EMAIL: akhmlu1@gmail.com (L. Akhmetshina); for___students@.ukr.net (A. Yegorov)
ORCID: 0000-0002-5802-0907 (L. Akhmetshina); 0000-0002-7558-785X (A. Yegorov)
© 2021 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
generated noises. In this case, in addition to randomness, which is described in accordance with the
theory of probability, it is necessary to take into consideration an uncertainty present in the images [2,
3, 4], which is an attribute of information [5]. The currently used approach to solving the problem of
image analysis is based on the use of fuzzy methods [6, 7, 8] because of inaccuracy and
incompleteness of initial data and ambiguity of processing algorithms (for example, when one
determines classes, regions/boundaries of the objects).
2. Review of Literature
One of the ways to solve the problem of improving quality of the images and reliability of their
analysis assumes transition of informative features formed by the method of spatial transformations
[10] to a new space. From the number of methods used for forming informative features, a specific
one should be chosen by physical nature and brightness characteristics of initial data and in
accordance with the task set [11]. The difficulty in determining an effective solution method is
explained by the fact that characteristics of the imaging system, as well as presence and parameters of
the object of interest are often unknown a priori.
Fuzzy sets of the type-1 (T1) make it possible to transform an uncertainty into the membership
function that has a numerical value within the range 0,1 . Nonlinearity of the fuzzy processing
methods increases influence of variations in the brightness properties of the analyzed images and
eliminates ambiguity of gray. However, the fuzzy sets of the type-1 (T1) do not take into
consideration uncertainties in the membership functions, since they are characterized by clear values
[12, 13, 14].
Fuzzy logic of the type-2 (T2) was introduced by Zadeh L. as a generalized concept of the theory
of ordinary fuzzy sets and make it possible to consider problems with a higher degree of uncertainty,
which is typical, in particular, for the methods of image representation and for the algorithms of the
image processing [15, 16]. The corresponding membership functions T2 are determined as a
generalized fuzzy set by introducing fuzzy intervals; such approach correlates with inaccuracy
perception by humans [6]. Fuzzy sets of the T2 feature fuzzy membership functions and are able to
simulate such uncertainties [8, 17]. Fuzzy membership functions Т2 is characterized by upper h x
and lower h x boundaries, each of them is determined by the lower (LMF) and upper (UMF)
membership functions Т1.
The usage of fuzzy logic makes it possible to minimize uncertainties: removing randomness in
fuzzy sets T1 leads to unambiguity, and removing uncertainty in T2 leads to the fuzzy sets T1.
Today, image processing algorithms with using T2 are proposed for solving problems of clustering
[18], filtering [19], boundary detection [20] and in the image classification [21]. In [22], informational
capabilities of the method of segmentation of low contrast grayscale images based on the fuzzy sets
T2 were investigated.
3. Problem statement
Purpose of this work was to study a method for improving quality of the grayscale images basing
on orthogonalization of fuzzy membership functions of the type-2 with using a method of singular
value decomposition and selective filtering of combined noise in conditions of a priori uncertainty of
the imaging system characteristics and in the absence of a priori information about the location of a
possible objects of interest.
4. Materials and Methods
The ideas of methods of projection into eigen subspaces as one of the tools for mathematical
processing of experimental data were presented in [23, 24]. From the theory, it is well known that
decorrelation makes it possible to separate information and improve accuracy of estimates, which,
from a fundamental point of view, opens up additional possibilities for heightening sensitivity of the
analysis procedure.
The Principal Component Analysis (PCA) algorithm for image processing problems representing
two-dimensional structures was first applied in practice in the 2000s [25]. Currently, the PCA and
other orthogonalization methods are used for solving such problems as compression of visual
information and feature extraction during object recognition and video image search.
In the tasks of processing an ensemble of K images represented by a matrix of brightness values
I and dimension dy dx with the help of orthogonalization methods, the main purpose is to
transform initial data into a new coordinate system, for which the following condition is satisfied: the
sample variance of the data along the values of the k-th coordinate should be maximum on the
assumption of orthogonality to the first k-1 coordinates.
In [26], an algorithm for improving sensitivity and reliability of the image segmentation is
described, which is based on the process of multi-stage processing, which includes: expansion of
space of the input features based on the initial data by using the fuzzy clustering method;
orthogonalization of the obtained fuzzy membership functions; and, on their basis, formation and
visualization of new informative features.
In [27, 28], informational capabilities of the method of automorphic imaging with implementation
of orthogonal decomposition are considered in relation to solving the problems of filtering of low
contrast grayscale images. The method is based on the use of dimension-increasing informative
features of locally adaptive transformations, and makes it possible to apply methods of
multidimensional data processing to the grayscale images.
Processing of images with the help of the fuzzy methods can be considered as a nonlinear type of
the initial data transformation, the feature of which is that it is performed for the functions of pixel
membership with predefined clusters [2].
One of the most common types of images formed by systems of various physical nature are the
grayscale images represented by the level of quantization of their energy characteristics I x, y ,
where x and y are the coordinates of a pixel, which usually takes a value within the range 0,1 or
0,255 .
An image of the size of dy dx with gray levels L can be considered as an array of the fuzzy
singletons (i.e. fuzzy sets with a single reference point), which display a value of the membership of
the fuzzy set T1 u x , y for each point of the image I x, y relative to its property to be analyzed (for
example, brightness, homogeneity, noise, etc.). Then, in order to describe an ambiguity of the type-2
inherent to the image, one can introduce the fuzzy membership functions of the type-2 (MFT2) basing
on the expression:
akt ,i u kh,i u kl ,i , (1)
where u h is the "upper" value of the membership function T1 (MFT1), and u l is the "lower" value of
the MFT1.
In Figure 1 graphical image of the membership functions T2 is shown, which is characterized by
upper and lower boundaries, each of them is determined by the lower (LMF) and upper (UMF)
membership functions of the type-1. The shaded area in right part of Figure 1 is called a footprint of
uncertainty (FOU) and displays uncertainty of the solution made in the range between the upper and
the lower boundaries. The distribution form is determined by nature of the problem uncertainty. The
membership estimation for each element of the T2 type is a fuzzy set with the value of the T2 within
the range 0,1 .
For the case of a gray-scale image, this approach allows forming a multidimensional matrix X
with dimensions dy dx K , which makes it possible to use an orthogonal singular transformation
during its processing.
From mathematical point of view, matrix X is decomposed into the product of three matrices:
X U S V T . (2)
Here, U (left singular vectors) is a matrix formed by the orthonormal eigenvectors u r of the
matrix X X T , which correspond to the values r :
X X T u r r u r , (3)
V (right singular vectors) is a matrix formed by the orthonormal eigenvectors vr of the matrix
XT X :
X T X v r r v r , (4)
and S (eigenvalues) is a positively-defined diagonal matrix whose elements are singular values
1 ... r 0 equal to the square roots of the eigenvalues r .
Figure 1: Type‐2 fuzzy set: lower (LMF) and upper (UMF) membership functions and its interval a
footprint of uncertainty (FOU).
As a result, this transformation provides a transition of informative features to the new orthogonal
space. The first singular vector is low-frequency and has a constant component not equal to zero. The
rest of the singular vectors can be interpreted as noise components.
The combination of singular eigenvectors obtained in the space of membership functions T2 for
gray levels of the grayscale image allows performing selective filtering of the noise.
Therefore, the proposed in this work algorithm for improving quality of the grayscale images
consists of the following steps.
1. Scaling of initial range of the input image brightness to the range 0,1 .
2. Preprocessing of initial image The need for this step arises due to the possible irregularity of
the background component in the image, which, in particular, leads to the appearance of light-
stuck or darkened areas. In this research, we performed this procedure by the methods of adaptive
power-law correction of brightness and local background subtraction [29]. In the first method, the
initial grayscale image I undergoes the following transformations:
I 1x , y I x , y 1 / 256, I x , y I 0.5 / 2, x 1, dy , y 1, dx , (5)
where I is a mean of the brightness of image I . The need of applying the expression (5) is
explained by necessity to use the power transformations. As a result, a slight decrease in brightness
occurs, and values equal to 1 are deleted. Then, the image I 1 is subjected to the following
transformations:
1sgn I1 I I1 I 0.5 I1 (6)
1 1 sgn I x , y I
I 1x , y
1 x,y x,y x,y
2
2
I x, y I x, y ,
1 I x2, y / 2
2
1 I x , y (7)
I x3, y I x2, y .
Transformation (6) proportionally reduces brightness of the light-stuck areas and increases
brightness of the dark areas. Transformation (7) proportionally reduces brightness of the entire
image.
For the case when brightness correction is made by the method of local background subtraction
[29], transformation of the initial image is performed with the help of non-overlapping windows
(in our experiments window size was 15 15 ), and brightness of all pixels in each window is
determined by the expression:
w1x , y w I / 2 . (8)
So, this is the way of how values of the image brightness I 1 . On its basis, the image I 3 is formed
by the formula:
I x3, y I x , y I 1x , y . (9)
After applying expression (9) the image I 3 is scaled to the range 0,1 and adaptive histogram
equalization is applied to the obtained image.
3. The image I 3 is interpreted as a fuzzy membership function. And for it, the MFT2 ( I 4 )is
formed on the basis of the "upper" ( I h3 ) and "lower" ( I l3 ) values of the MFT1 according to the
formula (1). In this case, I h3 и I l3 are calculated by the method of power transformations with
using the following formulas:
1 I x3, y
1 I x3, y / 2
(10)
I x3, y h I x3, y ,
1 I x3, y / 2
3
1 I x , y (11)
I x3, y l I x3, y .
This step is necessary for the subsequent application of the singular value decomposition.
4. An ensemble of images is formed ( I 3 , I 4 , I l3 when performing the step 2 with the use of
power-law correction, or I 4 , I l3 , I h3 in the case of preprocessing by the method of local
background subtraction). Various methods are required for forming an ensemble of images
because when the processed images are light-stuck the preprocessing method affects the brightness
level I h3 . In case of the use of adaptive power-law correction, brightness level I h3 слишком
высок, is too high, which makes this image uninformative in comparison with the preprocessed
initial picture.
For this ensemble, singular value decomposition is applied. The resulting matrix of the left
singular vectors U is interpreted as a multidimensional image I 5 with dimensions dy dx K ,
each spectral component of which is scaled to the range 0,1 .
5. On the basis of the matrix of right singular vectors V , a vector of coefficients C is
calculated (it is used for estimating significance of the components of the matrix of left singular
vectors) by the following formula:
K (12)
Ci Vi , j V j ,i / 2, i 1, K .
K
j 1
j 1
This vector is ordered in descending order, and its elements are normalized so that their sum is
equal to 1.
6. The vector dC is formed, which contains differences for each pair of adjacent elements of the
vector C .
7. The value dC a is calculated by the formula:
K 1 (13)
dC a dC j / K 1 dC min dC max / 2 / 2 ,
j 1
where dC min and dC max are the minimum and maximum elements of the vector dC , respectively.
8. On the basis of the value dC a , when scanning the elements of the vector dC starting from
the end such index imax , is selected, for which the following condition should be satisfied:
dCimax dC a . (14)
Then elements of the vector dC with indices from 1 to imax 1 are normalized so that their sum is
equal to 1.
9. The final image I 6 is formed as a weighted sum of the most significant components of the
matrix U according to the following formulas:
imax 1 (15)
I y6, x I x5, y , j dC j S j ,
j 1
ni ni (16)
S j sgn Vi , j V j ,i .
j 1 j 1
For the final image, histogram equalization is applied when its preprocessing is performed by the
method of adaptive power-law correction, and inversion, which is followed by adaptive histogram
equalization, is applied when the preprocessing is based on the local background subtraction
method.
5. Results and Discussion
Quantitative metallography is widely used for specifying characteristics of the alloy
microstructure, namely: volumetric content of phases, grain size, specific surface area of grain
boundaries, distance between similar particles or phases, and others.
To obtain high quality materials, experimental alloys are smelted and mechanism of forming their
structure and morphology, which determine their properties, is researched. For achieving this goal,
among others, methods that allow to form images of the test samples are used (for example, micro X-
ray or X-ray structural, metallographic analyzes, microscopy). Based on analysis of the images
brightness characteristics it is possible to determine such quantitative parameters as the average size
of phases, geometrical value of the external specific surface of the phase, and statistical data.
However, used digital images often feature an inadequate quality due to irregular background,
noise, aberration artifacts, poor contrast, etc. In order to obtain reliable quantitative information from
the pixel intensity values, it is necessary to apply correction methods to ensure good accuracy of
photometry and to eliminate common defects of the images.
In Figure 2а, an example of grayscale image of the phosphorus-containing alloy Fe–2%Р–
0,042%С is shown. This image was obtained with magnification of x250 on a metallographic
microscope GX51 with a digital image analysis system of the company "Olympus". To separate the
individual phases of the alloy during determining its structure and properties, chemical etching was
used at first. After that thermal etching at temperatures of 400-600 oС with natural air circulation was
performed. The atoms that make up the phases interact with oxygen; therefore, an oxide membrane of
different thickness is formed in different phases. When observing through an optical microscope and,
accordingly, in the image, the phases have different levels of intensity [30].
The analysis of image in Figure 2a is difficult, in particular, due to the nonuniformity of the
background and the presence of the light-stuck area.
To obtain experimental results we used Matlab 6.1. Source code for proposed methods was written
in internal Matlab language (except for standard functions such as svd, histeq, adapthisteq, mat2gray).
In Figure 2b and Figure 2c, the results of preprocessing of the initial picture by means of the
adaptive power-law correction and by method of local background subtraction are shown,
respectively. In both cases, we managed to reduce brightness of the light-stuck area and to preserve
the overall intensity level at the level acceptable for visual analysis. In the latter case, the image is
more detailed, but at the same time it contains a blur effect.
In Figure 3 and Figure 4, the results of MFT2 formation are shown, as well as the "lower" and
"upper" values of the MFT1 interpreted as grayscale images, for both preprocessing methods. It can
be easily seen that preprocessing based on the local background subtraction method gives higher
brightness in the generated images.
Figure 2: Image of the phosphorus‐containing alloy Fe–2%Р–0,042%С: a initial grayscale image
(142x186); preprocessing based on the b adaptive power‐law brightness correction; c method of
local background subtraction.
Figure 3: Calculation of the MFT2 for preprocessing based on the adaptive power‐law brightness
correction: a MFT2; b "lower" value of the MFT1; c "upper" value of the MFT1.
Figure 4: Calculation of the MFT2 for preprocessing based on the method pf local background
subtraction: a MFT2; b "lower" value of the MFT1; c "upper" value of the MFT1.
In Figure 5, formation of the resulting image for both preprocessing methods is shown. It should
be noted about different levels of detailing in the results. Use of the adaptive power-law correction led
to the lower level of the resulting image brightness, but, at the same time, gives better image
definition, which is preferable when determining quantitative parameters and highlighting the
contours of the objects and various areas within one object. Preprocessing based on the method of
local background subtraction resulted in a brighter and more contrasting image, which simplifies its
visual analysis, but there the blur effect is seen.
Figure 5: The resulting image after preprocessing by the: a adaptive power‐law intensity
correction; b local background subtraction method.
6. Conclusions
On the basis of analysis of the experimental results obtained, the following conclusions are made:
the usage of fuzzy sets of the T2 type allows to synthesize additional parameters on the
basis of the grayscale initial image with using of nonlinear functions based on the local
transformation of the brightness levels, and to obtain an ensemble of data, to which
methods of multidimensional information processing can be applied;
orthogonalization, the method of singular value decomposition in particular, applied to the
ensemble of images with taking into account components of the T2 set, makes it possible
to take into consideration original ambiguity and uncertainty of the initial data, to analyze
the ensemble as a whole and, at the same time, to interpret each new component as a result
of anisotropic filtering in two-dimensional plane of the spatial frequencies;
visualization of the parameters synthesized on the basis of the eigenvalues of the singular
value decomposition, makes it possible to increase the level of detailing, contrast and
resolution of the resulting image and, hence, to improve reliability of visual and automated
analysis;
promising areas for further researches are the usage of different orthogonal
transformations; and applying of various methods for preprocessing of initial data and
formation of sets of the T2 type.
7. Acknowledgements
The authors thank for the images used in experiments, which were kindly provided by candidate of
physical and mathematical sciences, docent of Dnieper National University Named by Oles Honchar
(Faculty of Physics, Electronics and Computer Systems) Nadezhda Karpenko.
8. References
[1] W.K. Pratt, Digital Image Processing, John Wiley and Sons Inc., New York, NY, 2001.
[2] J.C.Bezdek, J. Keller, R. Krishnapuram, N.R. Pal, Fuzzy Models and Algorithms for Pattern
Recognition and Image Processing, Handbooks of Fuzzy Sets series, Kluwer Academic
Publisher, Boston, 1999.
[3] H.R. Tizhoosh, H. HauBecker, Fuzzy Image Processing: An Overview, in: B. Jähne, H
HauBecker. and P. GeiBler, (Ed.). Handbook on Computer Vision and Applications, Academic
Press, Boston, 1999, volume 2, pp. 683-727.
[4] H. Bustince et al., (Ed.), Fuzzy Sets and Their Extensions: Representation, Aggregation and
Models, Springer, 2008.
[5] L.A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning,
Information Science 8 (1975) 199–249. doi: 10.1016/0020-0255(75)90036-5.
[6] Z. Chi, H. Yan, T. Pham, Fuzzy algorithms: With Applications to Image Processing and Pattern
Recognition, Singapore, New Jersey, London, Hong Kong, Word Scientific, 1998.
[7] L.A. Zadeh, Fuzzy sets and their application to pattern recognition and clustering analysis, in: J.
Van. Ryzin (Ed.), Classification and Clustering, Academic Press, 1977, рр. 251-299. doi:
10.1016/C2013-0-11644-3.
[8] L. Cinquea, G. Forestib, L. Lombardic, Clustering fuzzy approach for image segmentation.
Pattern Recognition 37 (2004) 1797-1807. doi: 10.1016/j.patcog.2003.04.001.
[9] L. Pham Dzung, Jerry L. Prince, Adaptive Fuzzy Segmentation of Magnetic Resonance Images,
IEEE Transactions on Medical Imaging 18 (1999) 737-751. doi: 10.1109/42.802752.
[10] D. Forsyth, J. Pons, Computer vision: a modern approach, 2-nd ed., Prentice Hall, 2012.
[11] A. Yegorov, L. Akhmetshina, Optimizatsiya yarkosti izobrazheniy na osnove neyro-fazzi
tekhnologiy, Lambert, 2015.
[12] O. Castillo, P. Melin, Type-2 Fuzzy Logic: Theory and Applications, Springer-Verlag, 2008.
[13] Wu Haoyang, Wu Yuyuan, Luo Jinping, An Interval Type-2 Fuzzy Rough Set Model for
Attribute Reduction, IEEE Transactions on Fuzzy Systems 17.2 (2009) 301 – 315.
doi: 10.1109/TFUZZ.2009.2013458.
[14] J.M. Mendel, R. John, Type 2 Fuzzy Sets Made Simple, IEEE Transactions On Fuzzy Systems
10.2 (2002) 117-127. doi: 10.1109/91.995115.
[15] J.M. Mendel et al, Interval Type 2 Fuzzy Logic Systems Made Simple, IEEE Transactions on
Fuzzy Systems 14.6 (2007) 808-821. doi: 10.1109/TFUZZ.2006.879986.
[16] W.B. Zhang, W.J. Liu, Fuzzy Clustering for Rule Extraction of Interval Type-2 Fuzzy Logic
System, in: Proceedings of 46th IEEE Conference on Decision and Control, 2007, pp. 5318-
5322.
[17] C. Hwang, F. Rhee, An interval type-2 fuzzy C spherical shells algorithm, in: Proceedings of
IEEE International Conference on Fuzzy Systems, 2004, pp. 1117–1122. doi:
10.1109/FUZZY.2004.1375568.
[18] Z. Ji, Y. Xia, Q. Sun, G. Cao, Interval-valued possibilistic fuzzy C-means clustering algorithm,
Fuzzy Sets and Systems 253 (2014) 138–156. doi: 10.1016/j.fss.2013.12.011.
[19] M. Yuksel, A. Basturk, Application of Type-2 Fuzzy Logic Filtering to Reduce Noise in Color
Images, IEEE Computational Intelligence Magazine 7.3 (2012) 25–35. doi:
10.1109/MCI.2012.2200624.
[20] P. Melin, C.I. Gonzalez, J.R. Castro, O. Mendoza, O. Castillo, Edge-Detection Method for Image
Processing Based on Generalized Type-2 Fuzzy Logic, IEEE Transactions on Fuzzy Systems 22
(2014) 1515–1525.
[21] L. Lucas, T. Centeno, M. Delgado, Land cover classification based on general type-2 fuzzy
classifiers, International Journal of Fuzzy Systems 10 (2008) 207–216.
[22] L. Akhmetshina, A. Yegorov, Low-Contrast Image Segmentation by using of the Type-2 Fuzzy
Clustering Based on the Membership Function Statistical Characteristics In: International
Scientific Conference Lecture Notes in Computational Intelligence and Decision Making. AISC,
2019, volume 1020. pp 689-700.
[23] K. Pearson, On lines and planes of closest fit to systems of points in space. The London,
Edinburgh and Dublin Philosophical Magazine and Journal of Sciences 6.2 (1901) 559-572.
[24] H. Hoteling, Analysis of complex variables into principal components, Journal of Educational
Psychology 24.6 (1933) 417-441.
[25] J. Yang, D. Zhang, A.F. Frangi, J.-Y. Yang, Two-dimensional PCA: A new approach to
appearance-based face representation and recognition, IEEE Transactions on Pattern Analysis
and Machine Intelligence 26.1 (2004) 131-137. doi: 10.1109/TPAMI.2004.1261097.
[26] L.G. Akhmetshina, A.A. Yegorov, Adaptivnaya nechetkaya segmentatsiya izobrazheniy na
osnove kombinirovannogo singulyarnogo razlozheniya, Vestnik KHNTU 3 (2016) 3 (2016) 198-
202.
[27] L.G. Akhmetshina, Povysheniye razreshayushchey sposobnosti izobrazheniy geofizicheskikh
poley na osnove metoda mnogomernoy ortogonal'noy adaptivnoy klasterizatsii, Naukovyy
Visnyk Natsionalʹnoho hirnychoho universytetu 10 (2003) 35-38.
[28] L.G. Akhmetshina Adaptivnaya fil'tratsiya shumov v signalakh i izobrazheniyakh: metod
selektivnogo singulyarnogo razlozheniya avtomorfnogo otobrazheniya, Iskusstvennyy intellekt 3
(2005) 328-335.
[29] A.A. Luk'yanitsa, A.G. Shishkin, Tsifrovaya obrabotka videoizobrazheniy, Ay-Es-Es Press,
Moskva, 2009.
[30] O.B. Sukhova, N.V Karpenko, The peculiarities in contact interaction processes and technology
to fabricate multi-layered composites, Metallofizika i Noveishie Tekhnologii 30 (2008) 585-594.