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.
Microphone arrays are the standard technology for acoustic field visualization in terrain
monitoring systems [
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 [
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 [
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” [
Following [
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 [
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.
determined as follows:
In order to ensure the safety of mass events which involve the participation of people and
equipment, the open area with coordinates 1 = 2 ∈ [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 [
0 ⋅ ∑ =1 4 [( 1− )2+( 2− )2] ,
(1) is the number of sources; , where 0 = 10−12 Wt/m2 is the intensity of the audibility threshold;
and are the coordinates and power of the -th sound source.
The real acoustic image ( 1, 2) was generated by = 300 sources with a sound power
range of ∈ [10−8, 10−1] Wt, = 1, , which corresponds to acoustic levels of ∈ [40,
110] dB [
The acoustic image (1) at the output of the microphone array is shown in Figure 2. The
growth of the distance from the center of the array.
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 =3615 points decreases with the
= 1, , it is necessary to restore the real image ( 1, 2) 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 [
∈ { 1, . . . , } is the decision class for estimating the 4.2.
To evaluate the risk of incorrect reconstruction of the acoustic surface, it is necessary to make a
distribution of rules according to the acoustic levels { 1, . . . , } [
is the number of rules demanding the acoustic level , that is = 1+. . . + ;
is the number of rules reconstructed by fuzzy inference for the acoustic level instead of the acoustic level , that is
= 1+. . . + .
For the power classes of sound sources, quality of reconstruction is evaluated as follows: 1 = , 0 = 1 − 1 = 1 ∑ =1 , ≠
For the acoustic surface, quality of reconstruction is evaluated as follows: incorrect reconstruction of the acoustic level . where 1 is the probability of correct reconstruction of the acoustic level ; 0 is the risk of 1 = 1
∑ =1 , 0 = 1 − 1 = 1
∑ =1 ∑ =1 , ≠ incorrect reconstruction of the acoustic surface. where 1 is the probability of correct reconstruction of the acoustic surface; 0 is the risk of 4.3.
estimating the coordinates of the sound field 1
and 2. matrices «location – sound energy level » { 11, . . . , 1 1, 21, . . . , 2 2}={ 1, . . . , }, = 1 + 2, is the set of fuzzy terms for The fuzzy knowledge base (2) is modeled using the system of one-dimensional relation The equivalent relation matrix ⊆ × =[
, , = 1,2, = 1, , = 1, ]. ⊆ × =[ , = 1, , = 1, ] defines the fuzzy distribution of the sound energy levels in the field. The element ∈ [0, 1] of the matrix is interpreted as the measure of manifestation of the sound energy level at the location С . of the variable y composition [29].
Given matrices , the acoustic surface can be described with the help of the SFRE [29]:
= 1 ∘ 1 ∩ 2 ∘ 2,
where ( ) = ( 1, . . . , ) is the vector of membership degrees of the variable to the
fuzzy locations , = 1,2, = 1, ; = ( 1, . . . , ) is the vector of membership degrees
to the sound energy levels , = 1, ; is the operation of max-min
Following [
( ) =
, ( ) = – acoustic energy level ”, = 1, .
SFRE [
( ) = 1( 1 ) ∘ 1 ∩ 2( 2) ∘ 2, = 1, ,
where = ( ̂ ( 1), . . . , ̂ ( )) and
=( 1, , . . . , , )
are the vector-columns of the
the membership function of the form [
To obtain the degree of membership of the coordinate to the fuzzy location , we will use ( ) =
1 1+ − 2 , 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: 4.4.
Method of acoustic surfaces reconstruction based on
solving composite SFRE
(7)
(8)
(9)
(10)
(11)
(12)
Following [
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 and the fuzzy relation matrix R, which provide the least distance between the model and the , , = ∑ =1[ ( 1 , 2, , , 1, 2) − ]2 =
, , 1, 2 .
When searching for the reconstruction set, the problem of solving the composite SFRE (8) is
formulated as follows [
= [ ], = 1, , = 1, , should be found which satisfies the constraints ∈ [0, 1] and provides the least distance between the model and the observed vectors of membership degrees to the sound energy levels ; that is, the minimum value of the criterion (9): = ∑ =1 ̂ ( 1 , 2, , , 1, 2) − ( )
2
1, 2 = , = 1, . composite SFRE where for each sound energy level , = 1, , fuzzy relations are restored by solving the = ( 1, 2, , , 1, 2), ( ) = ̂ ( 1 , 2, , , 1, 2 ), = 1, , obtained according to the microphone array measurement results ( 1 , 2, ), = 1, . the fuzzy locations 1, . . . , output connection, corresponding to formulas (3), (6) and (5), (6), respectively. are the operators of inputs
Following [
, ] ⊂ [
Rule : IF 1 = 1 AND 2 = 2 THEN = = , = 1, .
Here (
) are the lower (upper) bounds of the fuzzy relations in the sound field energy distribution; ( ) ∈ { 1, . . . , } are the decision classes for estimating the variables ( ) in the rule for the lower (upper) acoustic surfaces.
The null solution 0 = [ 0 ], = 1, , = 1, , of the optimization problem (9) allows to parallelize the search for upper and lower bounds of the intervals (11) for each sound energy level , where ∈ [ 0 , 1], solving the optimization problem (10). If ( ) = [ ( )] is some t -th solution of the optimization problem (10), then ( ( )) = ( 0). When forming the intervals (11), the search space is restricted by the intervals
( ) ∈ [ ( − 1), 1] for the upper bounds; [0, ( − 1)] for the lower bounds. The search for the intervals (11) will go on until ( ) ∈ ( ) ≠ ( − 1). If ( ) = ( − 1)), then ( )=
( ).
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 , and the fuzzy relations , = 1, , =
energy level , where the parameters are recoded within the search space [
The cross-over operation is performed by exchanging parts of the chromosomes in the vectors of fuzzy locations parameters
, 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 are used; and when searching for the reconstruction set, the recurrent relations ( + 1) = ( ) −
0 ( ) С ( + 1) = ( ) − ; ( + 1) = ( ) −
0 ( )
,
0 ;
( )
,
( )
( + 1) = ( ) −
are used [
0 = = 1 2 1 2 ( ̂ ( − ) ,
2 − )2. the t -th training step; η is a parameter of training. ̂ Here , are the observed and the model levels of acoustic energy at the t -th training step; are the observed and the model degrees of membership of field energy levels to the classes at the t -th training step; ( ) are the fuzzy relations at the t -th training step; ( ), ( ) are the parameters of membership functions for the fuzzy terms of sources locations at
For the discrete coordinate space of the microphone array, the partial derivatives included in (13), (14) are obtained using finite differences [31, 33, 35].
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 = 7, are formed as follows: [ , ]=[50, 57][57, 64][64, 70][70, 78][78, 85][85, 92] [92, 100]. 2
The sound field with coordinates 1 = 2 ∈ [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 1 = 2 = 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 ~150 m the acoustic level .
2 ~175 m ~200 m ~225 m 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 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 . Incorrect rules are divided into rules from the contiguous and remote power classes. In Table 3, the contiguous (remote) incorrect rules are In Table 1, the total number of rules = 81 is distributed according to the sound energy 1 = 12; 2 = 29; 3 = 16; 4 = 15; 5 = 5; 6 = 3; 7 = 1.
The risk of incorrect reconstruction of the acoustic levels is presented in Table 4, where 0( )) – is the risk of reconstruction of the contiguous (remote) acoustic levels instead of The probability of correct reconstruction of the acoustic surface is 1 = 69/81 = 0.85. The risk of incorrect reconstruction is 0 = 12/81 = 0.15, which is distributed to the risks of reconstruction of the contiguous (remote) acoustic levels
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. The rule set that describes the lower and upper reconstructed acoustic surfaces −
2 2−3 2−3 2 2 2−3 2−3 ~150 m
~175 m ~200 m ~225 m 1−2
2∗ 4−5 4−5 3−4 5−6 5−6 3 5 1−2 4−5 4−5 4−5 2∗−3 2∗−3 3∗ 3∗ Quality indicators of the reconstruction of acoustic levels 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 reconstruction of the remote acoustic levels does not exceed for the reconstruction of complex acoustic scenes on the terrain. 0( ) = 0.05, which is permissible 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 1 = 2 = 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.
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.
The paper was prepared within the 58–D–393 “Specialized AD systems for audio location and identification of objects on terrain” project. [9] K.Peeva, Y.Kyosev, Fuzzy Relational Calculus. Theory, Applications and Software, World
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