The non-factor fuzziness is considered as one of the forms of uncertainty modeling by detecting hidden knowledge contained in the original data array (cloud of data) and fuzzy set. It is shown that the objective component of the fuzziness consists of 2 parts: 1-subset of ordered pairs, calculated on the basis of tensor decomposition structured as a 2D or 3D tensor of the initial data array; 2 - the immersion of the interval of possible values of the fuzzy set into a special matrix (Toeplitz), followed by a tensor decomposition. The results obtained: it is shown that in the general case analysis of uncertainty must be performed using a multi-fuzzy set; with restrictions on the formation of the membership function it is suggested to use a subset of ordered pairs on the basis of objective fuzziness; the developed algorithm for reducing the multi-fuzzy set to FS-1 type. The following examples illustrate the effectiveness of the proposed methodology.
The theory of fuzzy sets (TFS) as a method and apparatus for solving problems in uncertain conditions has shown its high efficiency. However, at present, its application has encountered a number of difficulties due, in particular, to the following factors: the emergence of fundamentally new problems associated with the complication of technological processes, which limits the possibility of assigning a heuristic membership function (MF); the need to work with data of super large volumes and high measurements, which require consideration of their life cycle; philosophical interpretation of the phenomenon of fuzziness, which declares in the composition of this phenomenon the objective and subjective components, involves taking into account its influence on the structure and type of fuzzy set (FS), because the standard FS is constructed only on the basis of the subjective component - FS, that is, the standard type of FS is a special case; the initial data set (IDS), on the basis of which the universal set (US) is calculated, is the bearer of hidden knowledge, but at present, research on the influence of the structure of IDS on the form and structure of FS has not been carried out; IDS containing fuzziness (or inaccuracy), allows to distinguish several types of formal (objective) fuzziness in the form of subsets of ordered pairs (SOP).
A x, A (x) | x X , A [
In this regard, it is advisable to turn to the theory of non-factors [
In this context, only data fuzziness is considered. Separately, pay attention to
inaccuracy, in the work an example of modeling of inaccuracy is given by a subset of
ordered sequences. It is shown in [
M. Black [
Here are the main provisions of work [
The methodological basis for considering the phenomenon of fuzziness and the concept that it expresses is the formalization of intuitive notions of fuzziness. For this purpose, an analysis of the forms of the appearance of fuzzy bones is used to identify the components that make up its contents and the synthesis of the system of interconnections of the fuzziness of the object's property in the process of cognitive activity of the subject. For the formalization of intuitional notions of fuzziness, the method of determination through abstraction is used; the method of an equation is used in the process of understanding the ontological grounds of the phenomenon of fuzziness and the appearance of the relationship between its objective and subjective aspects.
One of the most important conclusions of the work [
Since fuzziness is a characteristic, first of all, the cognitive process, then the solution
to this issue is the most effective method from the establishment of consistency patterns
to the construction of a formal system based on them. In work [
The theory of non-factors proposed by A.S. Narinyany, devotes great attention to the analysis of non-precision, which is a multidimensional characteristic of fuzziness, being one of the most common forms of its expression. The content of the concept of fuzzy contains a consistent series of lower-level abstractions, which in turn, in each case, contain abstractions of an even lower level, both explicit and intuitive; the common thing for them is that they characterize the power of the real objects, phenomena, relations and concepts.
Touching TFS, we note that in the general case in the formation of FS (process of fuzzification) objective fuzziness is to calculate the interval US, subjective - in the choice (heuristic) MF, while it should be borne in mind that the basis of the methodological approach to the analysis of the relationship objective and subjective fuzziness is the rationale for choosing one of two aspects of the concept of fuzziness as defining in relation to another. In this context authors, objective inaccuracy is additionally determined by the artificial blurring of the US with the help of the immersion procedure in a special matrix.
In [14], it is shown that the problem of uncertainty (which, unfortunately, reduces only to one of the non-factors - fuzziness) inevitably arises in the simulation of complex systems, in which a person plays an active role. In order to obtain significant conclusions about the behavior of a complex system, it is necessary, in principle, not only to abandon the high standards of accuracy and severity which are characteristic of relatively simple systems, but to consider alternatives, not only varying with different standard FS, but also changing the approach to computation SOP.
Objective fuzziness is a description of the objects, properties and relations of the real world, not dependent on consciousness; subjective fuzziness, in turn, characterizes the process of knowledge, the human way of thinking and knowledge of objective fuzziness measures reflection of the internal patterns of the object in its external manifestations. Subjective fuzziness is defined as a measure of the reflection of the phenomena of the external world into the inner world of the individual, which allows for the personality of the human psyche to be taken into account. At least at the level of sensation and perception, subjective fuzziness has an objective basis, which, in turn, gives grounds to believe that objective obscurity is decisive in relation to subjective.
Subjective and objective types of fuzziness also differ in depth of reflection of the essence of processes and in relation to these types of basic categories of dialectics. Reflecting the contradictory process of cognition, objective and subjective fuzziness, while in a dialectical contradiction, adequately describe the phenomenon of fuzziness in its totality, indicating their dialectical unity. The study of various interpretations of fuzziness, their analysis and generalization showed that no interpretation completely does not cover the content of the concept under study and does not fully reflect its essence. Therefore, there are prerequisites for considering the fuzziness as a theoretical concept, in which any preliminary interpretation will act as a separate case. From a methodological point of view, the interpretation of fuzziness as a theoretical concept has the essential advantage that such an understanding of fuzziness does not allow to reduce any one interpretation of this concept into a general rank and, at the same time, deny any of them due to its incompleteness. 2.2
In table 1 is presented main abbreviations, that are used in the article. j-th column, i-row of matrix
Frobenius norm
Theory of fuzzy set
Initial data set Set of ordered pairs
Kronecker product Principal component analysis
Multi-fuzzy set Singular value decomposition 3
x x / x, x [
In TFS, the notion of the nearest (to fuzzy x ) crisp set is considered, the closeness is understood as the proximity of the norm, in particular, Euclid, Frobenius, or elsewhere, that is x x0 2
F min , where
n
x (x x )1 ,
x X , x [
F
1/2 n trace A AH , the sets x x and x 1 n x0 x 0 must be represented in the matrix form, in particular, in the form of the x 1 KP- x x T and x x0 T respectively.
The concept of the nearest object in the work is expanded by forming the tasks of determining the subsets of ordered pairs: a) nearest to the given FS; b) nearest to the structured IDS. Let's pay attention to the following. The nearest crisp set approximates the fuzzy, subset of ordered pairs nearest to the standard fuzzy can be used for comparison or estimation or to be more convenient for further processing, since it always has the same form and obtained on the basis of formal transformations, which excludes the participation of an expert. Accordingly, the following tasks are formulated.
An object in uncertainty is represented by an IDS (or array), in the general case, a data cloud, for example, an array of data obtained when measuring a certain value with devices with limited accuracy in conditions of interference. To work with this object, its representation is used in the form of some statement such as "approximately equal to ...", "close to ...", etc., which, in turn, can be presented to FS x x / x , x 0,1 , x X x , where x 0,1 is MF, which is appointed by the expert heuristically, as a rule, from the composition of the means of a specific application package, which is used in the work.
Need to show:
the existence, in addition to the standard FS
x( j) X of a collection of SOP that is endowed with properties of FS (having one
of the components as a weight function) are analyzed on the basis of formal
procedures and represent the object no less effectively than FS; it is important that the
specified SOP are obtained by detecting hidden knowledge (intellectual analysis)
that are found both in the components of the FS and in the IDS;
the possibility of using the SOP as an FS, if there are restrictions in the designation
of FS;
the expediency of submitting an object in conditions of the uncertainty of a multi of
fuzzy sets containing FS and SOP that characterize this object;
the possibility of using methods of fuzzy mathematics, developed for type FS-1
type, on a multi-fuzzy set (MFS);
real examples of the solution of the above tasks.
x( j) x( j) / x( j) , x( j) [
Definition 1. Subjective fuzziness is a subset of ordered pairs xi xi , one of its components of which xi 0,1 plays the role of weight function and obtained on the basis of mental activity of a person (in particular, common sense); objective fuzziness xi 0,1 obtained on the basis of formal calculations.
Formation of a subset of ordered pairs based on the blurring of the vector of input data [15]. One of the tasks that were posed by the authors was to use the logical and semantic proximity of tasks that arise in the processing of noisy (blurring) images and procedures of blurring (fuzzification) input values in the formation of FS. As it is known, the restoration of blurry and tired images, for example, magnetic resonance tomography (MRT) is the main problem in digital images. Some elements of the blind deconvolution method used to restore a crisp (sharp) image from blurring and noisy MRT images can be used to solve problems, in particular, to make decisions under uncertainty, using TFS.
Recall that blind deconvolution is a method of restoring a sharp (crisp) version of a
blurred image when the source of blurring is unknown. In some applications,
particularly in the physical imaging problems, the blurring process is not known and needs to
be restored with the image. A similar situation occurs when using TFS - the process of
fuzzification x xi / xi , xi [
It is known [17] that if one imagines a degradation (image, signal) in the form of a linear fashion, i.e. g (x, y) = f (x, y) h (x, y) + (x, y), where - a symbol of Kroneker's product, f (x, y) is the initial image, h (x, y) is the function of scattering points are unknown functions, then the restoration is reduced to the problem of blind deconvolution [18]. The expression for g (x, y) is also considered in the vector-matrix form: g H f , where the matrix [H] can be constructed from the function h of the discrete point distribution and has the structure of the so-called special matrix.
These can be the Toeplitz matrix, Hankel matrix or a combination of them: for example, a block matrix in Toeplitz form with Hankel-shaped blocks or the like. General view - the method of forming a Toeplitz matrix from a vector (in this case US) - is shown in Fig.1. This matrix-vector form can be useful in analyzing the problem of fuzziness, which is the basis for the formation of a SOP with the properties of the FS.
It is shown [19] that in most applications where using a blurred image B restore its sharp and clear version X, the most common model of the blurring process is given as B = A [X] + N, where A is the blurring process, be it linear or nonlinear process, and N denotes some additive noisy process. For digital imaging X and B are discrete vectors or arrays, and as a result, the linear blur A corresponds to a matrix-matrix operator. In the case when the blur operator A is invariant with respect to the displacement, the corresponding blurring matrix will be well structured and may be, for example, Toeplitz, circular or Hankel. The algebra of calculations with structured matrices is considered in the paper [20]; in this paper we consider the cut singular decomposition of the matrices because it allows the obtaining of the SOP, and the first component of which is the US, and the second - weight function, that is, SOP structurally and semantically coincides with FS, although calculated on the basis of formal relations.
Tensor models. A tensor is a d-linear form or d-dimensional array: A a i1i2...id Tensor has: dimensionality (order) d - number of indices (measurements, axes, directions), size n1. . .nd - the number of readings per axis [21]. An example of the sepa . ration of variables is singular decomposition A avaua . The canonical tensor a1.r decomposition also appears in terms of the Kronecker's product (KP) of the matrices. Case d 3 differs substantially from the case d = 2: the approximation of a tensor with a decrease in rank is a complex multidimensional optimization task. In [22], the efficiency of using tensor approximations for calculations is shown, and the d-dimensional tensor can be considered as a vector-column with d-dimensional indexation. An extreme case is the transformation of a vector of size N = 2d into a d-tensor of measurements 22 ... 2.
When maintaining the total number of elements of the tensor, the effectiveness of its use (representation of the tensor, the possibility of performing algebra operations, the formation of FS-2 or n-types, etc.) can be significantly increased by increasing the number of measurements and reducing the number of readings for each dimension. One of the specific characteristics of a tensor is its quantization-the introduction of additional (virtual) measurements. While maintaining the total number of elements of the tensor, the effectiveness of its presentation can be substantially increased by increasing the number of measurements and reducing the number of readings for each dimension.
The main object of the TFS - SOP is to some extent universal, but in a number of
cases it can be used in other. forms [23], in particular, when performing operations of
fuzzy mathematics and fuzzy logic than is customary in the TFS. FS-1 type
a a / a , a U , a [
min .
F
Note that in the TFS, in addition to the most common types of FS-1 type, designed
for the simulation of the simplest types of uncertainty, in which the values of the MF
from the interval [
The basis of matrix (tensor) decompositions. Since the appearance of matrix decompositions, later extended to tensors, they were treated as a decomposition on a basis and weighted coefficients [25], which are the bases of all decompositions, are most clearly visible in the methodology of Principal component analysis (PCA) and singular value decomposition (SVD) . But if one considers that one of the interpretations of the FS and MF is that the FS is a subset of ordered pairs, one of whose components is a universal set (basis in the notation of the PCA), the second is the heuristic significance of each element of the US in truth, in particular some assertions (weighted coefficients in the notation of the PCA, we can see the identity in the definition and semantics of the FS and the matrix (tensor) decomposition. SVD is a decomposition of a real (complex) matrix in order to bring it to a canonical form.
In the writings of authors it is shown that SVD can be used as a procedure for determining subsets of ordered pairs similar to FS. This allows us to expand the possibilities of singular expansion - from solving standard tasks - approaching the least-squares method, compression of images, etc. - to tasks in conditions of uncertainty. An example, when the matrix 33 is represented by a subset of ordered pairs (Tabl.2) in such way: the KP of a component of which allows a new matrix (initial approximation) with an accuracy of not less than 1%, is given below. Note that the absolute values are left and the right singular vectors have the meaning of weight functions, in addition, sum (v (:, 1)2)=sum (u (:, 1)2 ) = 1.
initial matrix
Decomposition of the 3rd order tensor in Fig. 2. A comparison of matrix and tensor theorems shows a clear analogy between two cases, in particular, the left and right singular vectors of the matrix are generalized as n-modal singular vectors nal matrix (n) R InIn and the column wise extended orthonormal matrix V (n) C In1In2INI1I2In1In are defined in accordance (n) diag 1(n) , 2(n) ,... I(n) ,
n V(n)H S(n) U (n1) U (n2)...U (N 1) U (1) U (2) ... U (n1) T , (3) (4) ~ in which S (n) - a normalized version S(n) with rows, varied according to the scale to one-dimensional length S(n) (n) S~ (n) .
Multi-fuzzy sets as a way of storing, analyzing and processing data in conditions of uncertainty. Given that in the general case, as a model of uncertainty (fuzziness), has an objective and subjective component, it should be recognized that the MFS will be more adequate to the model than the application of the FS. In addition, MFS is a unique way of representing, analyzing and processing large data, a subset of ordered sequences, the most natural form of representation of uncertainty. The theory of multiple fuzzy sets in terms of multidimensional membership functions is an extension of a series of related theories: fuzzy sets, L-fuzzy sets, intuitionistic fuzzy sets, which are discussed in [27], from which the basic definitions are taken.
Definition 2. Let X be a non-empty set and {Li: iP} be the family of complete lattices. Multi FS A in X is a set of ordered sequences:
A { x,1 (x),2 (x),...,i (x),... : x X } , where i LiX , for iP.
We recall that a complete lattice is a partially ordered set in which any non-empty subset has an exact upper and lower bound, usually called the union and intersection of the elements of a subset.
Remark 1. If the sequences of membership functions have only k-terms (the
end-ofnumber of terms), k is called dimensional A . If Li = [
Definition 3. Let k be a positive integer and let it go , in MkFS (X) such, that =( μ1, ..., μk ) = { x, μ1 (x), μ2 (x),..., μk (x): x X} and = ( 1,...,k ) ={ x, 1 (x), 2(x), ..., k (x): x X}, then we have the following relations and operations: 1. if and only if i i to all i=1,2,…,k; 2. = if and only if i =i to all i=1,2,…,k; 3. = = (1 1,…, k k)={ x, max (1(x),1(x)),…, max(k(x),k(x)): xX}; 4. = (1 1,…, k k)={ x, min (1(x),1(x)),…, min(k(x),k(x)): xX}; 5. + = = (1+1,…, k+k)={x,1(x)+1(x) - 1(x) 1(x),…,k(x)+k(x)-k(x) 1kx): x X };
Remark 2. Let μ = (μ1, μ2) be the fuzzy set of measure 2 and let μ1 (x), μ2 (x) be the gradation of the membership and non- membership of the quantity x in μ, respectively. If μ1(x) + μ2(x) ≤ 1, then μ is an intuitionist FS. Consequently, every intuitionist FS in X is a multi-fuzzy set in the X dimensionality of 2, and each intuitionist fuzzy operation is a multi-fuzzy mapping on the multi fuzzy sets. But multi-FS does not necessarily have to be an intuitionistic fuzzy set, for example, multi-FS μ = {(x, μ1 (x), μ2 (x)): μ1(x)= .9, μ2 (x) = .8, x X} is not an intuitionistic FS.
Arithmetic on multi-fuzzy sets:
A+B={(x, A+B(x), A+B(x))|xX}, where A+B(x)=A(x)+B(x)-A(x) B(x), A+B(x)= A(x)B(x),
AB={(x, AB(x), AB(x))|xX}, where AB(x)=A(x)B(x) , AB(x)= A(x) + B(x) - A(x)B(x).
We note that between the subjective and the objective fuzziness, which are characterized, above all, by the proximity of the F-norm and the value of the defuzzified value of the FS, there are certain interconnections, and there are cases when these values are close or coincide. 5
Assume that an object is an inaccurately measured set (array) of values, which is structured as a matrix of 88
The object obtained by the above-mentioned method (under uncertainty) is formalized: a) in the form of an FS with a heuristic designated MF using standard library MF; b) SOP №1, which is formed on the basis of immersion of the interval US in a special matrix; c) SOP №2, which is formed on the basis of IDS (Fig. 3). Recall that in the general case we have 3 channels of formalization of uncertainty:
standard (for TFS) - FS a a / A, A [
a / A(2) , A(2) [
2. .Formation FS: x {x / x} , x[
m=max(abs(U2(:,1)).* abs(U3(:,1))) sort([abs(U1(:,1)*S(1))*m,abs(U2(:,1))/max(abs(U2(:,1))), abs(U3(:,1))/max(abs(U3(:,1)))]), where m is the norm factor, introduced for compatibility of the results in the 2D-tensor. In the MFS notation we have: x=abs(U1(:,1)*S(1))*m, (1) =abs(U2(:,1))/max(abs(U2(:,1))), (2)=abs(U3(:,1))/max(abs(U3(:,1))),
Δ that is x {x/ μ(1),μ(2) } .
Due to the fact that the solution under uncertainty conditions is most often adopted
on the basis of the standard FS-1 type {x / }, x X, [
FS x
1
The second method is to immerse the set of frontal slices x (:,:, 1), x (:,:, 2) and x (:,:, 3) in the Toeplitz matrix: 1 = toeplitz ([x (:,: , 1), x (:,:, 2), x (:,:, 3)]) we have 1 = [x (:,:, 1) x (:,:, 2) x (:,:, 3); x (:,:, 2) x (:,:, 1) x (:,:, 2); x (:,:, 2) x (:,:, 1)]. The singular decomposing of matrix 1 allows one to obtain a SOP x {x / } which represents the integral characteristic of the object under uncertainty conditions.
Example. An object in conditions of uncertainty (inaccuracy of measurements, the effect of interference, etc.) is described in the form of a statement more than 3 and not more than 9 and represents one of the realizations of the expression X = 3 + rand (9) * 6; for its formalization by an expert, the proposed FS x = (x / x) with a triangular MF of the form: trimf = trimf (x, [3.00 6.00 9.00]), xX = [3.06 3.76 4.47 5.17 5.88 6.59 7.29 8.00 8.70], defuzzified value of FS, calculated by the center of gravity (CG) method and the F-norm are given below: def ( x ) = 5.98, x F =norm ( x , 'fro') = 18.54. In addition to heuristic FS, this object can be characterized by a further number of subsets of ordered pairs, endowed with properties of FS, which is accepted as a standard, the results of calculations are given in Table 3. || x ||F ...def (x)|| x ||F ...def (x) || x ||F ...def (x) || x ||F ...def (x)
Thus, in conditions of uncertainty, under which in this case we will understand the situation when a semantically uniquely identified object (for example, the result of measuring a certain value in the conditions of interference), the known initial data array, which allows structuring and determine the interval of values, which is accepted as a US for the formation of FS, an object in conditions of uncertainty can be represented by the totality of FS and SOP.
In this case, the standard FS x {x/ x }, x [
SOP2 x2 {x/ x2 }, x2 [
All SOP and FS have the properties: proximity to F-norm, defuzzified valuesdef (x) def (x1) def ( x2 ) takes place inclusion - X1X, X2 X. It should be noted separately that the integral model of uncertainty containing the of the Kronecker product components of FS, IDS, the matrix of the Toeplitz model of fuzziness, which is considered as a block vector, immersed in a special matrix, has the value || x ||F and def ( x ) that is practically the same as the FS, but the length of the interval US , on which the SOPs are defined, is much shorter than the corresponding magnitude for FS (1.27 versus 5.5). Failure to take into account this factor can have a significant effect on decision making as a result of the use of fuzzy mathematics. FS by its nature gives guaranteed partial overvaluation, it can be effectively used as a control parameter in decision-making.
Operations of fuzzy mathematics in the system of subsets of ordered pairs calculated on the basis of the singular decomposition of the Toeplitz matrices formed by immersing the real component of the FS into a special matrix.
Let 2 fuzzy number (or FS) be given: a {a / a}, a 0,1, aX; b {b / b} ;
b 0,1, bX; the principle of fuzzy expansion the result of a mathematical
(arithmetic) operation on them allows you to represent in the form c a *f b {c / c}
where *f{+, -, *, /}, c = a * b, c min( a ,b ),c [
Suppose that the FS a and b are represented in the form of FS with a triangular or trapezoidal MF, that is, trimf (x, [a, b, c]) or trapmf (x, [a,b,c,d]), the choice of these FS is due to the fact that they explicitly contain the interval of values for trimf x=Iac, for trapmf x = Iad. If the number of α-levels is n, then the FS as the SOP has the form of matrices with the dimension 2n, the 1-st vector-column (a1a2 ...an )T or (b1b2...bn )T used to form the Toeplitz matrix simulating the process of blurring (fuzzines) the interval. an(T ) ~ The procedure for b is performed similarly. The singular decomposition of the ~ (n) Toeplitz matrixes Ta and Tb allows us to calculate the SOP a(n) and b , which are ~ structurally and functionally similar to a and b , because || a(n) ||F || a ||F , The result of the operation a *f b is nearest to the F-norm and the defuzzification ~(n) value of the result a
~(n) *f b .
Table 4 shows the results of an arithmetic operation, which is performed on a FS (with expertly assigned MF) - a subjectivistic fuzziness, and over a SOP calculated on the basis of formal interval blurring models by immersion of interval (US) into the Toeplitz matrix.
Remark 3. Specificity of the Toeplitz matrix is such that it must be formed on the basis of a vector of US with a dimension of 1m, m = 2n, whereas the FS is based on a US measure of 1n. The SOP with the dimension of 2m, which is calculated as a result of the singular decomposition of the Toeplitz matrix, has the same content in two adjacent lines. In order to be able to compare on the basis of calculated F-norms and defuzzified values of FS and SOP, measuring 2n and 2m, respectively, SOP are cut to a dimension of 2n, using only odd lines. 7
Conclusions 1.The complication of tasks in conditions of uncertainty, due, above all, the complication of technological processes requiring automation, the need to take into account the data of super-large volumes and measurements, which are represented by rank-1 tensors, leads to the need to take into account the situation when the MF, cannot be fundamentally formed or have insurmountable difficulties in their formation. Another reason that determines the limited use of TFS for solving problems under uncertainty is that under real conditions, the object is described not only by the standard MF, which is designated heuristically, but also by a number of SOP, caused by the discovery of latent knowledge that takes into account the structural features of the initial data array and the value interval - US, on which the MF and SOP are formed.
2. The technology of blurring of the interval on which the FS is formed is proposed as a procedure for immersing this interval (vector) into a special matrix (Toeplitz), the singular decomposition of which allows one to calculate the SOP, which is one of the characteristics of uncertainty. It is shown that this SOP is the nearest (in the sense of the F-norm) to the heuristic FS intended for this range and can be to some extent a verification test for the expediency of using the adopted MF.
3.The necessity (in a number of cases) of the simultaneous consideration of the subjective component of the fuzziness in the form of the KP of the components of the FS (2D tensor mm), structured IDS and the blurred interval of FS (objective fuzziness), characterizing the uncertainty, by creating a block of the Toeplitz Matrix, the singular decomposition of which allows for a generalized SOP, which integrally characterizes the uncertainty. Over SOP, obtained by means of singular decompositions (2D and 3D tensors), it is possible to apply without limitation all processing methods developed for standard FS. 14. Perepelitsa V. A., Tebuyeva F.B.: Diskretnaya optimizatsiya i modelirovaniye v usloviyakh neopredelennosti dannykh . Moscou (2007). 15. Debnath, A., Rai H., Yadav C., Agarwal A: Deblurring and Denoising of Magnetic Resonance Images using Blind Deconvolution Method. In Journal of Computer Applications vol. 81, 7-12 (2013). 16. Cichocki, A. et al.: "Tensor decompositions for signal processing applications: From twoway to multiway component analysis. In Journal Signal Processing Magazine vol. 32, 145163 (2015). doi: 10.1109/MSP.2013.2297439 17. Gonsales R., Vuds R., Eddins S.: Tsifrovaya obrabotka izobrazheniy v srede MATLAB.
Russia (2006). 18. Chaudhuri, S., Velmurugan, R., Rameshan, R..: Blind Image Deconvolution: Methods and Convergence. Springer International Publishing, (2014). doi https://doi.org/10.1007/978-3319-10485-0 19. Hansen С., Nagy J. G., O’Leary D. P.: Deblurring images: Matrices, spectra, and filtering, by Per,SIAM, Philadelphia, PA, Arxiv. pp. 130 (2006). 20. Gray R. M.: Toeplitz and Circulant Matrices: A review. Department of Electrical Engineering Stan-ford University. Stanford 94305, USA. https://ee.stanford. edu/~gray/toeplitz. last accessed 2018/10/19. 21. Tyrtyshnikov Ye.: Metody chislennogo analiza na osnove tenzornykh predstavleniy dannykh http://mpamcs2012.jinr.ru/file/tyrtyshnikov.pdf. last accessed 2019/05/12. 22. Tyrtyshnikov Ye.: Tenzornyye approksimatsii matrits, porozhdennykh asimptoticheski gladkimi funktsiyami. In: Journal Matematichesky sbornik, , vol. 194, 147–160 (2003). doi: https://doi.org/10.4213/sm747 23. Minayev Yu.N., Filimonova O.Yu., Minayeva J.I.: Kronekerovy (tenzornyye) modeli nechetko-mnozhestvennykh granul. In: Journal Kibernetika i sistemnyy analiz, vol 50(4), 42-52 (2014). doi: https://doi.org/10.1007/s10559-014-9640-6 24. Zade, L.: Ponyatiye lingvisticheskoy peremennoy i yego primeneniye k prinyatiyu priblizhennykh resheniy. Moscou (1976). 25. Pajarola, R.., Ballester-Ripol,l R.: Tutorial: Tensor Decomposition Methods in Visual Computing. Tensor Decomposition Models. https://www.ifi.uzh.ch/dam/jcr:ded4873d-64d84ecf-b60f-2fb2742d9c16/TA_Tutorial_Part1.pdf. last accessed 2019/05/11. 26. De Lathauver, L., De Moor, B.,Vanderwalle, J.: A Multilinear Singulatr Value Decomposition. In: Journal SIAM Journal on Matrix Analysis and Applications, vol. 21(4), 1253–1278 (2000). 27. Sebastian, S., Ramakrishnan ,T.: Multi-Fuzzy Sets. In: Journal International Maths Forum
International Maths Forum, vol. 50, 2471 – 2476 (2010). 28. Costantini R., Sbaiz L., Susstrunk S., Higher order SVD analysis for dynamic texture synthesis. In: Journal IEEE Trans. Image Process. vol. 17(1), 42–52 ( 2008).