CC BY
26
An Integrated Approach to Navigation of Mobile Devices Indoors Based on Wi-Fi
and Image Objects
Erin A. A., Khomonenko A. D. Emperor Alexander I Petersburg State Transport University St. Petersburg, Russia [email protected], [email protected]
Abstract. An approach to navigation of mobile devices indoors using artificial neural networks to determine location by comparing photographs of the premises taken by the user with images of the premises in the database is proposed. A comparison with existing models of indoor navigation is fulfilled. An example of a neural network for the chosen model and its complexity is given. An example of generation of an optimal route using algorithm Dijkstra, on the basis of the premises of the Department "Computing systems" of Emperor Alexander I Petersburg State Transport University is given.
Keywords: Artificial neural networks, indoor navigation, image moment invariants, Dijkstra's algorithm, Wi-Fi.
Introduction
Currently, mobile devices are widely used for the various governmental purposes. One of the main features offered by mobile devices is navigation. The optimal route to your destination, the accommodation, the search for alternative paths - all this greatly facilitates modern life. However, many modern navigation methods are unsuitable for indoor navigation. Buildings have complex infrastructure, many buildings consist of several buildings, and often quite difficult to quickly navigate and find the right room. For these reasons, the development of navigation of mobile devices within the premises is an urgent task.
To date, the main technology used for navigation is the global navigation satellite system GPS/GLONASS [1]. Beacons GPS is built into nearly all mobile devices and with them in power is determined by the location of the device on the ground, laid the routes of long distances, etc. However, they will not be able to navigate inside, because the density and the materials from which made the building, greatly reduce the location accuracy that space is critical, and with their help it is impossible to determine on which floor the user device.
Therefore, for implementation of indoor navigation other technologies are used [2]. One of the technologies used to implement indoor navigation technology is a Wi-Fi wireless data transmission. With the help of trained Wi-Fi network can determine the location of the device on the floor plan and control movement along the route [3].
The main tasks that need to be addressed when implementing the navigation of mobile devices within the premises are: search device location on the plan of the building, a route from the start point to the end, follow-up monitoring device according to the route.
This article considers the existing methods of navigation for mobile devices indoors, as well as proposes an alternative meth-
od indoor navigation based on finding a location using photo comparison areas.
Existing navigation model of mobile devices indoor
Currently, there are several models of navigation of mobile devices indoors [4]. Consider two models that use Wi-Fi to determine the location of the mobile device.
The first model is based on measuring the signal strength from the source to the customer and its subsequent processing. The first stage of this model is to determine the signal strength RSSI from the source. The value of RSSI is defined as
RSSI = -10 • n • log(d)+A,
where d is the distance, A is the transmitter power, n is the propagation constant of the signal. However, due to the physical properties of waves and other factors, this formula does not give sufficient accuracy, so, to calculate the distance according to the value of the force signal is used model of attenuation:
P(r)dBm = P(ro)dB +10 • n • logr0,
r
where P(r)dBm is the value of RSSI on the distance r, n is the attenuation coefficient, r is the distance from the device to the transmitter, the distance from the device to the point where the measurement was carried out signal strength P(r0)dB [5, 6].
After was produced by measuring distances, it becomes possible to build a geometric solution of the problem of positioning using triangulation graphs.
The second model is used artificial neural network to memorize the configuration of the premises and the subsequent recognition of the premises as visible in them signals [7-9]. At the training stage, a map of the room based on radio fingerprints, performed physical collecting radio fingerprint for each point. Then all the collected radio-prints and additional information about each point are combined into one file for training a neural network. After this occurs the training of the neural network.
The finished app works on the following principle: the user includes beacon Wi-Fi, collect radio fingerprints Wi-Fi hotspots to which the device can connect, the imprint is sent to the trained neural network, which determines where the user is located, by comparing the received data from the card and finding the closest values. Thus, determined by the most probable Wi-Fi access point, which is about the user.
Model indoor navigation,
image-based and Wi-Fi network
Both models are considered based on the use of Wi-Fi networks to determine location of the device. However, the location accuracy via Wi-Fi network is affected by many factors, such as interference from devices operating on the same frequency; the obstacles that weaken signal strength and others. Therefore, alternatively, determining the initial location of the device we proposed to use the mechanism of comparison of photographs of the premises taken by the user.
The main idea of this method consists in using a mathematical model of an artificial neural network to determine the location of a mobile device made device pictures. It then executes the further building of the route from the starting location, on the photos to the end point selected by the user. To control the advance of the mobile device on the specified route uses a Wi-Fi network. Applied positioning method for assessing the signal strength of nearby access points Wi-Fi.
The software package consists of two components:
• a mobile application on the user's device, which carries out the survey of the premises, sends the image and the start location information about ending point of the route to the server, displays the constructed route to the user and monitors the progress on the route;
• the server where the database stores the image space, with their description and associated with PLA-us buildings and building plans in the form of graphs, and a server application that searches images using artificial neural network and builds the route.
Fig. 1. Architecture of the software
The algorithm works
mobile application
With the help of mobile applications having access to the device camera, the user photographs the room in which it is located and from which wants to route. The user then selects the end point of the route, a list with most places, the treatment, or noting the location on the floor plan, either by entering the room number or name in a search target row, then select. The resulting image and information about the endpoint sent to the server, where the processing and the construction of marsh root.
After a response received from server, building plan with marked on it the route displayed on the user's screen. Control of the movement route is via Wi-Fi module of the device. The Wi-Fi module in the background scans the network, receiving information about signal strength of Wi-Fi hotspots in the area which was hit by the device. Based on the information received, the application makes a conclusion in which the route point is a device and controls the correct route.
1. Obtaining the image and information on a fitial point from the user
Z Transfer of the image in the trained artificial neural network
Définition of initial location
4. Finding the shortest path from the start point to the end point on the graph
5. Creation of a route on the map ofrooms
6_ Transfer of the constructed route to the user
Fig. 2. The algorithm works mobile application
Using moment invariants of images
In many tasks of digital image processing has found a broad application of the torque characteristics of the images and counting them on the basis of moment invariants. The invariant is a value that remains unchanged under certain transformations. The invariant moments have become an essential tool for recognizing patterns irrespective of their particular position, orientation, viewing angle and other changes. The main advantage of moment invariants are insensitive to their rotation. This makes their use effective as features in the task of detection and recognition of image objects with an arbitrary orientation [10, 11].
In practice, recognizable images differ from each other appearance scale, rotation, and shear. For images of the same class in most of these cases come from the fact that a recognizable image was the result of a geometric transformation (scaling, rotation in the XY-plane and cyclic shift). If you consistently perform all the possible geometric transformation of the image and to compare the conversion result with a recognizable way, it is possible to register the parameters of the transformation, in which occurs the highest value of measures of similarity [11].
The invariant moments are the characteristics of the image based on exponential moments and describing the silhouette of an object. In accordance with its name, these features are invariant to affine transformations of the image. For processing digital images are discrete analogs of the torque characteristics. The formula for the moment of order (k, s) is written as follows [10]:
|ife = J J mknsx(m,n), k,s = 0,1,... (1)
Typically, in pattern recognition are used by the Central points having invariance to image shift. The corresponding Central moment is given by the formula:
|ks = J J (m -m)k(n -n)sx(m,n)dmdn, k,s = 0,1,.. (2)
where: m = m10 /m00;n = m01 /m00 are the coordinates of the center of gravity of the image. Central moments (2) expressed via moments (1) using the ratio:
iks=t t (-iy+'ccim-ins- jty, (3)
i=0 j=0
where C'k - binomial coefficients. For centered image values of the moments (1) and (2) coincide. Of Central moments (2) can be normalized to provide invariance to image scaling. Using the Central moments defined by features, invariant to image rotation (torque invariants). Having a set of characteristics, you can define the following seven invariant moments.
q>i =120 +I02, 92 = (l20 +l02)2 +
93 = (|30 + 3|i2)2 + (3|21 03^
94 = (l30 + 3l12)2 + (l 21 -l 03^
95 = (l30 + 3l12 )(l30 + 112 )[(l30 + l12)2 - 3(l21 +103)2] +
+(3l21 - I03)(l21 +103 ) X [3(l30 + l12)2 - (l21 + 103)2],
96 = (l20 + 102 )[(l30 +l12)2 - (l21 + 103 )2 ] + +4l11(l30 -l 12)(l21 + I03X
97 = (3l21 + 103 )(l30 + 112 )[(l30 + l12)2 - 3(l21 +I03 )2 ] --(l30 - 3I12)(l21 +l03) X [3(l30 +l12)2 - (l21 +l03)2].
These seven moments are invariant to shifts, rotations, axial symmetries, and strains and compressions [11]. In General, they are nonlinear combinations of Central moments, which are, in turn, functions of the geometric (initial) moments.
To assess the computational complexity of the invariants it is necessary to calculate the number of operations multiplications and summations when computing the invariants [12]. Formula for calculation:
бум (9) = Ебум (I j + « + P + (Y-1), (4)
j=1
всл (9) =Ебсл (l Pjj) + (S-1), (5)
j=1
were бум (| pjqj), вал (Ipjqj) is number of multiplications and additions at calculation of the moment entering functionality; J -amount of the moments entering functionality of an invariant ф; a - number of the multipliers which are subject to data in the second degree; p - number of the multiplications spent for multiplication of the moments (or their sums) on constant coefficient which module isn't equal to 1.
Calculation of computing complexity of invariants which results are taken out in table 1 is given in article [12].
Thus, it is possible to draw a conclusion that computing complexity increases in process of increase in an order of a moment invariant, computing complexity of an invariant depends on number of the moments entering functionality of an invariant and computing expenses are proportional to the image sizes.
Using of neural network
for comparison of photos
For information search about the room in the image it is offered to use artificial neural network. As one of the main properties of the photo is quality and not each device is capable to take the high-quality picture of the room, in order that there was an opportunity to find the necessary room according to the taken picture of any quality, it is offered to use neural network of Hopfild [13, 14].
As the neural network of Hopfild is applied generally to recovery of noisy images, it has to level quality of the taken picture. The neural network of Hopfild consists of N artificial neurons; the axon of each neuron is tied with dendrites of other neurons, forming feedback. Each neuron can be in one of two states:
x(t) e {-1;1},
where x(t) is a condition of neuron at the time of t. Corresponds to "excitement" of neuron +1, and to "braking"-1. Dynamics of a state in time i-oho of neuron in network from N neurons is described by discrete dynamic system:
N
xi(t) = sign [ Z wijxi(t - j e N,
j=1, j *i
Table 1
Computing complexity of invariants
Invariant Number of multiplications Q (ф) ^ум vt V Number additions 0сл ^i) Relative number of multiplications qym (ф;) Relative number of additions qon (ф; ) Full expenses еад Relative full expenses Q(<p, )
Ф1 24 5 1 1 29 1
Ф2 43 8 1,8 1,6 51 1,76
Фз 104 19 4,33 3,8 123 4,24
Ф4 102 19 4,25 3,8 121 4,17
Фз 112 27 4,66 5,4 139 4,79
Фб 146 30 6,1 6 176 6,1
Ф7 112 27 4,66 5,4 139 4,79
Fig. 3. Structure of Hopfield network
where w is weight coefficient between neurons of i and j, Xj (t -1) are values of exits of neuron of j in the previous time point [14].
Training of network of Hopfild in output images comes down to calculation of values of elements of a matrix wj. It is formally possible to describe training process as follows: let it is necessary to train neural network to distinguish the M images designated {X|,| = 1,...,M}. The entrance image represents:
Xin _ y in
^ - A ^
+ 8,
where e - the noise imposed on an initial image.
Calculation of a square matrix of scales is made by the Hebb rule:
1 M
- - Z [ x
ij N ^ ^
Training of a neural network is made on the images which are stored in the database on the server that further the trained neural network could define to what image there corresponds the taken picture.
As an example the neural network of Hopfild capable to recognize to what image which is stored in memory of a network is realized there corresponds the image transferred to processing. At the moment the network is capable to work with black-and-white images of a format 50*50. In case of such format the one-layer network of Hopfild has 2500 neurons which accept one or the other values {-1; 1}. The realized network showed the following results provided in table 2.
Table 2
Temporary characteristics of neural network
Time of processing of the image, s Time of network training, s Time of recognition of the normal image, s Time of recognition of the noisy image, s
20 the training examples, 50x50 in size 4,8 96 5,2 5,8
At the same time if in the training set there were no strongly similar images, then the network recognized all usual images correctly, and noisy if they have been distorted less than for 35 % that all are distinguished correctly, at a strong noisiness the network could make a mistake. If in the training set there were several similar images, then the network could recognize the image incorrectly.
Creation of the route on the column
Having defined starting and ending point of a route, it is necessary to lay out a route. Building plans are stored in the database on the server in the form of a nonoriented graph. Trailing peaks are finite locations, remaining peaks are other locations, and edges of a graph - transitions and corridors between locations. Lengths of edges are lengths of the appropriate routes. As weight of edges - is non-negative, and throughput isn't important, for a route spacer that is search of the shortest way in the graph, Dijkstra's algorithm is used [15].
Dijkstra's algorithm finds the shortest ways from one of the count's tops to all others. The algorithm works only for counts without edges of negative weight. The algorithm works step by step - on each step he "visits" one top and tries to reduce tags. Work of an algorithm comes to the end when all tops are visited [15-17].
Initialization. The tag of the top of a necessary equal 0, tags of other tops - infinity. It reflects that distances from top a to other tops are still unknown. All tops of the count are marked as not visited.
Algorithm step. If all tops are visited, the algorithm comes to the end. Otherwise, u top having the minimum tag gets out of yet not visited tops. We consider various routes in which top u are penultimate point. We will call tops in which conduct edges from u neighbors of this top. For each neighbor of top of u, we will consider the new length of a way equal to the sum of values of the current tag of u and length of the edge connecting u to this neighbor. If the received value of length is less than value of a tag of the neighbor, we will replace value of a tag with the received value of length. Having considered all neighbors, we will mark u top as visited and we will repeat an algorithm step.
As an example of work of an algorithm the count corresponding to premises of "Information and Computing Systems" department of PGUPS, presented in fig. 4 has been constructed.
Lengths of routes of the count decide on the help of the following matrix of the weights.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
1 - 4 6 9 - 5 12 5
2 4 - 4 6 - 7 9 3
3 6 4 - 4 - 9 6 5
4 9 6 4 - - 11 3 7
5 - - - 11 - - 3 - - - - - - - - - -
6 5 7 9 3 - - 15 10 - - - 45 65 - - - 62
7 12 9 6 7 3 15 - 8 - - 5 12 - - - - -
8 5 3 5 - - 10 8 - 3 3 - - - - - - -
9 - - - - - - - 3 - - - - - - - - -
10 - - - - - - - 3 - - - - - - - - -
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
11 - - - - - - 5 - - - - 8 - - - - -
12 - - - - - 45 12 - - - 8 - 15 - - - -
13 - - - - - 65 - - - - - 15 - 10 8 10 10
14 10 - 4 6 -
15 8 4 - 4 -
16 10 6 4 - -
17 - - - - - 62 - - - - - - 10 - - - -
Table 3
Resulting table of the shortest routes
№ of tops Shortest route № of tops Shortest route
2 1-2 10 1-8-10
3 1-3 11 1-7-11
4 1-4 12 1-7-12
5 1-7-5 13 1-7-12-13
6 1-6 14 1-7-12-13-14
7 1-7 15 1-7-12-13-15
8 1-8 16 1-7-12-13-16
9 1-8-9 17 1-7-12-13-17
Example of the algorithm: it is necessary to find all the shortest routes leading from vertex 1. The starting vertex from which the tree of shortest paths is constructed is vertex 1. Set the starting conditions: d(1) = 0, d(x) = We find the nearest vertex to the starting point, using the formula: d(x) = min{d(x); D(y) + + ay, x}:
d(2) = min{-;0+4} = 4 d(3) = min{-;0+6} = 6 d(4) = min{-;0+9} = 9 d(6) = min{-;0+5} = 5 d(7) = min{-;0+12} = 12
Mark the corresponding vertices with new weights, select the nearest smallest vertex and add the corresponding arc, as the shortest path to this vertex. In this case, vertex 2 and arc (1,2). Repeat the step of the algorithm, now for vertex 2. d(3) = min{6;4+4} = 6 d(4) = min{9;4+6} = 9 d(6) = min{5;4+7} = 5 d(7) = min{12;4+9} = 12
The weights of the vertices have not changed, therefore, take the next smallest vertex 6 and write its arc (1.6).
Repeat the algorithm steps until all vertices are visited. As a result, we obtain a table with all shortest routes from vertex 1.
Conclusion
In article the developed model for navigation of mobile devices in locations is provided, reasons for use of the selected technologies are given, information on the necessary software is provided.
On this basis the program complex realizing the developed navigation model is developed.
Further researches are supposed to be continued in the following directions on development of a program complex:
• improving of a neural network that the neural network could recognize color images of the big sizes;
• improving of system of a choice of finite location in the form of adding of an alternative possibility of an interactive choice of finite location on a building card;
• transfer of a mobile application on other operating systems, with the purpose to increase target audience.
• adding of a possibility of support of following along a route by means of network Wi-Fi.
References
1. Serapinas B. B. Global Positioning Systems: study guide [Global'nye sistemy positsionirovaniya: uchebnoye posobiye], Moscow, 2002. 106 p.
2. Yakovenko I. A. Indoor Navigation. Overview and Comparative Analysis of Technologies: GSM, Bluetooth, Wi-Fi, GPS, RFID, NFC [Navigatsiya vnutri pomeshcheniy. Obzor i sravnitelniy analiz tekhnologiy: GSM, Bluetooth, Wi-Fi, GPS, RFID, NFC], Youth scientific and technical bulletin [Molodezh-niy nauchno-tekhnicheskiy vestnik], 2015, no. 6.
3. Evennou F., Marx F. Advanced Integration of WiFi and Inertial Navigation Systems for Indoor Mobile Positioning, EURASIP J. on Applied Signal Proc, 2006, Article ID 86706, pp. 1-11.
4. Ovsyannikov A. A., Novikov P. A. Implementation of the Indoor Navigation Algorithm Based on Bluetooth Low Energy 4.0. Technology [Modeli realizatsii navigatsii vnutri pomesh-
Fig. 4. The count corresponding to the map of rooms
cheniy pri pomoshchi analiza besprovodnikh istochnikov dan-nykh], Computer Tools in Education [Kompyuternye instrumen-ty v obrazovanii], 2015, no. 4, pp. 37-51.
5. Srinivasan K., Levis Ph. RSSI is Under Appre-ciated. Available at: https://sing.stanford.edu/pubs/rssi-emnets06.pdf (accessed 24 April 2017).
6. Park J. J., Yang L. T., Lee C. Future Information Technology, 6th Int. Conf. Future Inform. Technol., FutureTech 2011, Crete, Greece, June 28-30, 2011. Proc. Springer, 2011, pp. 89-90.
7. Mok E., Cheung Bernard K. S. An Improved Neural Network Training Algorithm for Wi-Fi Fingerprinting Positioning, ISPRSInt. J. Geo-Inf, 2013, no. 2, pp. 854-868.
8. Novikov P. A., Khomonenko A. D., Yakovlev E. L. Software for Mobile Indoor Navigation using Neural Networks [Kompleks programm dlya navigatsii mobilnikh ustroystv vnutri pomeshcheniy s pomoshchi neironnykh setey], Information and Control Systems [Informatsionno-upravlyayushchie sistemy], 2016, no. 1 (80), pp. 32-39.
9. Novikov P. A., Khomonenko A. D., Yakovlev E. L. Justification of the choice of neural networks learning algorithms for indoor mobile positioning, Proc. CEE-SECR '15 Proc. 11th Central & Eastern European Software Eng. Conf. Russia Article, no. 9. NY, ACM, 2015.
10. Glumov N. I. The Construction and Application of Moment Invariants for Image Processing in a Sliding Window [Postro-enie i primenenie momentnikh invariantov dlya obrabotki izo-brazheniy v skolzyashchem okne], KO, 1995, no. 1, pp. 14-15.
Available at: http://cyberleninka.ru/article/n/postroenie-i-prime-nenie-momentnyh-invariantov-dlya-obrabotki-izobrazheniy-v-skolzyaschem-okne (accessed 13 March 2017).
11. Gonzalez R. C., Woods R. E., Steven L. Eddins Digital Image Processing Using MATLAB [Tsifrovaya obrabotka izobra-zheniy v srede Matlab], Moscow, Tekhnosfera, 2006, 616 p.
12. Medvedik A. D., Verchenko V.A., Babak P. E. Evaluation of the Computational Complexity of Moment Invariants Used in Pattern Recognition Problems [Otsenka vichislitelnoy slozhnosti momentnikh invariantov, ispolzuemykh v zadachakh raspozna-vaniya], Radioelectronic and Computer Systems [Radioelektron-nye i comp'yuternye sistemy], 2015, no. 4 (74), pp. 124-130.
13. Kruglov V. V, Borisov V. V. Artificial Neural Networks. Theory and Practice [Iskusstvennye neyronnye seti. Teoriya i praktika], Moscow, Goryachaya Liniya Telekom, 2002, 382 p.
14. Wasserman Ph. D. Neural Computing. Theory and Practice [Neirokompyuternaya tekhnika: teoriya i praktika], Moscow, Mir, 1992, 240 p.
15. Cormen T. H., Leiserson Ch. E., Rivest R. L., Stein C. Introduction to Algorithms [Algoritmy: postroyenie i analiz], Moscow, Vilyams, 2006, 1296 p.
16. Levitin A. V. Algoritmy. Algorithms. Introduction to Development and Analysis [Vvedenie v razrabotku i analiz], Moscow, Vilyams, 2006, 576 p.
17. Asanov M. O., Baranskiy V A., Rasin V V Discrete Mathematics: Graphs, Matroids, Algorithms [Diskretnaya matematika: grafy, matroidy, algoritmy], Izhevsk, NITS "Regularnaya i kha-oticheskaya dinamika", 2001, 288 p.
Комплексный подход к навигации мобильных устройств внутри помещений на основе Wi-Fi и изображений объектов
Ерин А. А., Хомоненко А. Д. Петербургский государственный университет путей сообщения Императора Александра I
Санкт-Петербург, Россия [email protected], [email protected]
Аннотация. Предлагается подход к навигации мобильных устройств внутри помещений с использованием искусственной нейронной сети для определения местоположения путем сравнения фотографий помещения, сделанных пользователем с изображениями помещений в базе данных. Проводится их сравнение с существующими моделями навигации внутри помещений. Рассматривается пример нейронной сети для выбранной модели и его трудоемкость. Приводится пример построения оптимального маршрута с помощью алгоритма Дейкстры на основе помещений кафедры «Информационно вычислительные системы» Петербургского государственного университета путей сообщения Императора Александра I.
Ключевые слова: искусственные нейронные сети, навигация внутри помещений, инварианты моментов изображений, алгоритм Дейкстры, Wi-Fi.
Литература
1. Серапинас Б. Б. Глобальные системы позиционирования: учеб. пособие / Б. Б. Серапинас. М.: Каталог, 2002. 106 с.
2. Яковенко И. А. Навигация внутри помещений. Обзор и сравнительный анализ технологий: GSM, Bluetooth, Wi-Fi, GPS, RFID, NFC / И. А. Яковенко // Молодеж. науч.-технич. вестн., 2015, № 6.
3. Evennou F. Advanced Integration of WiFi and Inertial Navigation Systems for Indoor Mobile Positioning / F. Evennou, F. Marx // EURASIP J. on Applied Signal Proc., 2006. Article ID 86706, р. 1-11.
4. Овсянников А. А. Модели реализации навигации внутри помещения при помощи анализа беспроводных источников данных / А. А. Овсянников, П. А. Новиков // Компьютерные инструменты в образовании, 2015, № 4.
5. Srinivasan K. RSSI is Under Appreciated / K. Srinivasan, Ph. Levis. URL: https://sing.stanford.edu/pubs/rssi-emnets06. pdf (дата обращения 24.04.2017).
6. Park J. J. Future Information Technology / J. J. Park, L. T. Yang, C. Lee // 6th Int. Conf. on Future Inform. Technol., Future Tech 2011, Crete, Greece, June 28-30, 2011. Proc. Springer, 2011, pp. 89-90.
7. Mok E. An Improved Neural Network Training Algorithm for Wi-Fi Fingerprinting Positioning / E. Mok, B. K. S. Cheung // ISPRS Int. J. Geo-Inf, 2013, № 2. P. 854-868.
8. Новиков П. А. Комплекс программ для навигации мобильных устройств внутри помещений с помощью нейронных сетей / П. А. Новиков, А. Д. Хомоненко, Е. Л. Яковлев // Информационно-управляющие системы, 2016, № 1 (80). С. 32-39.
9. Novikov P. A. Justification of the choice of neural networks learning algorithms for indoor mobile positioning / P.A. Novikov, A. D. Khomonenko, E. L. Yakovlev // Proc. CEE-SECR '15 Proc. 11th Central & Eastern European Software Eng. Conf. in Russia Article № 9. NY: ACM, 2015.
10. Глумов Н. И. Построение и применение моментных инвариантов для обработки изображений в скользящем окне / Н. И. Глумов // КО, 1995, № 1. С. 14-15.
11. Гонсалес Р. Цифровая обработка изображений в среде Matlab / Р. Гонсалес, Р. Вудс, Р. Эддинс. М.: Техносфера, 2006. 616 с.
12. Медведик А. Д. Оценка вычислительной сложности моментных инвариантов, используемых в задачах распознавания / А. Д. Медведик, В. А. Верченко, П. Е. Бабак // Радиоэлектронные и компьютерные системы, 2015, № 4(74). С. 124-130.
13. Круглов В. В. Искусственные нейронные сети. Теория и практика / В. В. Круглов, В. В. Борисов. М.: Горячая линия-Телеком, 2002. 382 с.
14. Уоссермен Ф. Нейрокомпьютерная техника: Теория и практика / Ф. Уоссермен. М.: Мир, 1992. 240 с.
15. Кормен Т. Х. Алгоритмы: построение и анализ / Т. Х. Кормен, Ч. И. Лейзерсон, Р. Л. Ривест, К. Штайн. 2-е изд. М.: Вильямс, 2006. 1296 с.
16. Левитин А. В. Алгоритмы. Введение в разработку и анализ / А. В. Левитин. М.: Вильямс, 2006. 576 с.
17. Асанов М. О. Дискретная математика: графы, матрои-ды, алгоритмы / М. О. Асанов, В. А. Баранский, В. В. Расин. Ижевск: НИЦ «Регулярная и хаотическая динамика», 2001. 288 с.
