=Paper=
{{Paper
|id=Vol-3101/Paper14
|storemode=property
|title=Reconstruction of acoustic surfaces from incomplete data as an identification problem based on fuzzy relations
|pdfUrl=https://ceur-ws.org/Vol-3101/Paper14.pdf
|volume=Vol-3101
|authors=Olexiy Azarov,Leonid Krupelnitskyi,Hanna Rakytyanska,Jan Fesl
|dblpUrl=https://dblp.org/rec/conf/citrisk/AzarovKRF21
}}
==Reconstruction of acoustic surfaces from incomplete data as an identification problem based on fuzzy relations==
https://ceur-ws.org/Vol-3101/Paper14.pdf
Reconstruction of Acoustic Surfaces from Incomplete Data
as an Identification Problem Based on Fuzzy Relations
Olexiy Azarov1, Leonid Krupelnitskyi1, Hanna Rakytyanska1 and Jan Fesl2
1
Vinnytsa National Technical University, Khmelnitske Shosse 95, Vinnytsa, 21021, Ukraine
2Czech Technical University in Prague, Jugoslávských partyzánů 1580/3, Prague 6 Dejvice, 160 00, Czech Republic
Abstract
The model of the acoustic surface in the form of the system of fuzzy relational equations (SFRE) is
proposed. The relationship matrix connects fuzzy locations of sources or their groups and sound
energy levels. The problem of acoustic surface reconstruction from incomplete data is reduced to the
problem of identifying the matrix of fuzzy relations by solving the composite SFRE. Properties of the
solution set allow avoiding the generation and selection of the source distribution parameters. The
method for reconstructing acoustic surfaces by solving the composite SFRE is proposed. To
reconstruct the set of solutions in the form of fuzzy if-then rules, the genetic-gradient algorithm is
used. The reconstruction process is simplified due to ability to parallelize the process of numerical
solution of the composite SFRE, that allows to increase the frequency of reconstruction when
processing acoustic data streams. To minimize processing time, the number of microphones is limited,
provided that the risk of incorrect reconstruction remains acceptable. For the testing set of acoustic
images, the risk of incorrect reconstruction is evaluated by the comparison of the extracted rules and
the rules which describe the real acoustic surface. The risk of incorrect reconstruction of the acoustic
level is defined as the ratio of the number of rules from the contiguous and remote power classes to
the total number of rules in the actual power class. The risk of incorrect reconstruction of the acoustic
surface is defined as the average risk of incorrect reconstruction over all sound energy levels.
Keywords 1
Inverse problems in acoustics, risk of incorrect reconstruction of the acoustic surface, identification
based on fuzzy relations, solving fuzzy relational equations
1. Introduction
Microphone arrays are the standard technology for acoustic field visualization in terrain
monitoring systems [1]. Physical principles of the construction of arrays with a limited number
of microphones cause the problem of sparse data through the irregular distribution of focal points
at the intersection of rays. Reconstruction of the acoustic field from incomplete data is based on
the retrospective propagation of sound pressure by solving the inverse problem [2]. The problem
CITRisk’2021: 2nd International Workshop on Computational & Information Technologies for Risk-Informed Systems, September
16–17, 2021, Kherson, Ukraine
EMAIL: azarov2@vntu.edu.ua (O.Azarov); krupost@gmail.com (L.Krupelnitskyi); rakit@vntu.edu.ua (H.Rakytyanska);
fesl@post.cz (J.Fesl)
ORCID: 0000-0002-8501-1379 (O.Azarov); 0000-0001-7370-9772 (L.Krupelnitskyi); 0000-0001-5863-3730 (H.Rakytyanska);
0000-0001-7192-4460 (J.Fesl)
© 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)
of increasing the acoustic image resolution consists in finding the coordinates and powers of
sound sources, provided that the number of sources and their configuration are unknown. The
model of acoustic field is built on the basis of the fundamental laws of sound energy propagation
[2]. The use of classical regularization methods for solving the inverse problem of sound field
reconstruction is limited to cases of a known number of sparse sources [3, 4]. Under data
uncertainty, the source parameters are estimated using statistical methods [5, 6]. Such methods
require significant computational costs for conducting a series of expensive experiments in order
to specify the position of sources and their powers.
To minimize processing time, the number of microphones is limited, provided that the risk of
incorrect reconstruction remains acceptable. The problem of acoustic field reconstruction from
incomplete data can be considered as an identification problem based on fuzzy relations [7, 8].
The acoustic surface is described by the fuzzy rule base, which is modeled by the fuzzy
relationship matrix. In the theory of fuzzy relational equations, the problem of extraction of the
relationship matrix from experimental data belongs to the class of inverse problems [9]. The set
of solutions to the inverse problem corresponds to the set of variants of the acoustic field
reconstruction in the form of the set of if-then rules [8]. The risk of incorrect reconstruction of
the acoustic surface is defined as the ratio of the number of incorrect rules to the total number of
rules. Therefore, it is advisable to use intelligent identification technologies that will provide the
permissible risk of incorrect reconstruction of the acoustic field without resorting to a series of
costly experiments [7, 8].
2. Literature Review
The problem of incomplete data is solved by increasing the number of measurements, where
estimates of source parameters are determined by beamforming methods [10]. In contrast to the
classical beamforming, a series of experiments are performed from different positions of the
array with a variable distribution of sensors (microphones). In [10], the sequence of array
positions around the sources is chosen randomly or experiments are performed with
deterministic patterns of measurement positions. The concept of forming the average beam,
which consists in choosing a stationary field based on the results from different measurement
positions, is proposed in [11]. In [12], the method of beamforming based on the most frequently
repeated observations is derived. In [12, 13], the optimal array position and beamforming mode
are determined on the basis of the Monte Carlo method.
Further processing of the experimental results is aimed to estimate the position of the sources
and their contribution to the overall distribution of sound energy of the field by statistical
inference methods [14]. The inference method [14] is based on maximizing the likelihood
function, which can be interpreted as the measure of agreement between the statistical model and
uncertain measurement data. To estimate the parameters of sound sources, an algorithm of
expectation maximization is used, which iteratively maximizes the criterion by sequentially
estimating the contribution of each source to the overall distribution of the sound field energy
[14]. The reliability of the source parameters estimation is determined by the degree of data
sparseness [15]. In general, the contribution of each source is estimated by simulation modeling
[16]. The candidate source field is removed from the field generated by the cumulative effect of
the sources until the noise level is less than the threshold value [14, 16, 17].
Refinement of the acoustic image by the method of Bayesian inference is called Bayesian
focusing [10, 18, 19]. Resolution of the reconstruction based on the Bayesian model is increased
due to a priori information on the sources distribution. This makes it possible to consistently
estimate the position of sound sources by selecting a model of sound energy distribution
according to the measured data. As a result of solving the inverse problem, the scan parameters
are automatically estimated together with the source distribution parameters. In this case, the
current estimates of the source parameters are used to update the tuning functions at the next
iteration.
Statistically optimized field reconstruction requires significant costs. The cost of experiments
rises with increasing requirements for equipment [10, 13, 20], which in contrast to empirical
tuning provides optimal beamforming parameters in accordance with source distribution
parameters. However, processing of acoustic stream data eliminates the possibility of conducting
a series of experiments for multiple initialization of iterative search for sound sources
parameters. In this case, derivation of a three-dimensional model of the acoustic field is carried
out using alternative technologies. Separation of closely located sources on the terrain is not
advisable in terms of computational costs, as excessive detail does not improve the general idea
of real events in the acoustic field. Therefore, 3D visualization of complex acoustic scenes uses
the method of an equivalent source, which reproduces the sound pressure similar to that
measured in some virtual acoustic volume using the method of least squares [15, 16, 21–23]. In
this case, spatial reconstruction is simplified and adapted to real measurement scenarios without
the need to reconstruct the parameters of all sources in the acoustic field [10].
In conditions of incomplete data, the equivalent source method leads to the development of
the sound field models with interval or fuzzy parameters [24, 25]. The location of the sources is
estimated by fuzzy clustering methods based on the analysis of the density of the peak values in
the sound field [26–28]. The contribution of an individual source or a group of sources to the
total sound energy level is modeled by relationship matrices. Finally, for the found number of
sources or their clusters, the field parameters are restored by regularization methods [3, 4].
Reconstruction of real acoustic scenes on the terrain is carried out by generating possible
scenarios based on cluster analysis, followed by genetic selection of sources that provide a
minimum distance between the model and observed sound pressure levels [28]. As a result,
reconstruction of the field generated by multiple sources requires significant computational
costs, as it is reduced to the generation and selection of relationship matrices that determine the
distribution of the sources in the sound field.
3. Problem Statement
This paper proposes a model of the acoustic surface based on the system of fuzzy relational
equations (SFRE). The coordinates of a sound source or a group of sources are described by
fuzzy terms, and the matrix of fuzzy relations connects the locations of the sources and the sound
energy levels of the field. The SFRE connects membership functions of the source locations and
the sound energy levels using the compositional rule of inference [29]. Then the problem of
acoustic surface reconstruction is reduced to solving the composite SFRE obtained for the
available measurement results “coordinates - field strength” [8, 9]. The method for extracting
fuzzy relations from experimental data by solving the composite SFRE was proposed in [8, 30,
31]. The method [8, 30, 31] is based on the numerical resolution of the SFRE using genetic and
neural technology aimed at adapting the solution as new experimental data becomes available
[32, 33]. Following [8, 30, 31], solving the composite SFRE is accomplished in two stages. At
the first stage, the null solution for the relationship matrix and the membership functions
parameters is determined. The null solution ensures the minimum difference between the results
of linguistic approximation and experimental data. At the second stage, the null solution allows
to organize the parallel search for the solution set in the form of the lower and upper bounds of
fuzzy relations. Linguistic interpretation involves the transition from a set of solutions for the
matrix of fuzzy relations to a set of fuzzy IF-THEN rules [34].
Following [8, 30, 31], at the first stage of reconstruction, the null distribution of sound energy
levels in the form of the relational matrix is obtained. The parameters of membership functions
of the fuzzy terms, which describe the coordinates of sound sources, are determined
simultaneously with the null distribution of the field energy. At the second stage, the set of
solutions in the form of boundary sound energy levels of the field for the matrix of fuzzy
relations is determined. In this case, the results of fuzzy logic inference are the lower and upper
acoustic surfaces. The number of variants for sound field reconstruction is defined by
transforming the relationship matrix to a set of IF-THEN rules describing the acoustic surface.
Risk of incorrect reconstruction is evaluated by the comparison of the extracted rules and the
rules which describe the real acoustic surface. The total number of rules is distributed according
to the sound energy levels. Due to incomplete data, the interval rule is considered correct if the
actual acoustic level is embedded within the lower and upper acoustic surfaces. The probability
of correct reconstruction of the acoustic level is defined as the ratio of the number of correct
rules to the total number of rules in the power class. The rule is considered incorrect if a different
acoustic level is reconstructed instead of the actual acoustic level. Incorrect rules are divided into
rules from the contiguous and remote power classes. The risk of incorrect reconstruction of the
acoustic level is defined as the ratio of the number of incorrect rules to the total number of rules
in the certain power class. The probability of correct reconstruction of the acoustic surface is
defined as the average probability of correct reconstruction over all sound energy levels, and the
risk of incorrect reconstruction – as the average risk of reconstruction of the contiguous and
remote acoustic levels.
Properties of the solution set of the composite SFRE allow avoiding the generation and
selection of the source distribution parameters based on relationship matrices “location - sound
energy level” [26–28]. The process of recovering the field generated by multiple sources is
simplified due to ability to parallelize the process of numerical resolution of the composite
SFRE, that allows to increase the frequency of reconstruction when processing acoustic data
streams. The genetic-neural algorithm of solving the SFRE for the problems of acoustic field
reconstruction was proposed in [35], where the field model was built on the basis of the
fundamental laws of the sound theory [1, 2], and the source parameters were determined by crisp
values. Unlike [35], in this work the number of sound sources is not limited. An acoustic surface
in the form of the fuzzy knowledge base is subject to reconstruction, where the number of input
terms is limited by the size of the controlled area. The proposed approach does not require a
series of cost experiments and allows to restrict experimental conditions with equipment that
implements classical beamforming methods for processing acoustic data streams under
incomplete measurement results.
The aim of the work is to develop the method based on solving the composite SFRE for
reconstructing acoustic surfaces from incomplete data. The method should provide the minimum
processing time while preserving the permissible risk of incorrect reconstruction of the acoustic
field.
4. Model and Method for Sound Surfaces Reconstruction
4.1. Problem of knowledge extraction for recovering acoustic
surfaces
In order to ensure the safety of mass events which involve the participation of people and
equipment, the open area with coordinates 𝑥𝑥1 = 𝑥𝑥2 ∈ [0, 250] m is controlled by acoustic vision.
It is assumed that there is no effect of repeated reflection of acoustic signals. Following the
fundamental laws of the sound theory [1, 2], the levels of sound field energy 𝑦𝑦(𝑥𝑥1 , 𝑥𝑥2 ) are
determined as follows:
1 𝑤𝑤
𝑦𝑦(𝑥𝑥1 , 𝑥𝑥2 ) = 10 𝑙𝑙𝑙𝑙 � ⋅ ∑𝑛𝑛𝑝𝑝=1 4𝜋𝜋 [(𝑥𝑥 −𝑢𝑢 )2𝑝𝑝+(𝑥𝑥 −𝑣𝑣 )2 ]�, (1)
𝐼𝐼0 1 𝑝𝑝 2 𝑝𝑝
−12
where 𝐼𝐼0 = 10 Wt/m is the intensity of the audibility threshold; 𝑛𝑛 is the number of sources;
2
𝑢𝑢𝑝𝑝 , 𝑣𝑣𝑝𝑝 and 𝑤𝑤𝑝𝑝 are the coordinates and power of the 𝑝𝑝-th sound source.
The real acoustic image 𝑦𝑦(𝑥𝑥1 , 𝑥𝑥2 ) was generated by 𝑛𝑛 = 300 sources with a sound power
range of 𝑤𝑤𝑝𝑝 ∈ [10−8 , 10−1 ] Wt, 𝑝𝑝 = 1, 𝑛𝑛, which corresponds to acoustic levels of 𝑦𝑦 ∈ [40,
110] dB [1, 2]. The real acoustic image (1) at the input of the microphone array is shown in
Figure 1.
The acoustic image (1) at the output of the microphone array is shown in Figure 2. The
microphone array is formed by the matrix of 32*32 microphones with the distance of 25 m and
the scanning step of 10 [35]. Image resolution which is 𝑄𝑄 =3615 points decreases with the
growth of the distance from the center of the array.
Figure 1: The real acoustic image for 𝑛𝑛 = 300
Figure 2: The observed acoustic image at the output of the microphone array
The problem of sound field reconstruction consists in the following. For the observed image in
the form of the 𝑄𝑄 =3615 measurement results “coordinates 𝑥𝑥�1𝑠𝑠 , 𝑥𝑥�2𝑠𝑠 - level of acoustic energy 𝑦𝑦�𝑠𝑠 ”,
𝑠𝑠 = 1, 𝑄𝑄, it is necessary to restore the real image 𝑦𝑦(𝑥𝑥1 , 𝑥𝑥2 ) at the input of the microphone array
in order to increase the resolution to 250⨯250 = 62500 points. For this purpose, using the
available measurement results, it is necessary to extract knowledge about the acoustic surface in
the form of IF-THEN rules [7, 8]:
Rule 𝐾𝐾: IF 𝑥𝑥1 = 𝑎𝑎1𝐾𝐾 AND 𝑥𝑥2 = 𝑎𝑎2𝐾𝐾 THEN 𝑦𝑦 = 𝑑𝑑𝐾𝐾 , 𝐾𝐾 = 1, 𝑍𝑍, (2)
where 𝑎𝑎1𝐾𝐾 ∈ {𝑐𝑐11 , . . . , 𝑐𝑐1𝑘𝑘1 } and 𝑎𝑎2𝐾𝐾 ∈ {𝑐𝑐21 , . . . , 𝑐𝑐2𝑘𝑘2 } are the fuzzy terms for estimating the
variables 𝑥𝑥1 and 𝑥𝑥2 in the rule 𝐾𝐾; 𝑑𝑑𝐾𝐾 ∈ {𝐸𝐸1 , . . . , 𝐸𝐸𝑀𝑀 } is the decision class for estimating the
variable 𝑦𝑦 in the rule 𝐾𝐾; 𝑍𝑍 is the number of rules.
4.2. Assessing the risk of incorrect reconstruction of the
acoustic surface
To evaluate the risk of incorrect reconstruction of the acoustic surface, it is necessary to make a
distribution of 𝑍𝑍 rules according to the acoustic levels {𝐸𝐸1 , . . . , 𝐸𝐸𝑀𝑀 } [7].
We shall denote:
𝑧𝑧𝐽𝐽 is the number of rules demanding the acoustic level 𝐸𝐸𝐽𝐽 , that is 𝑍𝑍 = 𝑧𝑧1 +. . . +𝑧𝑧𝑀𝑀 ;
𝑧𝑧𝐽𝐽𝐽𝐽 is the number of rules reconstructed by fuzzy inference for the acoustic level 𝐸𝐸𝐿𝐿 instead
of the acoustic level 𝐸𝐸𝐽𝐽 , that is 𝑧𝑧𝐽𝐽 = 𝑧𝑧𝐽𝐽1 +. . . +𝑧𝑧𝐽𝐽𝐽𝐽 .
For the power classes of sound sources, quality of reconstruction is evaluated as follows:
𝑧𝑧 1
𝑃𝑃𝐽𝐽1 = 𝐽𝐽𝐽𝐽 , 𝑃𝑃𝐽𝐽0 = 1 − 𝑃𝑃𝐽𝐽1 = ∑𝑀𝑀
𝐿𝐿=1 𝑧𝑧𝐽𝐽𝐽𝐽 ,
𝑧𝑧𝐽𝐽 𝑧𝑧𝐽𝐽
𝐿𝐿≠𝐽𝐽
where 𝑃𝑃𝐽𝐽 is the probability of correct reconstruction of the acoustic level 𝐸𝐸𝐽𝐽 ; 𝑃𝑃𝐽𝐽0 is the risk of
1
incorrect reconstruction of the acoustic level 𝐸𝐸𝐽𝐽 .
For the acoustic surface, quality of reconstruction is evaluated as follows:
1 1 𝑀𝑀 𝑀𝑀
𝑃𝑃1 = ∑𝑀𝑀 0 1
𝐽𝐽=1 𝑧𝑧𝐽𝐽𝐽𝐽 , 𝑃𝑃 = 1 − 𝑃𝑃 = ∑𝐽𝐽=1 ∑𝐿𝐿=1 𝑧𝑧𝐽𝐽𝐽𝐽 ,
𝑍𝑍 𝑍𝑍
𝐿𝐿≠𝐽𝐽
where 𝑃𝑃1 is the probability of correct reconstruction of the acoustic surface; 𝑃𝑃0 is the risk of
incorrect reconstruction of the acoustic surface.
4.3. Fuzzy relational model of the acoustic surface
We shall redenote:
{𝑐𝑐11 , . . . , 𝑐𝑐1𝑘𝑘1 , 𝑐𝑐21 , . . . , 𝑐𝑐2𝑘𝑘2 }={𝐶𝐶1 , . . . , 𝐶𝐶𝑁𝑁 }, 𝑁𝑁 = 𝑘𝑘1 + 𝑘𝑘2 , is the set of fuzzy terms for
estimating the coordinates of the sound field 𝑥𝑥1 and 𝑥𝑥2 .
The fuzzy knowledge base (2) is modeled using the system of one-dimensional relation
matrices «location 𝑐𝑐𝑖𝑖𝑖𝑖 – sound energy level 𝐸𝐸𝐽𝐽 »
𝑹𝑹𝑖𝑖 ⊆ 𝑐𝑐𝑖𝑖𝑖𝑖 × 𝐸𝐸𝐽𝐽 =[𝑟𝑟𝑖𝑖𝑖𝑖,𝐽𝐽 , 𝑖𝑖 = 1,2, 𝑙𝑙 = 1,𝑘𝑘𝑖𝑖 , 𝐽𝐽 = 1,𝑀𝑀].
The equivalent relation matrix
𝑹𝑹 ⊆ 𝐶𝐶𝐼𝐼 × 𝐸𝐸𝐽𝐽 =[𝑟𝑟𝐼𝐼𝐼𝐼 , 𝐼𝐼 = 1,𝑁𝑁, 𝐽𝐽 = 1,𝑀𝑀]
defines the fuzzy distribution of the sound energy levels in the field. The element 𝑟𝑟𝐼𝐼𝐼𝐼 ∈ [0, 1] of
the matrix 𝑹𝑹 is interpreted as the measure of manifestation of the sound energy level 𝐸𝐸𝐽𝐽 at the
location С𝐼𝐼 .
Given matrices 𝑹𝑹𝑖𝑖 , the acoustic surface can be described with the help of the SFRE [29]:
𝝁𝝁𝐸𝐸 = 𝝁𝝁𝐴𝐴1 ∘ 𝑹𝑹1 ∩ 𝝁𝝁𝐴𝐴2 ∘ 𝑹𝑹2 , (3)
𝐴𝐴𝑖𝑖 𝑐𝑐𝑖𝑖1 𝑐𝑐𝑖𝑖𝑘𝑘𝑖𝑖
where 𝝁𝝁 (𝑥𝑥𝑖𝑖 ) = (𝜇𝜇 , . . . , 𝜇𝜇 ) is the vector of membership degrees of the variable 𝑥𝑥𝑖𝑖 to the
fuzzy locations 𝑐𝑐𝑖𝑖𝑖𝑖 , 𝑖𝑖 = 1,2, 𝑙𝑙 = 1,𝑘𝑘𝑖𝑖 ; 𝝁𝝁𝐸𝐸 = (𝜇𝜇 𝐸𝐸1 , . . . , 𝜇𝜇 𝐸𝐸𝑀𝑀 ) is the vector of membership degrees
of the variable y to the sound energy levels 𝐸𝐸𝐽𝐽 , 𝐽𝐽 = 1,𝑀𝑀; is the operation of max-min
composition [29].
Following [8, 30, 31], the matrices of membership degrees
𝜇𝜇̂ 𝑐𝑐𝑖𝑖1 (𝑥𝑥�𝑖𝑖1 ) . . . 𝜇𝜇̂ 𝑐𝑐𝑖𝑖𝑘𝑘𝑖𝑖 (𝑥𝑥�𝑖𝑖1 ) 𝜇𝜇̂ 𝐸𝐸1 (𝑦𝑦�1 ) . . . 𝜇𝜇̂ 𝐸𝐸𝑀𝑀 (𝑦𝑦�1 )
� 𝐴𝐴𝑖𝑖 (𝑥𝑥�𝑖𝑖𝑠𝑠 ) = � . . .
𝝁𝝁 ... . . . � , 𝝁𝝁 � 𝐸𝐸 (𝑦𝑦�𝑠𝑠 ) = � . . . ... ... �
𝑐𝑐𝑖𝑖1 𝑄𝑄 𝑐𝑐𝑖𝑖𝑘𝑘𝑖𝑖 𝑄𝑄
𝜇𝜇̂ 𝐸𝐸1 (𝑦𝑦�𝑄𝑄 ) . . . 𝜇𝜇̂ 𝐸𝐸𝑀𝑀 (𝑦𝑦�𝑄𝑄 )
𝜇𝜇̂ (𝑥𝑥�𝑖𝑖 ) . . . 𝜇𝜇̂ (𝑥𝑥�𝑖𝑖 )
can be obtained according to the microphone array measurement results “field coordinates 𝑥𝑥�1𝑠𝑠 , 𝑥𝑥�2𝑠𝑠
– acoustic energy level 𝑦𝑦�𝑠𝑠 ”, 𝑠𝑠 = 1, 𝑄𝑄.
Given matrices 𝝁𝝁 � 𝐴𝐴𝑖𝑖 , 𝝁𝝁 �𝐸𝐸 , the acoustic surface can be described with the help of the composite
SFRE [8, 30, 31]:
� 𝐴𝐴1 (𝑥𝑥�1𝑠𝑠 ) ∘ 𝑹𝑹1 ∩ 𝝁𝝁
� 𝐸𝐸 (𝑦𝑦�𝑠𝑠 ) = 𝝁𝝁
𝝁𝝁 � 𝐴𝐴2 (𝑥𝑥�2𝑠𝑠 ) ∘ 𝑹𝑹2 . (4)
For each sound energy level 𝐸𝐸𝐽𝐽 , 𝐽𝐽 = 1,𝑀𝑀, the SFRE (4) can be rewritten in the form [9]:
𝐸𝐸
� 𝐴𝐴1 (𝑥𝑥�1𝑠𝑠 ) ∘ 𝒓𝒓1𝐽𝐽 ∩ 𝝁𝝁
� 𝐽𝐽 (𝑦𝑦�𝑠𝑠 ) = 𝝁𝝁
𝝁𝝁 � 𝐴𝐴2 (𝑥𝑥�2𝑠𝑠 ) ∘ 𝒓𝒓2𝐽𝐽 , 𝑠𝑠 = 1,𝑄𝑄, (5)
𝐸𝐸𝐽𝐽
where 𝝁𝝁 𝐸𝐸𝐽𝐽
� = (𝜇𝜇̂ (𝑦𝑦�1 ), . . . , 𝜇𝜇̂ (𝑦𝑦�𝑄𝑄 ))𝐸𝐸𝐽𝐽 𝑇𝑇
and 𝒓𝒓𝑖𝑖𝐽𝐽 =(𝑟𝑟𝑖𝑖1,𝐽𝐽 , . . . , 𝑟𝑟𝑖𝑖𝑘𝑘𝑖𝑖,𝐽𝐽 )𝑇𝑇 are the vector-columns of the
matrix of observed values 𝝁𝝁 � 𝐸𝐸 and the fuzzy relation matrix 𝑹𝑹𝑖𝑖 for the sound energy level 𝐸𝐸𝐽𝐽 .
To obtain the degree of membership of the coordinate 𝑥𝑥 to the fuzzy location 𝑐𝑐, we will use
the membership function of the form [7]:
1
𝜇𝜇𝑐𝑐 (𝑥𝑥) = 𝑥𝑥−𝛽𝛽 2
, (6)
1+� �
𝜎𝜎
where β is the coordinate of the function maximum; σ is the concentration parameter.
To obtain the crisp values of acoustic energy, the defuzzification operation is performed
according to the centroid method [29].
Correlations (3)–(6) define the fuzzy model of the acoustic surface as follows:
𝑦𝑦 = 𝑓𝑓𝑅𝑅 (𝑥𝑥1 , 𝑥𝑥2 , 𝜝𝜝𝐶𝐶 , 𝜴𝜴𝐶𝐶 , 𝑹𝑹1 , 𝑹𝑹2 ), (7)
where for each sound energy level 𝐸𝐸𝐽𝐽 , 𝐽𝐽 = 1,𝑀𝑀, fuzzy relations are restored by solving the
composite SFRE
𝐸𝐸
� 𝐽𝐽 (𝑦𝑦�𝑠𝑠 ) = 𝑓𝑓̂𝑅𝑅𝐽𝐽 (𝑥𝑥�1𝑠𝑠 , 𝑥𝑥�2𝑠𝑠 , 𝜝𝜝𝐶𝐶 , 𝜴𝜴𝐶𝐶 , 𝒓𝒓1𝐽𝐽 , 𝒓𝒓2𝐽𝐽 ), 𝑠𝑠 = 1,𝑄𝑄,
𝝁𝝁 (8)
obtained according to the microphone array measurement results (𝑥𝑥�1𝑠𝑠 , 𝑥𝑥�2𝑠𝑠 , 𝑦𝑦�𝑠𝑠 ), 𝑠𝑠 = 1, 𝑄𝑄.
Here 𝜝𝜝𝐶𝐶 = (𝛽𝛽 𝐶𝐶1 , . . . , 𝛽𝛽 𝐶𝐶𝑁𝑁 ) and 𝜴𝜴𝐶𝐶 = (𝜎𝜎 𝐶𝐶1 , . . . , 𝜎𝜎 𝐶𝐶𝑁𝑁 ) are the vectors of β- and σ- parameters for
𝐽𝐽
the fuzzy locations 𝐶𝐶1 , . . . , 𝐶𝐶𝑁𝑁 membership functions; 𝑓𝑓𝑅𝑅 and 𝑓𝑓̂𝑅𝑅 are the operators of inputs-
output connection, corresponding to formulas (3), (6) and (5), (6), respectively.
4.4. Method of acoustic surfaces reconstruction based on
solving composite SFRE
Following [8, 30, 31], the problem of acoustic surface reconstruction is reduced to finding the
null solution and the solution set for the fuzzy matrix of energy distribution R.
When searching for the null distribution, the problem of tuning the fuzzy model (7) is
formulated as follows. It is necessary to find the vectors of fuzzy locations parameters 𝜝𝜝𝐶𝐶 , 𝜴𝜴𝐶𝐶 ,
and the fuzzy relation matrix R, which provide the least distance between the model and the
observed acoustic images:
𝐹𝐹 = ∑𝑄𝑄 �1𝑠𝑠 , 𝑥𝑥�2𝑠𝑠 , 𝜝𝜝𝐶𝐶 , 𝜴𝜴𝐶𝐶 , 𝑹𝑹1 , 𝑹𝑹2 ) − 𝑦𝑦�𝑠𝑠 ]2 =
𝑠𝑠=1[𝑓𝑓𝑅𝑅 (𝑥𝑥 𝑚𝑚𝑚𝑚𝑚𝑚 . (9)
𝜝𝜝𝐶𝐶 ,𝜴𝜴𝐶𝐶 ,𝑹𝑹1 ,𝑹𝑹2
When searching for the reconstruction set, the problem of solving the composite SFRE (8) is
formulated as follows [8, 30, 31]. Given fuzzy locations parameters 𝜝𝜝𝐶𝐶 , 𝜴𝜴𝐶𝐶 , the fuzzy relation
matrix 𝑹𝑹 = [𝑟𝑟𝐼𝐼𝐼𝐼 ], 𝐼𝐼 = 1,𝑁𝑁, 𝐽𝐽 = 1,𝑀𝑀, should be found which satisfies the constraints 𝑟𝑟𝐼𝐼𝐼𝐼 ∈ [0, 1]
and provides the least distance between the model and the observed vectors of membership
degrees to the sound energy levels 𝐸𝐸𝐽𝐽 ; that is, the minimum value of the criterion (9):
𝐸𝐸 2
𝐽𝐽 𝐽𝐽 𝐽𝐽
̂ �1𝑠𝑠 , 𝑥𝑥�2𝑠𝑠 , 𝜝𝜝𝐶𝐶 , 𝜴𝜴𝐶𝐶 , 𝒓𝒓 , 𝒓𝒓 ) − 𝝁𝝁
𝐹𝐹 = ∑𝑀𝑀
𝐽𝐽=1 �𝑓𝑓𝑅𝑅 (𝑥𝑥 1 2 � 𝐽𝐽 (𝑦𝑦�𝑠𝑠 )� = 𝑚𝑚𝑚𝑚𝑚𝑚
𝐽𝐽 𝐽𝐽
, 𝑠𝑠 = 1,𝑄𝑄. (10)
𝒓𝒓1 ,𝒓𝒓2
Following [8, 30, 31], the composite SFRE (8) has the solution set, that defines the set of
variants for the sound field reconstruction in the form of the lower and upper acoustic surfaces.
The solution to the SFRE (8) can be represented in the form of intervals [32, 33]:
𝑟𝑟𝐼𝐼𝐼𝐼 = [𝑟𝑟𝐼𝐼𝐼𝐼 , 𝑟𝑟𝐼𝐼𝐼𝐼 ] ⊂ [0,1], 𝐼𝐼 = 1,𝑁𝑁, 𝐽𝐽 = 1,𝑀𝑀, (11)
which correspond to the set of IF-THEN rules
Rule 𝐾𝐾: IF 𝑥𝑥1 = 𝑎𝑎1𝐾𝐾 AND 𝑥𝑥2 = 𝑎𝑎2𝐾𝐾 THEN 𝑦𝑦 = 𝑑𝑑𝐾𝐾 𝐴𝐴𝐴𝐴𝐴𝐴 𝑦𝑦 = 𝑑𝑑𝐾𝐾 , 𝐾𝐾 = 1, 𝑍𝑍. (12)
Here 𝑟𝑟𝐼𝐼𝐼𝐼 (𝑟𝑟𝐼𝐼𝐼𝐼 ) are the lower (upper) bounds of the fuzzy relations 𝑟𝑟𝐼𝐼𝐼𝐼 in the sound field energy
distribution; 𝑑𝑑𝐾𝐾 (𝑑𝑑𝐾𝐾 ) ∈ {𝐸𝐸1 , . . . , 𝐸𝐸𝑀𝑀 } are the decision classes for estimating the variables 𝑦𝑦(𝑦𝑦) in
the rule 𝐾𝐾 for the lower (upper) acoustic surfaces.
The null solution 𝑹𝑹0 = [𝑟𝑟𝐼𝐼𝐼𝐼0 ], 𝐼𝐼 = 1,𝑁𝑁, 𝐽𝐽 = 1,𝑀𝑀, of the optimization problem (9) allows to
parallelize the search for upper and lower bounds of the intervals (11) for each sound energy
level 𝐸𝐸𝐽𝐽 , where
𝑟𝑟𝐼𝐼𝐼𝐼 ∈ [𝑟𝑟𝐼𝐼𝐼𝐼0 , 1], 𝑟𝑟𝐼𝐼𝐼𝐼 ∈ [0, 𝑟𝑟𝐼𝐼𝐼𝐼0 ].
Following [8, 30–33], restoration of the acoustic image is accomplished by way of multiple
solving the optimization problem (10). If 𝑹𝑹(𝑡𝑡) = [𝑟𝑟𝐼𝐼𝐼𝐼 (𝑡𝑡)] is some t-th solution of the
optimization problem (10), then 𝐹𝐹(𝑹𝑹(𝑡𝑡)) = 𝐹𝐹(𝑹𝑹0 ). When forming the intervals (11), the search
space is restricted by the intervals 𝑟𝑟𝐼𝐼𝐼𝐼 (𝑡𝑡) ∈ [𝑟𝑟𝐼𝐼𝐼𝐼 (𝑡𝑡 − 1), 1] for the upper bounds; 𝑟𝑟𝐼𝐼𝐼𝐼 (𝑡𝑡) ∈
[0, 𝑟𝑟𝐼𝐼𝐼𝐼 (𝑡𝑡 − 1)] for the lower bounds. The search for the intervals (11) will go on until 𝑟𝑟𝐼𝐼𝐼𝐼 (𝑡𝑡) ≠
𝑟𝑟𝐼𝐼𝐼𝐼 (𝑡𝑡 − 1). If 𝑟𝑟𝐼𝐼𝐼𝐼 (𝑡𝑡) = 𝑟𝑟𝐼𝐼𝐼𝐼 (𝑡𝑡 − 1)), then 𝑟𝑟𝐼𝐼𝐼𝐼 (𝑟𝑟𝐼𝐼𝐼𝐼 )=𝑟𝑟𝐼𝐼𝐼𝐼 (𝑡𝑡).
The genetic-gradient approach is proposed for solving the optimization problems (9), (10)
[30–33]. When searching for the null distribution, the chromosome is defined as a string of
binary codes of the fuzzy locations parameters 𝛽𝛽 𝐶𝐶𝐼𝐼 , 𝜎𝜎 𝐶𝐶𝐼𝐼 and the fuzzy relations 𝑟𝑟𝐼𝐼𝐼𝐼 , 𝐼𝐼 = 1,𝑁𝑁, 𝐽𝐽 =
1,𝑀𝑀. When searching for the reconstruction set, the chromosome is separated for each sound
energy level 𝐸𝐸𝐽𝐽 , where the parameters 𝑟𝑟𝐼𝐼𝐼𝐼 are recoded within the search space [8, 30, 31].
The cross-over operation is performed by exchanging parts of the chromosomes in the vectors
of fuzzy locations parameters 𝜝𝜝𝐶𝐶 , 𝜴𝜴𝐶𝐶 and the matrix of fuzzy relations R. The fitness function is
based on the criteria (9), (10). The criterion for stopping the algorithm is the absence of new
upper and lower bounds for energy distribution (12) within a given time window of the
microphone array [35].
When searching for the null distribution, the recurrent relations
𝜕𝜕𝜀𝜀𝑡𝑡0
𝑟𝑟𝐼𝐼𝐼𝐼 (𝑡𝑡 + 1) = 𝑟𝑟𝐼𝐼𝐼𝐼 (𝑡𝑡) − 𝜂𝜂 ;
𝜕𝜕𝑟𝑟𝐼𝐼𝐼𝐼 (𝑡𝑡)
𝜕𝜕𝜀𝜀 0 𝜕𝜕𝜀𝜀 0
𝛽𝛽 С𝐼𝐼 (𝑡𝑡 + 1) = 𝛽𝛽 𝐶𝐶𝐼𝐼 (𝑡𝑡) − 𝜂𝜂 𝜕𝜕𝛽𝛽𝐶𝐶𝐼𝐼𝑡𝑡(𝑡𝑡); 𝜎𝜎 𝐶𝐶𝐼𝐼 (𝑡𝑡 + 1) = 𝜎𝜎 𝐶𝐶𝐼𝐼 (𝑡𝑡) − 𝜂𝜂 𝜕𝜕𝜎𝜎𝐶𝐶𝐼𝐼𝑡𝑡(𝑡𝑡), (13)
are used; and when searching for the reconstruction set, the recurrent relations
𝜕𝜕𝜀𝜀
𝑡𝑡
𝑟𝑟𝐼𝐼𝐼𝐼 (𝑡𝑡 + 1) = 𝑟𝑟𝐼𝐼𝐼𝐼 (𝑡𝑡) − 𝜂𝜂 𝜕𝜕𝑟𝑟 (𝑡𝑡) , (14)
𝐼𝐼𝐼𝐼
are used [8, 30, 31], which minimize the criteria
1
𝜀𝜀𝑡𝑡0 = (𝑦𝑦�𝑡𝑡 − 𝑦𝑦𝑡𝑡 )2 ,
2
1 𝐸𝐸 𝐸𝐸
𝜀𝜀𝑡𝑡 = (𝜇𝜇̂ 𝑡𝑡 𝐽𝐽 − 𝜇𝜇𝑡𝑡 𝐽𝐽 )2 .
2
Here 𝑦𝑦�𝑡𝑡 , 𝑦𝑦𝑡𝑡 are the observed and the model levels of acoustic energy at the t-th training step;
𝐸𝐸 𝐸𝐸
𝜇𝜇̂ 𝑡𝑡 𝐽𝐽 , 𝜇𝜇𝑡𝑡 𝐽𝐽 are the observed and the model degrees of membership of field energy levels to the
classes 𝐸𝐸𝐽𝐽 at the t-th training step; 𝑟𝑟𝐼𝐼𝐼𝐼 (𝑡𝑡) are the fuzzy relations at the t-th training step; 𝛽𝛽 𝐶𝐶𝐼𝐼 (𝑡𝑡),
𝜎𝜎 𝐶𝐶𝐼𝐼 (𝑡𝑡) are the parameters of membership functions for the fuzzy terms of sources locations at
the t-th training step; η is a parameter of training.
For the discrete coordinate space of the microphone array, the partial derivatives included in
(13), (14) are obtained using finite differences [31, 33, 35].
5. Results of the Acoustic Surface Reconstruction
Terrain monitoring is carried out in order to detect zones of acoustic activity caused by emission
of the sources or their groups belonging to certain power classes.
The output classes, the number of which is limited to 𝑀𝑀 = 7, are formed as follows:
[𝑦𝑦, 𝑦𝑦�]=[50, ��[57,
���57] ���� �][64,
64 ���� �][70,
70 ���� �][78,
78 ���� �][85,
85 ���� �] [92,
92 100].
�����
𝐸𝐸1 𝐸𝐸2 𝐸𝐸3 𝐸𝐸4 𝐸𝐸5 𝐸𝐸6 𝐸𝐸7
The sound field with coordinates 𝑥𝑥1 = 𝑥𝑥2 ∈ [0,250] m is divided into sections with a step of
25 m. In this case, the number of input fuzzy terms is limited to 𝑘𝑘1 = 𝑘𝑘2 = 9, where
с1,1÷9 = с2,1÷9 =near 25, 50, 75, 100, 125, 150, 175, 200, 225 m.
The real acoustic surface (Figure 1) is described using the set of rules presented in Table 1.
For the observed data (Figure 2), the solution set of the composite SFRE (8) is presented in
Table 2.
Table 1
The rule set that describes the real acoustic surface
𝑥𝑥1
𝑥𝑥2
~25 m ~50 m ~75 m ~100 m ~125 m ~150 m ~175 m ~200 m ~225 m
~25 m 𝑬𝑬𝟏𝟏 𝐸𝐸2 𝐸𝐸2 𝐸𝐸1 𝐸𝐸1 𝐸𝐸1 𝐸𝐸1 𝐸𝐸1 𝐸𝐸1
~50 m 𝐸𝐸2 𝐸𝐸4 𝐸𝐸3 𝐸𝐸2 𝐸𝐸2 𝐸𝐸1 𝐸𝐸4 𝐸𝐸3 𝐸𝐸2
~75 m 𝐸𝐸2 𝐸𝐸4 𝐸𝐸3 𝐸𝐸2 𝐸𝐸2 𝐸𝐸4 𝐸𝐸4 𝐸𝐸3 𝐸𝐸3
~100 m 𝐸𝐸2 𝐸𝐸4 𝐸𝐸4 𝐸𝐸2 𝐸𝐸2 𝐸𝐸5 𝐸𝐸5 𝐸𝐸2 𝐸𝐸2
~125 m 𝐸𝐸2 𝐸𝐸3 𝐸𝐸2 𝐸𝐸3 𝐸𝐸3 𝐸𝐸3 𝐸𝐸4 𝐸𝐸2 𝐸𝐸2
~150 m 𝐸𝐸2 𝐸𝐸4 𝐸𝐸2 𝐸𝐸3 𝐸𝐸3 𝐸𝐸5 𝐸𝐸4 𝐸𝐸2 𝐸𝐸2
~175 m 𝐸𝐸3 𝐸𝐸3 𝐸𝐸2 𝐸𝐸6 𝐸𝐸4 𝐸𝐸6 𝐸𝐸2 𝐸𝐸1 𝐸𝐸2
~200 m 𝐸𝐸3 𝐸𝐸4 𝐸𝐸5 𝐸𝐸6 𝐸𝐸7 𝐸𝐸3 𝐸𝐸2 𝐸𝐸1 𝐸𝐸2
~225 m 𝐸𝐸3 𝐸𝐸2 𝐸𝐸4 𝐸𝐸4 𝐸𝐸4 𝐸𝐸5 𝐸𝐸2 𝐸𝐸1 𝐸𝐸1
Table 2
Solution set of the composite SFRE
THEN 𝑦𝑦
IF
𝐸𝐸1 𝐸𝐸2 𝐸𝐸3 𝐸𝐸4 𝐸𝐸5 𝐸𝐸6 𝐸𝐸7
𝑐𝑐11 [0, 0.80] 0.69 0.44 0.12 0 0 0
𝑐𝑐12 0 0.85 [0.22, 0.67] 0.48 [0, 0.60] 0 0
𝑐𝑐13 [0, 0.73] [0.50, 1] [0, 0.46] 0.89 [0.45, 0.71] 0 0
𝑐𝑐14 [0, 0.64] 0.78 0.34 [0.23, 0.69] 0.08 [0.85, 1] 0.72
𝑥𝑥1 𝑐𝑐15 [0, 0.60] 0.95 [0.41, 0.56] 0.70 0 0 0.91
𝑐𝑐16 [0, 0.53] 0.84 0.49 [0.15, 0.68] 0.73 [0, 0.67] 0
𝑐𝑐17 [0, 0.79] 0.90 [0.34, 0.53] 0.72 [0.49, 0.65] 0 0
𝑐𝑐18 [0, 0.84] [0.71, 1] 0.82 [0, 0.67] 0.10 0 0
𝑐𝑐19 [0, 0.75] 0.86 [0.18, 0.61] 0.45 0 0 0
𝑐𝑐21 [0, 0.62] [0.85, 1] 0.49 0.16 0 0 0
𝑐𝑐22 [0, 0.80] [0.39, 0.73] 0.45 0.68 [0, 0.62] 0 0
𝑐𝑐23 [0, 0.54] 0.76 [0.21, 0.60] [0, 0.53] 0.82 0 0
𝑥𝑥2 𝑐𝑐24 [0, 0.46] [0.27, 0.69] 0.62 0.78 [0.36, 0.74] 0 0
𝑐𝑐25 0 0.57 0.77 [0.43, 0.64] 0.25 0 0
𝑐𝑐26 0 [0.54, 1] 0.41 0.55 [0.19, 0.68] 0.84 0
𝑐𝑐27 0 0.78 [0, 0.49] [0.25, 1] 0.81 0.65 0
𝑐𝑐28 [0, 0.58] [0.35, 0.60] 0.39 [0, 0.64] [0.75, 1] 0.92 0.86
𝑐𝑐29 [0, 0.80] 0.64 0.45 [0.38, 1] 0.81 0 0
The solution set for the relational matrix corresponds to the rule set that defines the variants of
acoustic field reconstruction presented in Table 3. Due to incomplete data, the interval rule is
considered correct if the actual acoustic level 𝐸𝐸𝐽𝐽 in Table 1 is embedded within the lower and
upper acoustic levels in Table 3. The rule is considered incorrect if a different acoustic level is
reconstructed instead of the acoustic level 𝐸𝐸𝐽𝐽 . Incorrect rules are divided into rules from the
contiguous and remote power classes. In Table 3, the contiguous (remote) incorrect rules are
marked with * (**).
In Table 1, the total number of rules 𝑍𝑍 = 81 is distributed according to the sound energy
levels as follows:
𝑧𝑧1 = 12; 𝑧𝑧2 = 29; 𝑧𝑧3 = 16; 𝑧𝑧4 = 15; 𝑧𝑧5 = 5; 𝑧𝑧6 = 3; 𝑧𝑧7 = 1.
The risk of incorrect reconstruction of the acoustic levels is presented in Table 4, where
0(𝑐𝑐) 0(𝑟𝑟)
𝑃𝑃𝐽𝐽 (𝑃𝑃𝐽𝐽 ) – is the risk of reconstruction of the contiguous (remote) acoustic levels instead of
the acoustic level 𝐸𝐸𝐽𝐽 .
The probability of correct reconstruction of the acoustic surface is 𝑃𝑃1 = 69/81 = 0.85. The
risk of incorrect reconstruction is 𝑃𝑃0 = 12/81 = 0.15, which is distributed to the risks of
0(𝑐𝑐) 0(𝑟𝑟)
reconstruction of the contiguous (remote) acoustic levels 𝑃𝑃 = 10/81 = 0.12 (𝑃𝑃 =
2/81 = 0.03).
The obtained solutions provide the reconstruction of the acoustic field in the form of the
lower and upper surfaces, which are shown in Figure 3 together with the real acoustic image.
Table 3
The rule set that describes the lower and upper reconstructed acoustic surfaces
𝑥𝑥1
𝑥𝑥2
~25 m ~50 m ~75 m ~100 m ~125 m ~150 m ~175 m ~200 m ~225 m
~25 m 𝑬𝑬𝟏𝟏−𝟐𝟐 𝐸𝐸2−3 𝐸𝐸1−2 𝐸𝐸1−2 𝐸𝐸1−2 𝐸𝐸1−2 𝐸𝐸1−2 𝐸𝐸1−2 𝐸𝐸1−2
∗ ∗ ∗
~50 m 𝐸𝐸2 𝐸𝐸3−4 𝐸𝐸2 𝐸𝐸3−4 𝐸𝐸1−2 𝐸𝐸2 𝐸𝐸4−5 𝐸𝐸3−4 𝐸𝐸4∗∗
~75 m 𝐸𝐸2−3 𝐸𝐸4−5 𝐸𝐸3 𝐸𝐸2−3 𝐸𝐸1−2 𝐸𝐸4−5 𝐸𝐸4−5 𝐸𝐸3−4 𝐸𝐸2−3
~100 m 𝐸𝐸2−3 𝐸𝐸3−4 𝐸𝐸4−5 𝐸𝐸2 𝐸𝐸1−2 𝐸𝐸4−5 𝐸𝐸4−5 𝐸𝐸4∗∗ 𝐸𝐸2
~125 m ∗ ∗
𝐸𝐸2 𝐸𝐸2 𝐸𝐸2 𝐸𝐸3 𝐸𝐸2−3 𝐸𝐸3−4 𝐸𝐸2−3 𝐸𝐸2 𝐸𝐸2
~150 m ∗
𝐸𝐸2 𝐸𝐸4−5 𝐸𝐸2 𝐸𝐸3 𝐸𝐸2−3 𝐸𝐸5−6 𝐸𝐸2−3 𝐸𝐸2 𝐸𝐸2
~175 m 𝐸𝐸2−3 𝐸𝐸3−4 𝐸𝐸2 𝐸𝐸6 𝐸𝐸3−4 𝐸𝐸5−6 𝐸𝐸3∗ 𝐸𝐸2∗ 𝐸𝐸2
~200 m 𝐸𝐸2−3 ∗
𝐸𝐸4 𝐸𝐸4−5 𝐸𝐸6−7 𝐸𝐸7 𝐸𝐸3 𝐸𝐸3 𝐸𝐸1−2 𝐸𝐸1−2
~225 m 𝐸𝐸2−3 𝐸𝐸2−3 𝐸𝐸3∗ 𝐸𝐸3−4 𝐸𝐸3−4 𝐸𝐸5 𝐸𝐸2−3 𝐸𝐸1−2 𝐸𝐸1−2
Table 4
Quality indicators of the reconstruction of acoustic levels
Acoustic levels
Indicator
𝐸𝐸1 𝐸𝐸2 𝐸𝐸3 𝐸𝐸4 𝐸𝐸5 𝐸𝐸6 𝐸𝐸7
1
𝑃𝑃𝐽𝐽 10/12=0.83 24/29=0.83 14/16=0.88 12/15=0.80 1 1 1
0(𝑐𝑐)
𝑃𝑃𝐽𝐽 2/12=0.17 3/29=0.10 2/16=0.12 3/15=0.20 0 0 0
0(𝑟𝑟)
𝑃𝑃𝐽𝐽 0 2/29=0.07 0 0 0 0 0
To minimize processing time, the number of microphones is limited, provided that the risk of
reconstruction remains acceptable. For the testing set of 170 acoustic images, the risk of
0(𝑟𝑟)
reconstruction of the remote acoustic levels does not exceed 𝑃𝑃 = 0.05, which is permissible
for the reconstruction of complex acoustic scenes on the terrain.
a)
b)
Figure 3: A comparison of the real and reconstructed image in the form of the lower (a) and upper (b)
acoustic surfaces
6. Discussion of the Results of Effectiveness Estimation for
Reconstruction of Acoustic Surfaces
The experiment was conducted for equipment with the classical method of beamforming, which
eliminates the multiple initialization for the location of sources or their groups. The comparison
of the proposed method was carried out with the methods of acoustic field reconstruction [26–
28]. In [26–28] under similar measurement conditions, the contribution of each source (group of
sources) to the total field energy is estimated on the basis of the genetic selection of the
relational data model. Each variant of field reconstruction is described by the relational matrix,
the search for which requires restarting the genetic algorithm. The number of sources is not
limited. Instead, groups of sources are considered in some virtual acoustic volume, and the
dimension of the relational matrix is determined by the number of such groups.
The principal difference of the given method is the possibility of simultaneous search for the
lower and upper bounds of fuzzy relations for each power class of sound sources, that allows
reducing the computational complexity.
Implementation of the models [26–28] with adjustment of the relational matrix requires
solving the sequence of V optimization problems with NM parameters, where V is the number of
variants of the field reconstruction. Reconstruction of the acoustic surface in the form of
solutions of the composite SFRE requires solving the sequence of 2VM optimization problems
with N parameters for the lower and upper bounds of fuzzy relations. Generation of the null
distribution additionally requires solving the optimization problem with NM+2N variables for
two-parameter membership functions.
The reduction in computational complexity allows to obtain the following time estimates. The
time of acoustic field reconstruction was estimated for the maximum number of input terms
𝑘𝑘1 = 𝑘𝑘2 = 9 according to the given size of the controlled area. For detailed reconstruction of the
acoustic surface, the method can be applied to individual areas of the terrain. The time of
generation for the lower and upper bounds of solutions using the principles of parallel computing
does not exceed 3 s, which provides on-line reconstruction of the acoustic data stream (Intel
Core i5-7400 3.0 Ghz processor). Reconstruction of the acoustic surface by the methods [26–28]
is carried out with the delay of 7–8 s. Thus, the proposed method allows to halve the time
window of the microphone array, i.e. double the frequency of reconstruction, that increases the
reliability of terrain monitoring without attracting additional computing resources.
7. Conclusions
For the acoustic surface generated by many sources, the model based on fuzzy rules and relations
is proposed. The number of sources in the sound field is not limited. Instead, the number of input
terms is limited by the size of the controlled area. For the available measurement data, the
problem of acoustic surface reconstruction is reduced to the problem of identifying the matrix of
fuzzy relations. In fuzzy relational calculus [9], this problem belongs to the class of inverse
problems and requires solving the composite SFRE. Properties of the solution set allow avoiding
the generation and selection of the source distribution parameters. The solution set is interpreted
in the form of the set of if-then rules “fuzzy location – sound energy level”.
For reconstructing the acoustic surface from incomplete data, the method based on solving
the composite SFRE is proposed. The method provides the linguistic approximation of the
acoustic image in the form of the lower and upper surfaces, where the number of reconstruction
variants is determined by the set of solutions for the relational matrix. To solve the inverse
reconstruction problem, the genetic-gradient algorithm is used. Simplification of the
reconstruction process is achieved due to the simultaneous search for the lower and upper
bounds of solutions for each power class, that allows to increase the frequency of reconstruction
when processing acoustic data streams. The method provides the minimum processing time
while preserving the permissible risk of incorrect reconstruction of the acoustic field. For the
testing set of acoustic images, the risk of incorrect reconstruction is evaluated by the comparison
of the extracted rules and the rules which describe the real acoustic surface. The risk of incorrect
reconstruction of the acoustic level is defined as the ratio of the number of rules from the
contiguous and remote power classes to the total number of rules in the actual power class. The
risk of incorrect reconstruction of the acoustic surface is defined as the average risk of incorrect
reconstruction over all sound energy levels.
A further area of research is the development of a method for intelligent focusing of acoustic
images by optimizing the fuzzy knowledge base that describes the acoustic surface. The problem
is to choose the number of input terms, output classes and rules that provide the necessary or
extreme levels of accuracy and reconstruction time.
8. Acknowledgements
The paper was prepared within the 58–D–393 “Specialized AD systems for audio location and
identification of objects on terrain” project.
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