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 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].

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][22][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][27][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.

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][27][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.

Model and Method for Sound Surfaces Reconstruction

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 𝑥𝑥 1 = 𝑥𝑥 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 [1,2], the levels of sound field energy 𝑦𝑦(𝑥𝑥 1 , 𝑥𝑥 2 ) are determined as follows:

𝑦𝑦(𝑥𝑥 1 , 𝑥𝑥 2 ) = 10 𝑙𝑙𝑙𝑙 � 1 𝐼𝐼 0 ⋅ ∑ 𝑤𝑤 𝑝𝑝 4𝜋𝜋 [(𝑥𝑥 1 −𝑢𝑢 𝑝𝑝 ) 2 +(𝑥𝑥 2 −𝑣𝑣 𝑝𝑝 ) 2 ] 𝑛𝑛 𝑝𝑝=1 �,(1)

where 𝐼𝐼 0 = 10 −12 Wt/m 2 is the intensity of the audibility threshold; 𝑛𝑛 is the number of sources; 𝑢𝑢 𝑝𝑝 , 𝑣𝑣 𝑝𝑝 and 𝑤𝑤 𝑝𝑝 are the coordinates and power of the 𝑝𝑝-th sound source.

The real acoustic image 𝑦𝑦(𝑥𝑥 1 , 𝑥𝑥 2 ) was generated by 𝑛𝑛 = 300 sources with a sound power range of 𝑤𝑤 𝑝𝑝 ∈ [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 1 0 [35]. Image resolution which is 𝑄𝑄 =3615 points decreases with the growth of the distance from the center of the array.

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 {𝐸𝐸 1 , . . . , 𝐸𝐸 𝑀𝑀 } [7]. We shall denote: 𝑧𝑧 𝐽𝐽 is the number of rules demanding the acoustic level 𝐸𝐸 𝐽𝐽 , that is 𝑍𝑍 = 𝑧𝑧 1 +. . . +𝑧𝑧 𝑀𝑀 ; 𝑧𝑧 𝐽𝐽𝐽𝐽 is the number of rules reconstructed by fuzzy inference for the acoustic level 𝐸𝐸 𝐽𝐽 instead of the acoustic level 𝐸𝐸 𝐽𝐽 , that is 𝑧𝑧 𝐽𝐽 = 𝑧𝑧 𝐽𝐽1 +. . . +𝑧𝑧 𝐽𝐽𝑀𝑀 .

For the power classes of sound sources, quality of reconstruction is evaluated as follows:

𝑃𝑃 𝐽𝐽 1 = 𝑧𝑧 𝐽𝐽𝐽𝐽 𝑧𝑧 𝐽𝐽 , 𝑃𝑃 𝐽𝐽 0 = 1 − 𝑃𝑃 𝐽𝐽 1 = 1 𝑧𝑧 𝐽𝐽 ∑ 𝑧𝑧 𝐽𝐽𝐽𝐽 𝑀𝑀 𝐽𝐽=1 𝐽𝐽≠𝐽𝐽 ,

where 𝑃𝑃 𝐽𝐽 1 is the probability of correct reconstruction of the acoustic level 𝐸𝐸 𝐽𝐽 ; 𝑃𝑃 𝐽𝐽 0 is the risk of incorrect reconstruction of the acoustic level 𝐸𝐸 𝐽𝐽 .

For the acoustic surface, quality of reconstruction is evaluated as follows:

𝑃𝑃 1 = 1 𝑍𝑍 ∑ 𝑧𝑧 𝐽𝐽𝐽𝐽 𝑀𝑀 𝐽𝐽=1 , 𝑃𝑃 0 = 1 − 𝑃𝑃 1 = 1 𝑍𝑍 ∑ ∑ 𝑧𝑧 𝐽𝐽𝐽𝐽 𝑀𝑀 𝐽𝐽=1 𝐽𝐽≠𝐽𝐽 𝑀𝑀 𝐽𝐽=1

, where 𝑃𝑃 1 is the probability of correct reconstruction of the acoustic surface; 𝑃𝑃 0 is the risk of incorrect reconstruction of the acoustic surface.

Fuzzy relational model of the acoustic surface

We shall redenote: {𝑐𝑐 11 , . . . , 𝑐𝑐 1𝑘𝑘 1 , 𝑐𝑐 21 , . . . , 𝑐𝑐 2𝑘𝑘 2 }={𝐶𝐶 1 , . . . , 𝐶𝐶 𝑁𝑁 }, 𝑁𝑁 = 𝑘𝑘 1 + 𝑘𝑘 2 , is the set of fuzzy terms for estimating the coordinates of the sound field 𝑥𝑥 1 and 𝑥𝑥 2 .

The fuzzy knowledge base (2) is modeled using the system of one-dimensional relation matrices «location 𝑐𝑐 𝑖𝑖𝑖𝑖 -sound energy level

𝐸𝐸 𝐽𝐽 » 𝑹𝑹 𝑖𝑖 ⊆ 𝑐𝑐 𝑖𝑖𝑖𝑖 × 𝐸𝐸 𝐽𝐽 =[𝑟𝑟 𝑖𝑖𝑖𝑖,𝐽𝐽 , 𝑖𝑖 = 1,2, 𝑙𝑙 = 1,𝑘𝑘 𝑖𝑖 , 𝐽𝐽 = 1,𝑀𝑀]. The equivalent relation matrix 𝑹𝑹 ⊆ 𝐶𝐶 𝐼𝐼 × 𝐸𝐸 𝐽𝐽 =[𝑟𝑟 𝐼𝐼𝐽𝐽 , 𝐼𝐼 = 1,𝑁𝑁, 𝐽𝐽 = 1

,𝑀𝑀] defines the fuzzy distribution of the sound energy levels in the field. The element 𝑟𝑟 𝐼𝐼𝐽𝐽 ∈ [0, 1] of the matrix 𝑹𝑹 is interpreted as the measure of manifestation of the sound energy level 𝐸𝐸 𝐽𝐽 at the location С 𝐼𝐼 .

Given matrices 𝑹𝑹 𝑖𝑖 , the acoustic surface can be described with the help of the SFRE [29]:

𝝁𝝁 𝐸𝐸 = 𝝁𝝁 𝐴𝐴 1 ∘ 𝑹𝑹 1 ∩ 𝝁𝝁 𝐴𝐴 2 ∘ 𝑹𝑹 2 ,(3)

where 𝝁𝝁 𝐴𝐴 𝑖𝑖 (𝑥𝑥 𝑖𝑖 ) = (𝜇𝜇 𝑐𝑐 𝑖𝑖1 , . . . , 𝜇𝜇 𝑐𝑐 𝑖𝑖𝑘𝑘 𝑖𝑖 ) is the vector of membership degrees of the variable 𝑥𝑥 𝑖𝑖 to the fuzzy locations 𝑐𝑐 𝑖𝑖𝑖𝑖 , 𝑖𝑖 = 1,2, 𝑙𝑙 = 1,𝑘𝑘 𝑖𝑖 ; 𝝁𝝁 𝐸𝐸 = (𝜇𝜇 𝐸𝐸 1 , . . . , 𝜇𝜇 𝐸𝐸 𝑀𝑀 ) is the vector of membership degrees of the variable y to the sound energy levels 𝐸𝐸 𝐽𝐽 , 𝐽𝐽 = 1,𝑀𝑀;  is the operation of max-min composition [29].

Following [8,30,31], the matrices of membership degrees

𝝁𝝁 � 𝐴𝐴 𝑖𝑖 (𝑥𝑥 � 𝑖𝑖 𝑠𝑠 ) = � 𝜇𝜇̂𝑐𝑐 𝑖𝑖1 (𝑥𝑥 � 𝑖𝑖 1 ) . . . 𝜇𝜇̂𝑐𝑐 𝑖𝑖𝑘𝑘 𝑖𝑖 (𝑥𝑥 � 𝑖𝑖 1 ) . . . . . . . . . 𝜇𝜇̂𝑐𝑐 𝑖𝑖1 (𝑥𝑥 � 𝑖𝑖 𝑄𝑄 ) . . . 𝜇𝜇̂𝑐𝑐 𝑖𝑖𝑘𝑘 𝑖𝑖 (𝑥𝑥 � 𝑖𝑖 𝑄𝑄 ) � , 𝝁𝝁 � 𝐸𝐸 (𝑦𝑦 � 𝑠𝑠 ) = � 𝜇𝜇̂𝐸𝐸 1 (𝑦𝑦 � 1 ) . . . 𝜇𝜇̂𝐸𝐸 𝑀𝑀 (𝑦𝑦 � 1 ) . . . . . . . . . 𝜇𝜇̂𝐸𝐸 1 (𝑦𝑦 � 𝑄𝑄 ) . . . 𝜇𝜇̂𝐸𝐸 𝑀𝑀 (𝑦𝑦 � 𝑄𝑄 )

� can be obtained according to the microphone array measurement results "field coordinates 𝑥𝑥 � 1 𝑠𝑠 , 𝑥𝑥 � 2

𝑠𝑠

-acoustic energy level 𝑦𝑦 � 𝑠𝑠 ", 𝑠𝑠 = 1, 𝑄𝑄.

Given matrices 𝝁𝝁 � 𝐴𝐴 𝑖𝑖 , 𝝁𝝁 � 𝐸𝐸 , the acoustic surface can be described with the help of the composite SFRE [8,30,31]:

𝝁𝝁 � 𝐸𝐸 (𝑦𝑦 � 𝑠𝑠 ) = 𝝁𝝁 � 𝐴𝐴 1 (𝑥𝑥 � 1 𝑠𝑠 ) ∘ 𝑹𝑹 1 ∩ 𝝁𝝁 � 𝐴𝐴 2 (𝑥𝑥 � 2 𝑠𝑠 ) ∘ 𝑹𝑹 2 . (4)

For each sound energy level 𝐸𝐸 𝐽𝐽 , 𝐽𝐽 = 1,𝑀𝑀, the SFRE (4) can be rewritten in the form [9]:

𝝁𝝁 � 𝐸𝐸 𝐽𝐽 (𝑦𝑦 � 𝑠𝑠 ) = 𝝁𝝁 � 𝐴𝐴 1 (𝑥𝑥 � 1 𝑠𝑠 ) ∘ 𝒓𝒓 1 𝐽𝐽 ∩ 𝝁𝝁 � 𝐴𝐴 2 (𝑥𝑥 � 2 𝑠𝑠 ) ∘ 𝒓𝒓 2 𝐽𝐽 , 𝑠𝑠 = 1,𝑄𝑄,(5)

where 𝝁𝝁 � 𝐸𝐸 𝐽𝐽 = (𝜇𝜇̂𝐸𝐸 𝐽𝐽 (𝑦𝑦 � 1 ), . . . , 𝜇𝜇̂𝐸𝐸 𝐽𝐽 (𝑦𝑦 � 𝑄𝑄 )) 𝑇𝑇 and 𝒓𝒓 𝑖𝑖 𝐽𝐽 =(𝑟𝑟 𝑖𝑖1,𝐽𝐽 , . . . , 𝑟𝑟 𝑖𝑖𝑘𝑘 𝑖𝑖 ,𝐽𝐽 ) 𝑇𝑇 are the vector-columns of the matrix of observed values 𝝁𝝁 � 𝐸𝐸 and the fuzzy relation matrix 𝑹𝑹 𝑖𝑖 for the sound energy level 𝐸𝐸 𝐽𝐽 .

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]:

𝜇𝜇 𝑐𝑐 (𝑥𝑥) = 1 1+� 𝑥𝑥−𝛽𝛽 𝜎𝜎 � 2 , (6)

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: Here 𝜝𝜝 𝐶𝐶 = (𝛽𝛽 𝐶𝐶 1 , . . . , 𝛽𝛽 𝐶𝐶 𝑁𝑁 ) and 𝜴𝜴 𝐶𝐶 = (𝜎𝜎 𝐶𝐶 1 , . . . , 𝜎𝜎 𝐶𝐶 𝑁𝑁 ) are the vectors of βand σparameters for the fuzzy locations 𝐶𝐶 1 , . . . , 𝐶𝐶 𝑁𝑁 membership functions; 𝑓𝑓 𝑅𝑅 and 𝑓𝑓 ̂𝑅𝑅 𝐽𝐽 are the operators of inputsoutput connection, corresponding to formulas (3), ( 6) and ( 5), (6), respectively.

𝑦𝑦 = 𝑓𝑓 𝑅𝑅 (𝑥𝑥 1 , 𝑥𝑥 2 , 𝜝𝜝 𝐶𝐶 , 𝜴𝜴 𝐶𝐶 , 𝑹𝑹 1 , 𝑹𝑹 2 ),(7

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 𝜝𝜝 𝐶𝐶 , 𝜴𝜴 𝐶𝐶 , and the fuzzy relation matrix R, which provide the least distance between the model and the observed acoustic images:

𝐹𝐹 = ∑ [𝑓𝑓 𝑅𝑅 (𝑥𝑥 � 1 𝑠𝑠 , 𝑥𝑥 � 2 𝑠𝑠 , 𝜝𝜝 𝐶𝐶 , 𝜴𝜴 𝐶𝐶 , 𝑹𝑹 1 , 𝑹𝑹 2 ) − 𝑦𝑦 � 𝑠𝑠 ] 2 𝑄𝑄 𝑠𝑠=1 = 𝑚𝑚𝑖𝑖𝑛𝑛 𝜝𝜝 𝐶𝐶 ,𝜴𝜴 𝐶𝐶 ,𝑹𝑹 1 ,𝑹𝑹 2 . (9)

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 𝜝𝜝 𝐶𝐶 , 𝜴𝜴 𝐶𝐶 , the fuzzy relation matrix 𝑹𝑹 = [𝑟𝑟 𝐼𝐼𝐽𝐽 ], 𝐼𝐼 = 1,𝑁𝑁, 𝐽𝐽 = 1,𝑀𝑀, should be found which satisfies the constraints 𝑟𝑟 𝐼𝐼𝐽𝐽 ∈ [0, 1] and provides the least distance between the model and the observed vectors of membership degrees to the sound energy levels 𝐸𝐸 𝐽𝐽 ; that is, the minimum value of the criterion (9):

𝐹𝐹 = ∑ �𝑓𝑓 ̂𝑅𝑅 𝐽𝐽 (𝑥𝑥 � 1 𝑠𝑠 , 𝑥𝑥 � 2 𝑠𝑠 , 𝜝𝜝 𝐶𝐶 , 𝜴𝜴 𝐶𝐶 , 𝒓𝒓 1 𝐽𝐽 , 𝒓𝒓 2 𝐽𝐽 ) − 𝝁𝝁 � 𝐸𝐸 𝐽𝐽 (𝑦𝑦 � 𝑠𝑠 )� 2 𝑀𝑀 𝐽𝐽=1 = 𝑚𝑚𝑖𝑖𝑛𝑛 𝒓𝒓 1 𝐽𝐽 ,𝒓𝒓 2 𝐽𝐽 , 𝑠𝑠 = 1,𝑄𝑄.(10)

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]:

𝑟𝑟 𝐼𝐼𝐽𝐽 = [𝑟𝑟 𝐼𝐼𝐽𝐽 , 𝑟𝑟 𝐼𝐼𝐽𝐽 ] ⊂ [0,1], 𝐼𝐼 = 1,𝑁𝑁, 𝐽𝐽 = 1,𝑀𝑀,(11)

which correspond to the set of IF-THEN rules Rule 𝐾𝐾:

IF 𝑥𝑥 1 = 𝑎𝑎 1𝐾𝐾 AND 𝑥𝑥 2 = 𝑎𝑎 2𝐾𝐾 THEN 𝑦𝑦 = 𝑑𝑑 𝐾𝐾 𝐴𝐴𝑁𝑁𝐴𝐴 𝑦𝑦 = 𝑑𝑑 𝐾𝐾 , 𝐾𝐾 = 1, 𝑍𝑍. (12)

Here 𝑟𝑟 𝐼𝐼𝐽𝐽 (𝑟𝑟 𝐼𝐼𝐽𝐽 ) are the lower (upper) bounds of the fuzzy relations 𝑟𝑟 𝐼𝐼𝐽𝐽 in the sound field energy distribution; 𝑑𝑑 𝐾𝐾 (𝑑𝑑 𝐾𝐾 ) ∈ {𝐸𝐸 1 , . . . , 𝐸𝐸 𝑀𝑀 } are the decision classes for estimating the variables 𝑦𝑦(𝑦𝑦) in the rule 𝐾𝐾 for the lower (upper) acoustic surfaces. The null solution 𝑹𝑹 0 = [𝑟𝑟 𝐼𝐼𝐽𝐽 0 ], 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 𝐸𝐸 𝐽𝐽 , where

𝑟𝑟 𝐼𝐼𝐽𝐽 ∈ [𝑟𝑟 𝐼𝐼𝐽𝐽 0 , 1], 𝑟𝑟 𝐼𝐼𝐽𝐽 ∈ [0, 𝑟𝑟 𝐼𝐼𝐽𝐽 0 ].

Following [8,[30][31][32][33], restoration of the acoustic image is accomplished by way of multiple solving the optimization problem (10). If 𝑹𝑹(𝑡𝑡) = [𝑟𝑟 𝐼𝐼𝐽𝐽 (𝑡𝑡)] is some t-th solution of the optimization problem (10), then 𝐹𝐹(𝑹𝑹(𝑡𝑡)) = 𝐹𝐹(𝑹𝑹 0 ). When forming the intervals (11), the search space is restricted by the intervals 𝑟𝑟 𝐼𝐼𝐽𝐽 (𝑡𝑡) ∈ [𝑟𝑟 𝐼𝐼𝐽𝐽 (𝑡𝑡 − 1), 1] for the upper bounds; 𝑟𝑟 𝐼𝐼𝐽𝐽 (𝑡𝑡) ∈ [0, 𝑟𝑟 𝐼𝐼𝐽𝐽 (𝑡𝑡 − 1)] for the lower bounds. The search for the intervals (11) will go on until 𝑟𝑟 𝐼𝐼𝐽𝐽 (𝑡𝑡) ≠ 𝑟𝑟 𝐼𝐼𝐽𝐽 (𝑡𝑡 − 1). If 𝑟𝑟 𝐼𝐼𝐽𝐽 (𝑡𝑡) = 𝑟𝑟 𝐼𝐼𝐽𝐽 (𝑡𝑡 − 1)), then 𝑟𝑟 𝐼𝐼𝐽𝐽 (𝑟𝑟 𝐼𝐼𝐽𝐽 )=𝑟𝑟 𝐼𝐼𝐽𝐽 (𝑡𝑡).

The genetic-gradient approach is proposed for solving the optimization problems ( 9), (10) [30][31][32][33]. When searching for the null distribution, the chromosome is defined as a string of binary codes of the fuzzy locations parameters 𝛽𝛽 𝐶𝐶 𝐼𝐼 , 𝜎𝜎 𝐶𝐶 𝐼𝐼 and the fuzzy relations 𝑟𝑟 𝐼𝐼𝐽𝐽 , 𝐼𝐼 = 1,𝑁𝑁, 𝐽𝐽 = 1,𝑀𝑀. When searching for the reconstruction set, the chromosome is separated for each sound energy level 𝐸𝐸 𝐽𝐽 , where the parameters 𝑟𝑟 𝐼𝐼𝐽𝐽 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 𝜝𝜝 𝐶𝐶 , 𝜴𝜴 𝐶𝐶 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

𝑟𝑟 𝐼𝐼𝐽𝐽 (𝑡𝑡 + 1) = 𝑟𝑟 𝐼𝐼𝐽𝐽 (𝑡𝑡) − 𝜂𝜂 𝜕𝜕𝜀𝜀 𝑡𝑡 0 𝜕𝜕𝑟𝑟 𝐼𝐼𝐽𝐽 (𝑡𝑡)

;

𝛽𝛽 С 𝐼𝐼 (𝑡𝑡 + 1) = 𝛽𝛽 𝐶𝐶 𝐼𝐼 (𝑡𝑡) − 𝜂𝜂 𝜕𝜕𝜀𝜀 𝑡𝑡 0 𝜕𝜕𝛽𝛽 𝐶𝐶 𝐼𝐼(𝑡𝑡) ; 𝜎𝜎 𝐶𝐶 𝐼𝐼 (𝑡𝑡 + 1) = 𝜎𝜎 𝐶𝐶 𝐼𝐼 (𝑡𝑡) − 𝜂𝜂 𝜕𝜕𝜀𝜀 𝑡𝑡 0 𝜕𝜕𝜎𝜎 𝐶𝐶 𝐼𝐼 (𝑡𝑡) ,(13)

are used; and when searching for the reconstruction set, the recurrent relations

𝑟𝑟 𝐼𝐼𝐽𝐽 (𝑡𝑡 + 1) = 𝑟𝑟 𝐼𝐼𝐽𝐽 (𝑡𝑡) − 𝜂𝜂 𝜕𝜕𝜀𝜀 𝑡𝑡 𝜕𝜕𝑟𝑟 𝐼𝐼𝐽𝐽 (𝑡𝑡) ,(14)

are used [8,30,31], which minimize the criteria

𝜀𝜀 𝑡𝑡 0 = 1 2 (𝑦𝑦 � 𝑡𝑡 − 𝑦𝑦 𝑡𝑡 ) 2 , 𝜀𝜀 𝑡𝑡 = 1 2 (𝜇𝜇̂𝑡𝑡 𝐸𝐸 𝐽𝐽 − 𝜇𝜇 𝑡𝑡 𝐸𝐸 𝐽𝐽 ) 2 .

Here 𝑦𝑦 � 𝑡𝑡 , 𝑦𝑦 𝑡𝑡 are the observed and the model levels of acoustic energy at the t-th training step; 𝜇𝜇̂𝑡𝑡 𝐸𝐸 𝐽𝐽 , 𝜇𝜇 𝑡𝑡 𝐸𝐸 𝐽𝐽 are the observed and the model degrees of membership of field energy levels to the classes 𝐸𝐸 𝐽𝐽 at the t-th training step; 𝑟𝑟 𝐼𝐼𝐽𝐽 (𝑡𝑡) are the fuzzy relations at the t-th training step; 𝛽𝛽 𝐶𝐶 𝐼𝐼 (𝑡𝑡), 𝜎𝜎 𝐶𝐶 𝐼𝐼 (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].

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: ].

The sound field with coordinates 𝑥𝑥 1 = 𝑥𝑥 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 𝑘𝑘 1 = 𝑘𝑘 2 = 9, where с 1,1÷9 = с 2,1÷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 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 𝐸𝐸 𝐽𝐽 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 𝐸𝐸 𝐽𝐽 . Incorrect rules are divided into rules from the contiguous and remote power classes. In Table 3, the contiguous (remote) incorrect rules are marked with * (**).

𝑥𝑥 2 𝑥𝑥 1 ~25 m ~50 m ~75 m ~100 m ~125 m ~150 m ~175 m ~200 m ~225 m ~25 m 𝑬𝑬 𝟏𝟏 𝐸𝐸 2 𝐸𝐸 2 𝐸𝐸 1 𝐸𝐸 1 𝐸𝐸 1 𝐸𝐸 1 𝐸𝐸 1 𝐸𝐸 1 ~50 m 𝐸𝐸 2 𝐸𝐸 4 𝐸𝐸 3 𝐸𝐸 2 𝐸𝐸

In Table 1, the total number of rules 𝑍𝑍 = 81 is distributed according to the sound energy levels as follows: The risk of incorrect reconstruction of the acoustic levels is presented in Table 4, where 𝑃𝑃 𝐽𝐽 0(𝑐𝑐) (𝑃𝑃 𝐽𝐽 0(𝑟𝑟) ) -is the risk of reconstruction of the contiguous (remote) acoustic levels instead of the acoustic level 𝐸𝐸 𝐽𝐽 . The probability of correct reconstruction of the acoustic surface is 𝑃𝑃 1 = 69/81 = 0.85. The risk of incorrect reconstruction is 𝑃𝑃 0 = 12/81 = 0.15, which is distributed to the risks of reconstruction of the contiguous (remote) acoustic levels 𝑃𝑃 0(𝑐𝑐) = 10/81 = 0.12 (𝑃𝑃 0(𝑟𝑟) = 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 2/12=0.17

𝑥𝑥 2 𝑥𝑥 1 ~25 m ~50 m ~75 m ~100 m ~125 m ~150 m ~175 m ~200 m ~225 m ~25 m 𝑬𝑬 𝟏𝟏−𝟐𝟐 𝐸𝐸 2−3 𝐸𝐸 1−2 𝐸𝐸 1−2 𝐸𝐸 1−2 𝐸𝐸 1−2 𝐸𝐸 1−2 𝐸𝐸 1−𝐸𝐸 3 * 𝐸𝐸 1−2 𝐸𝐸 1−2 ~225 m 𝐸𝐸 2−3 𝐸𝐸 2−3 𝐸𝐸 3 * 𝐸𝐸 3−4 𝐸𝐸 3−4 𝐸𝐸 5 𝐸𝐸 2−3 𝐸𝐸 1−2 𝐸𝐸 1−2

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][27][28]. In [26][27][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][27][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 𝑘𝑘 1 = 𝑘𝑘 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][27][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.

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.