The paper deals with the problem of reconstructing the parameters of physical fields using distributed information and measurement systems for cases of incomplete laying of measurement lines. High-performance computing is typically used for solving advanced problems and performing research activities through computer modeling. The rise of Big Data has changed the entire perspective of data and data handling. Ever growing analytical needs for Big Data can be satisfied with extremely high-performance computing models. A new combined algorithm is presented, which is concluded in the “optimization of the geometry” of the measuring network with a view to further applying the complex of neural networks. The possibility of choosing and using the appropriate neural network from a complex of several pre-trained.
Computerization of almost all areas of modern life has a great impact on a human, on the majority of his activities and it contributes to development of information technology. In accordance with the modern tendency in the development of measuring instruments, in case of large amount of collected and processed information it is necessary to use not a lot of measuring instruments, but rather complex devices such as information and measuring systems [1].
Information measuring systems (IMS) is used to solve a wide range of applied problems however the main purpose is to provide continuous monitoring of large-scale and spatially inhomogeneous multidimensional physical fields [2].
What is important in the work of the IMS is the process of collecting information, on which the type of measuring network depends. Topology of the communication system depends on the choice of network technology and as a consequence the scope of application, types of input signals, types of measurements and functional properties of components. Examples of IMS with fundamentally different network topology are: • Fiber-optic measuring networks (FOMN) based on fiber-optic information and measuring networks; • Information-measuring systems based on wireless sensor networks (WSN).
Present day science intensive production cannot do without constant monitoring and control over the behavior dynamics of parameters range of distributed physical fields (PFs). The distributed information-measuring systems are called upon to solve this problem.
Information and measuring systems based on wireless sensor networks have great potential. This sort of lowpower communication devices can be deployed over the entire area of almost any physical space, ensuring continuous monitoring of physical phenomena in real time, processing and transferring collected information, and coordinating actions with other nodes of the network. However, it is impossible to fully deploy intelligent distributed IMS using wireless technologies in critical infrastructures, primarily due to the lack of network technologies that meet information security requirements [3].
FOMN are of greatest interest. These systems have various topologies and organization, and are constructed, for example, on fiber-optical element base [1]. One of the fundamental parts of the distributed fiber optical measuring system (DFOMS) is distributed fiber optical measuring network responsible for collecting measuring information regarding the PF parameters under study [2].
FOMN represents a set of fiber-optical measuring lines (MLs) [1, 2] stacked in accordance with a certain setup on the surface studied. Thus, reconstructing distributed PFs’ parameters against characteristics of optical radiation passing through FOMN assigns this mathematical problem to tomography [4]. To restore distributed function of PFs by means of FOMN, MLs were stacked along 2-4 directions.
In case of full data, in other words, with sufficient quality projections on all 180 degrees of angular range, high quality reconstructions are known to be obtained [4-6]. For comparison, to receive quality images in industry tomography, the necessary number of directions is p = 102-103.
Standard analytical methods are unacceptable for a fiber-optical tomography as direct application of inverse operator does not provide a unique stable solution. It is characteristic of any low angle problem in tomography. So, there is a good reason to consider other algorithms, perhaps, their synthesis as well. Restoring PFs’ functions by using FOMN can be broken into several steps: sampling, receiving and processing projection data, and back projecting.
The existing success and great prospects for the development of information and measuring systems are largely due to the fact that sensor networks built on the same principles can be used in completely different areas of human activity. However, wireless sensors have a number of limitations that have a negative impact on the provision of information security in the transmission of data within and outside the network. 2
Let's f (x1, x2) is the function of distributed PFs’ parameter on a planar surface Throughout this paper we will assume that f is infinitely differentiable and has a compact support. The 2D Radon transform maps a density function f as its line integrals. The objective of tomography is to produce an accurate image of an object interior based on a finite number of scanned views.
Mathematically, the problem is to reconstruct f, given the measurements of g(, s) on . The short-term objective will be focused on a comprehensive description of a projection function g = f.
Let index i determine an i-th direction of scanning i, and index j determine the samples sij in the selected i-th
direction. In this case a pair of indexes (i, j) corresponds to a straight line Lij along which the area is scanned. Then a
projection value along the straight line Lij can be written as
(
Lij S =
N S . k =1 k where is the Radon transform of function f, and dl is a gain along the straight line Lij. Pairs of numbers (i, sij) determine the parallel setup of scanning on a plane.
Let's break the area of research S R2 into smaller sites, so as
We consider the function f constant in each cell Sk and equal to fk, the symbol f also denotes the matrix corresponding to this partition: f1 f = fm+1 fn(m−1)+1 f2 fm+ 2 fn(m−1)+ 2 fm
f2m F = ( f1 fnm f2 fm fm+1 f2m
Т fnm ) .
Elementary cells Sk are referred to as image elements. Let’s assume function f as constant in each site Sk and equal
to fk. Symbol f also denotes a matrix corresponding to this decomposition. Then the integrated equations (
The specificity of the fiber-optic tomography tasks is the presence of an FOMN ultra-small survey data acquisition
scheme. As a rule, in such FOMN, the number of measuring lines is less than the number of monitored areas. SLAE
is underdetermined here (
Due to the fact that the input data has a large dimension, it is necessary to perform processing that allows you to
select the most significant parameters by reducing the number of free variables in SLAE (
In this context, FOMN optimization consists in deleting rows and columns along the edges of the matrix f (in the
“trimming” of matrix f), the sum of the elements of which has values equal to zero. Knowing the size of the matrix f
and the values of the column of the projection data, one can always check whether the matrix f has such a row or
column. Then the rows and columns with this feature are removed from the matrix f. Next, the matrix itself (
As a result of the Trimming Algorithm, a new matrix f of size n '× m' is formed, with n'≤n, m'≤m. Thus, when executing the algorithm described above, candidate areas in which the required “objects” are located are selected.
Let us give an example of applying the above method to a specific distribution of a parameter of a physical field, which in analytical form is given by a function: z( x, y) = e -0.5((x-6)2 +( y-11)2 ) + e -0.5((x-16)2 +( y-7)2 ) nK ,mK
Let FOMN, after applying the procedure described in clause 2, have dimensions n '× m', n'≤n, m'≤m. In general, the sizes n and m should decrease (n'≤n, m'≤m). This happens in most cases, because the spatial frequency b = π imposes restrictions on the size of the objects under study. In extreme cases, you will have to use the neural network for the entire area.
It is not known in advance what sizes will be there. Therefore, the question naturally arises: what exactly is the size of the neural network to use?
The answer to the question posed is contained in the approach proposed in this paper. It consists in the following: 1. We will train in parallel (independently of each other) several neural networks of different sizes.
Denote by NN (ni, mi) a neural network of size ni × mi, i.e. neural network, which is intended for FOMN processing of the appropriate size.
Through SN denote we set of all K pre-trained neural networks of the form NN (ni, mi):
SN = NN ( n1 , m1 ) , , NN ( ni , mi ) , ..., NN ( nK , mK ) , n1 ni nK , m1 mi mK ,
( ni , mi ) ( n j , m j ) ; i j, 1 i, j K .
2. For processing the projection data of the n '× m', n'≤n, m'≤m posting from FOMN, we choose in the SN set of
the form (
ni−1 n ' ni , mi−1 m ' mi , 1 i, j K .
From conditions (
In the work, neural networks of radial basis type are used in the work of neural networks. Earlier in the article [10], the authors have already investigated the possibility of using radial basis neural networks (RBFNN).
The information generated by the network, represented by vector G, was a set of topographic data for which the neural network must reconstruct the vector F. Thus, the neural network must perform the transformation F = A-1(G), having previously been trained on a set of training pairs {(G, F)}.
To create a training page, the author used Reinforcement method of selecting training pairs in which pairs of the form (Gi, Fi) were considered, where AFi = Gi.
When creating RBFNN training pairs in [10], Gaussian-type functions were used, and the parameters were selected as lattice points of the corresponding scanning scheme and Gaussian pairs.
For example, for the 5x5 field, a training page was created, consisting of 3325 training pairs, on which the RBFNN network was trained. It was experimentally shown that the constructed network makes it possible to restore the functions of the spatial distribution of the studied physical quantity with an error at a single point of no more than 2%, and has good predictive capabilities. However, it was noted that with this method of recovery in high-dimensional tasks using FOMN, there are serious difficulties in training the network due to the very large amount of training pages. Therefore, it became necessary to search for optimal paths when using neural networks.
One of the ways to optimize the processing of information is the use of a set of previously-trained neural networks of various dimensions.
The choice of reference functions should depend on the width of the spectrum b of the function f (x, y) under study. Functions of a Gaussian type
z ( x, y ) = e− ai (( x−ci )2 +( y −bi )2 ) can be used here, since they take non-zero values only in the zone around a certain center.
To analyze the neural network method for solving the problem using RBFNN, this work considered the tomographic task of restoring the functions of the FP according to the information coming from the informationmeasuring system of size 30x30.
It was assumed that the reference effect on the field has the form of a smooth function with a limited effective width of the spectrum b equal to the conditional spectral unit p. It is considered that all values of the function are nonnegative and normalized.
In this work, the same three types of reference distributions of a physical quantity are used as in [10]. The first and second types refer to the regular method, and the third refers to the random method. We describe them in more details.
Type I. The reference field distributions in this case are single Gaussians of the form (
Type II. Analytically, these functions can be represented as:
z( x, y) = e-a1 (x-c1 )2 +( y-b1 )2 + e-a2 (x-c2 )2 +( y-b2 )2
provided that the carriers are at least 2π / b. These are Gaussian couples with non-intersecting carriers.
(
Type III. Reference distributions of this type were obtained using a randomization process with normalization.
Each integer random set a1, а2, b1, c1, b2, c2= 1, N was assigned a function of the form (
To analyze the neural network method of solving the problem using the RBFNN complex, the informationmeasuring network 30x30 was considered.
In Table 1 for each neural network of a radial-basic type belonging to the set of SN, the following characteristics are presented:
- dimensions (ni, mi) correspond to the geometry of the measuring network, which is processed by a neural network of radial basis type NN(ni, mi), - the total amount of the training page (TP) includes types I - III, - the average training time for the results of a series of several (from 10 to 15) computational experiments, - values of the normalized mean square error (MSE) across the entire training page, - the number of impacts recognized by the neural network NN(ni, mi), - predicting capabilities, i.e. recognition by a network of types of effects that do not belong to the training page, - the quality of training is averaged characteristic, associated, including with the presence of artifacts as a result of insufficient amount of the training page.
From the results in Table 1, it can be seen that as the size of the network grows, the quality of learning decreases. From the above results, it follows that processing with the help of Trimming Algorithm an area with subsequent collective processing by neural networks yields a large gain in accuracy. This is explained by the localization of the site of impact on the network and processing using a neural network, as a rule, of a lower dimension, which is trained more efficiently and quickly. At the same time, the error of the standard deviation for the elements from the training page drops from 15 to 20 times.
Figure 2 shows the results of processing the species exposure z ( x, y ) = e −0.5(( x−6)2 +( y −19)2 ) .
After localization of the impact, a 3 × 3 size similar region was obtained, which is processed by a well-trained neural network NN (3.3). Finally, the original dimensions of the area were restored (Fig. 3b). In fig. 3c shows the result of the restoration of the function under study using a neural network of maximum size NN (30.30). The quality 1. 2. 3. 4. 5. 6. 7.
1. 2. 3.
Size (ni, mi) 3×3 5×5 7×7 10×10 15×15 20×20 30×30
Volume
TP of training NN (30,30) is low, which explains the appearance of artifacts even with the restoration of a single impact on the measuring network.
Tables 2 and 3 present the results of the work of the proposed method for the reference effects on the measuring
network of single effects (
−0.5((x−6)2 +( y−19)2 )
Information driven economy relies on the actionable insights extracted from data analytics. The era of data revolution has paved way to the need of convergence of paradigms like High Performance Computing and Big Data processing. The amalgamation of these paradigms is a herculean task involving various aspects like data management and computing efficiency. This has given rise to evolution of the data storage technologies and computing models.
The article presents a new combined projection data processing algorithm for reconstructing information received from fiber-optic measurement lines distributed by FOMN.
This algorithm consists in the sequential execution of two processes: 1. Pre-processing of measurement information by localizing the locations of impact on FOMN, 2. Application of a complex of neural networks for processing measuring systems of various geometries. From the above results it follows that: 1. Processing using the area cropping procedure with subsequent collective processing by neural networks gives a gain in accuracy largely due to the localization of the site of impact on the network and processing using the neural network, as a rule, of lower dimensionality, which is trained more qualitatively and quickly. At the same time, the value of the mse error for the elements from the training page drops 15 to 20 times.
2. Reducing the mse error and shortening the processing time mainly depends on how radically the computational process has been optimized as a result of the preprocessing and on the complexity of the function being restored.