c Allerton Press, Inc., 2012. ISSN 8756-6990, Optoelectronics, Instrumentation and Data Processing, 2012, Vol. 48, No. 5, pp. 522–528. c D.O. Sokolova, A.A. Spektor, 2012, published in Avtometriya, 2012, Vol. 48, No. 5, pp. 112–119. Original Russian Text
ANALYSIS AND SYNTHESIS OF SIGNALS AND IMAGES
Classification of Moving Objects Based on Spectral Features of Seismic Signals D. O. Sokolova and A. A. Spektor Novosibirsk State Technical University, pr. Karla Marksa 20, Novosibirsk, 630092 Russia E-mail:
[email protected],
[email protected] Received October 10, 2011 Abstract—The problem of object classification in seismic guard systems on the basis of statistical analysis of spectral moments used as features for classification is considered. An adaptive algorithm is developed, and its efficiency is experimentally determined. Keywords: seismic guard system, object classification, maximum likelihood principle, spectral moments. DOI: 10.3103/S8756699012050135
INTRODUCTION The seismic principle is now frequently used in developing modern guard systems [1–3]. Seismic guard systems (SGS) based on this principle offer significant potential capabilities of automatic detection of seismically active objects, estimation of their trajectories, and determination of their types (solving the classification problem). An important advantage of such systems is almost complete secrecy because the SGS operation principle is passive, and seismic sensors and connecting wires are usually submerged into soil. Solving the above-mentioned problems associated with SGS design requires mathematical methods, algorithms, and codes implementing necessary functions of signal processing to be developed. There are some few publications [4–9] on detection of seismic signals and trajectory estimations, but the problem of object classification with the use of SGS has not be adequately studied [10–12]. The present activities are aimed at developing a method of classification of seismically active objects, based on using the spectral analysis of signals. BAYESIAN CLASSIFICATION OF SEISMIC SIGNALS ON THE BASIS OF MOMENT CHARACTERISTICS OF SPECTRA It is known that signals detected by seismic sensors are formed as a result of superposition of seismic waves propagating in soil over multipath trajectories. Therefore, they follow the Gaussian distribution, and the spectral-correlation properties allow the use of the Markov models based on recurrent linear mechanisms of prediction [4, 6]. A seismic background signal in a limited time interval is a stationary random process. The presence of seismically active objects is manifested both in the character of signals and in their spectral-correlation characteristics. For instance, the seismic signal from a walking person is a pulsed random process, which can be described in the pulsed duration interval (like the background signal) by the Markov model of linear prediction, but with different values of parameters. This means that the spectral-correlation characteristics in regions where the pulses are present differ from the corresponding characteristics of the background, i.e., regions where the pulses are absent. The spectral-correlation characteristics of signals having a greater length and including one or several periods of steps occupy an intermediate position, because such a seismic signal is a mixture of the valid pulsed process and a background continuous random process. On the average, the presence of a group of walking people leads to a greater number of regions possessing properties of the pulsed signal in the observation interval. For this reason, the spectral-correlation characteristics become more different from the background 522
CLASSIFICATION OF MOVING OBJECTS BASED ON SPECTRAL FEATURES
G(f)
0
50
100
G(f)
0
G(f)
(a)
150
200
250 f, Hz
100
150
(b)
50
100
G(f)
(c)
50
0
200
250 f, Hz
0
523
150
200
250 f, Hz
200
250 f, Hz
(d)
50
100
150
Fig. 1. SPD of whitened (based on the background parameters) seismic signals: from an airplane (a), a car (b), a person (c), and a background signal (d).
signal characteristics. Thus, the presence of a person (or several persons) changes the spectral-correlation properties of detected seismic signals. Different types of seismic objects have appreciably different specific multipath mechanisms of signal formation and, therefore, spectral characteristics. This is a potential feature for distinguishing the objects. In detecting seismically active objects and estimating the parameters and trajectories of their motion, it is proposed to perform adaptive decorrelation (whitening) of the background signal as a primary processing procedure [4–7]. The learning procedure applied in regions where only the background signal is present reduces to measurement of its correlation function. After that, prediction coefficients, which describe the linear recurrent mechanism of background generation from white noise, are determined. These coefficients are also responsible for the whitening mechanism, which forms white Gaussian noise in background regions. In regions containing a mixture of the valid signal and background, the valid signal component also subjected to whitening (but with the background model parameters) remains colored by the Gaussian process retaining individual specific features of the spectral-correlation characteristics typical for this or that type of seismically active objects. The measure of the spectral differences in seismic signals induced by the action of different objects can be seen from Fig. 1. Note that the spectral power densities (SPD) presented in Fig. 1 are random realizations experiencing random changes from one observation to another. The approach proposed in this paper is based on using the moment characteristics of the spectrum of the signal whitened on the basis of the background model parameters (for brevity, we will talk about a transformed signal in what follows) for object classification. Let Gi (f ) be the SPD of the transformed seismic signal corresponding to a seismically active object of the ith class. We assume that the SPD Gi (f ) is normalized to the total power, i.e., the following equality is satisfied: Z∞ Gi (f ) df = 1,
∀ i.
0
As, in addition, the SPD Gi (f ) is greater than or equal to zero, its exact presentation can be based on using the set of initial moments Z∞ xk =
f k Gi (f ) df,
k = 1, 2, . . . ,
0
OPTOELECTRONICS, INSTRUMENTATION AND DATA PROCESSING
Vol. 48
No. 5
2012
524
SOKOLOVA, SPEKTOR
or the combination of initial and central moments [13] Z∞ x k = (f − x1 )k Gi (f ) df.
_
0
With the final set of moments X = kx1 , x2 , . . . , xK kT ,
k = 1, K,
whose vector can serve as a feature for classification of seismically active objects, we obtain an approximate presentation of the SPD. In what follows, we consider a classification on the basis of a four-dimensional vector of features Y = ky1 , y2 , y3 , y4 kT whose elements are y1 = x1 (first-order initial moment of the SPD Gi (f )), y2 = x2 − .q 3 _ _ _ x21 = x 2 (second-order central moment), y3 = = x 3 x 2 (coefficient of skewness of the function Gi (f )), _
_2
and y4 = x 4 / x 2 − 3 (coefficient of kurtosis). Let us emphasize that the elements of the vector of features are random owing to the random character of SPD shown in Fig. 1. Thus, for instance, for a comparatively narrow SPD of the whitened signal generated by a person (Fig. 1c), the value of the mean frequency (feature y1 ) can change within rather wide limits. Because of the multifactor character of the seismic medium, it is reasonable to use the greatest possible number of the learning sets of signals obtained in available versatile observation conditions for constructing statistical reference signals. One important factor is the distance from the point of the seismic action to the seismic sensor. As this distance increases, the upper frequencies of the spectrum are attenuated, which alters the spectrum of the whitened signals. It should be noted, however, that the influence of the latter factor should not be overestimated: as the range of seismic observation in guard systems is usually not very large (15–30 m), the signals can be effectively analyzed at these distances. Though the spectral differences in the signals are manifested at such distances, they do not exert any crucial effect on the statistical behavior of the features. Let we have a set of I reference probability distribution densities (PDD) P (Y | Ai ), i = 1, I, of the vector feature Y for I classes Ai of objects to be classified. The statistical classifier operating on the maximum likelihood principle makes a decision in accordance with the rule A∗ = arg max P (Y | Ai ).
(1)
Ai
The maximum classification efficiency is reached if the decision is made after the end of the signal containing the seismic action of the object. Nevertheless, SGS belong to real-time systems where no delays in decision making are allowed. The objects should be classified during the time when the signals are fed to a computer responsible for decision making. In this case, a strategy of stage-by-stage refinement of the decision seems to be acceptable. This strategy can be realized by dividing the entire time of signal analysis into comparatively short observation intervals and making decisions at the end of each interval. In contrast to previous decisions, each new decision is based on a more complete data set, it is statistically more exact, and, therefore, neutralizes all previous decisions of the system. At the same time, the SGS operator has important information about all previous decisions, which not only reflects the fact of making this decision, but also statistically converges to a correct decision. Let m = 1, 2, . . . , M be the number of the local interval under analysis, M be the total number of intervals, and P (Ym | Ai ), m = 1, M , i = 1, I be the PDD of the vector of features Ym for the mth observation interval in the presence of an object of the type Ai . As the durations of individual observation intervals are appreciably greater than the signal correlation interval, the local values of the vectors Ym can be assumed to be independent, and the compatible distribution of probabilities for the current number Mcurr of intervals is determined by the expression
P (Ycurr | Ai ) =
MY curr
P (Ym | Ai ),
i = 1, I,
(2)
m=1
OPTOELECTRONICS, INSTRUMENTATION AND DATA PROCESSING
Vol. 48
No. 5
2012
CLASSIFICATION OF MOVING OBJECTS BASED ON SPECTRAL FEATURES
525
where Y0curr = kY01 , Y02 , . . . , Y0Mcurr k is the total (for the current set of intervals) vector of features composed from local vectors. Taking into account the accumulative character of formation of moment estimates, we use the Gaussian distribution of the four-dimensional vector Ym : P (Ym | Ai ) = (2π)−2 |Ki |−1/2 exp
n
−
o 1 ¯ i )0 K−1 (Ym − Y ¯ i) , (Ym − Y i 2
m = 1, M , i = 1, I,
(3)
¯ i , Ki , and |Ki | are the mathematical expectation, the covariation matrix, and its determinant of where Y the vector of features of objects of the ith class Ai . The use of the PDD of features (3) allows the use of simple procedures of adaptation and classification, which is an undoubted advantage of the developed method under the condition of a sufficiently high (sufficient for applications) efficiency of the synthesized classification algorithms. A more rigorous approach can be based on non-Gaussian models whose adaptation, however, requires in the learning process to form estimates of four-dimensional non-Gaussian distributions of the vector of features [14], which appreciably complicates the processing procedure. Solving these problems can be the subject of another study. ¯ i of the mean If Eq. (3) is used, the learning procedure reduces to obtaining estimates of the vectors Y values of features for all classes of objects Ai and their covariation matrices Ki . The classification procedure proper, performed on the basis of Eqs. (1)–(3), is easily implemented by the computing mechanisms of SGS. RESULTS OF THE ANALYSIS OF THE BAYESIAN CLASSIFIER ON THE BASIS OF SPECTRAL MOMENTS The learning procedure is performed on the basis of a classified observation with the known type of the object that induced the observed seismic signal. After adaptive whitening (based on the parameters of the mathematical model of the background signal), the signal used for learning is divided into intervals of the same length as the intervals in the SGS operation during object classification. For each of them, the (r) normalized spectrum Gi (f ) is determined; as previously, i is the number of the class of the object to which the SPD corresponds and r is the number of the interval in the signal used for learning. After that, the above-given formulas are used to calculate the necessary spectral moments and the components of the vector (r) of features Yi , r = 1, R, for all R learning intervals. The estimates of the mathematical expectation and covariation matrix are found in a usual manner as R X (r) ¯i = 1 Y Yi , R r=1
Ki =
R 1 X (r) ¯ (r) ¯ i )0 . (Yi − Yi )(Yi − Y R r=1
The classification efficiency was estimated experimentally with allowance for the seismic signal records obtained from seismic sensors with a frequency band of 120 Hz and digitized with the sampling frequency of 600 Hz. The duration of the local interval for signal analysis was 1000 realizations, which corresponds to 1.67 s of real time. The volume of the learning sample in our experiments was R = 50. As an example, Fig. 2 shows the equiprobability ellipses at the levels of 0.5 and 0.1 (the equiprobability ellipse is understood as an elliptical region into which points with random coordinates distributed in accordance with the normal law can fall with an identical probability [13]) for two-dimensional distributions of different paired combinations of the elements of the vector of features and different classes of objects. It is seen that the multidimensional distributions are characterized by weakly expressed differences of the “shift” type, which leads to overlapping of elliptical regions (sometimes this overlapping can be quite significant). At the same time, there are significant differences in the values of the correlation matrices, which are manifested in the sizes of the elliptical regions and in the positions of their axes. Statistical criteria guarantee that differences of all types are used, and their efficiency increases if it is possible to use multiple observations with accumulation of local results. In the proposed method, accumulation is achieved owing to dividing the total time of analysis and obtaining local results in individual intervals. The basic results of the experiments are summarized in Tables 1–5. Each table corresponds to the actual presence of the object of one of the classes (indicated in the table title). The data on the decisions in favor of various classes of objects indicated in the left column are given in the tables (as percentage of the total number of experiments). The total number of local cycles M used to make the decision varied from 1 to 15, which corresponded to the real time of analysis from 1.67 to 25.05 s. OPTOELECTRONICS, INSTRUMENTATION AND DATA PROCESSING
Vol. 48
No. 5
2012
526
SOKOLOVA, SPEKTOR
(b)
2
0
-2
(c)
2
Coefficient of skewness
4 Coefficient of skewness
Coefficient of kurtosis
(a)
1 0 -1
40 60 80 100 Mathematical expectation
40
80 100 60 Mathematical expectation
2 1 0 -1 -2
2 4 0 Coefficient of kurtosis
Fig. 2. Equiprobability ellipses for different paired combinations of features: person (solid curve), airplane (dotted curve), car (dashed curve), and background (dot-and-dashed curve).
Table 1. Results of classification for the actual object “person” Total number of local cycles M
Type of the alternative object 1
5
10
15
Person
0.367
0.721
0.735
0.824
Group of two persons
0.015
0
0.088
0.073
Group of three persons
0
0
0
0
Large animal
0.030
0.029
0.030
0.030
Small animal
0.411
0.176
0.147
0.073
Car
0.147
0.059
0
0
Airplane
0
0
0
0
Background
0.030
0.015
0
0
Table 2. Results of classification for the actual object “group of three persons” Total number of local cycles M
Type of the alternative object Person
1
5
10
15
0.014
0.056
0.092
0.083
Group of two persons
0.097
0.111
0.092
0.069
Group of three persons
0.194
0.264
0.403
0.500
Large animal
0.056
0
1
0.014
Small animal
0.347
0.361
0.264
0.208
Car
0.181
0.139
0.083
0.069
Airplane
0
0
0
0.028
Background
0.111
0.069
0.065
0.028
Group of persons
0.291
0.375
0.495
0.569
OPTOELECTRONICS, INSTRUMENTATION AND DATA PROCESSING
Vol. 48
No. 5
2012
CLASSIFICATION OF MOVING OBJECTS BASED ON SPECTRAL FEATURES
527
Table 3. Results of classification for the actual object “large animal” Total number of local cycles M
Type of the alternative object 1
5
10
15
Person
0.056
0.083
0.082
0.028
Group of two persons
0.250
0.361
0.250
0.278
Group of three persons
0.139
0.222
0.139
0.110
Large animal
0.139
0.167
0.418
0.528
Small animal
0.305
0.139
0.111
0.056
Car
0.083
0.028
0
0
Airplane
0
0
0
0
Background
0.028
0
0
0
Table 4. Results of classification for the actual object “small animal” Total number of local cycles M
Type of the alternative object 1
5
10
15
Person
0
0.059
0
0
Group of two persons
0
0
0
0
Group of three persons
0
0
0
0
Large animal
0
0
0
0
Small animal
0.588
0.824
0.882
0.823
Car
0.059
0
0
0
Airplane
0
0
0.059
0.059
Background
0.353
0.117
0.059
0.118
Table 5. Results of classification for the actual object “car” Total number of local cycles M
Type of the alternative object 1
5
10
15
Person
0.267
0.133
0.102
0
Group of two persons
0.067
0
0
0
Group of three persons
0.200
0.067
0.067
0
Large animal
0
0
0
0
Small animal
0.133
0.200
0.147
0.133
Car
0.200
0.533
0.617
0.867
Airplane
0
0
0
0
Background
0.133
0.067
0.067
0
OPTOELECTRONICS, INSTRUMENTATION AND DATA PROCESSING
Vol. 48
No. 5
2012
528
SOKOLOVA, SPEKTOR
If the task of determining the number of persons in a group is not posed in solving the classification problem, then the data for two classes of objects (group of two persons and group of three persons) are united, which is seen in the last row of Table 2. It is seen from the Tables that the accuracy of the decision made increases with increasing time of analysis. Thus, for instance, in the presence of a person, the number of correct decisions increases from 73 to 84% as the number of local observation cycles used for analysis increases from 5 to 15. As a whole, the results obtained testify to the possibility of using this method in seismic guard systems because the fraction of correct decisions satisfies the requirements actually imposed for such systems. CONCLUSIONS Even in few publications dealing with the seismic principle of guard systems, the least attention is paid to the problem of object classification. The approach proposed in this paper offers a possibility of real-time analysis of the seismic situation. The results obtained show that the use of spectral moments as features makes it possible to calculate the classification characteristics suitable for practice.
REFERENCES 1. R. G. Magauenov, Intrusion Protection Systems: Fundamentals of the Theory and Design Principles (Goryachaya Liniya – Telekom, Moscow, 2008) [in Russian]. 2. G. K. Chistova, Models and Methods of Processing of Seismic Signals in Recognition Systems (Izd. Penz. Gos. Univ., Penza, 2003) [in Russian]. 3. Yu. N. Kurochkin, V. E. Ivanov, A. V. Pryshchak, et al., “Vibroseismic Detection Device to be Used in Integrated Means of Physical Protection,” in: Problems of Object Protection (Izd. Penz. Gos. Univ., Penza, 2000), pp. 61–68 [in Russian]. 4. M. A. Raifeld and A. A. Spektor, “Nonparametric Method for Detecting Signals from Seismically Active Objects,” Avtometriya 41 (6), 88–97 (2005) [Optoelectr., Instrum. Data Process. 41 (6), 73–80 (2005)]. 5. A. L. Markhakshinov and A. A. Spektor, “Estimating Local Characteristics of Object Motion in a Seismic Monitoring System,” Avtometriya 45 (5), 48–53 (2009) [Optoelectr., Instrum. Data Process. 45 (5) 419–423 (2009)]. 6. A. A. Spektor and S. G. Filatova, “Estimating the time position of pulses in seismic observation systems based on Markovian filtering,” Avtometriya 44 (4), 68–74 (2008) [Optoelectr., Instrum. Data Process. 44 (4) 337–341 (2008)]. 7. A. A. Spektor and E. A. Tonkonogov, “Recurrent Estimation of Varying Parameters of Dynamic Systems,” Nauch. Vestnik NGTU, No. 1, 87–93 (2009). 8. G. Succi, G. Prado, R. Gampert, et al., “Problems in Seismic Detection and Tracking,” Proc. SPIE 4040, 165–173 (2000). 9. R. A. Gramann, M. B. Bennett, and T. D. O’Brien, “Vehicle and Personnel Detection Using Seismic Sensors,” Proc. SPIE 3577, 74–85 (1999). 10. L. I. Dvoiris and V. A. Gerashchenkov, “Formation of the Space of Features of Seismic Signals in the Frequency Range,” Radiotekhnika, No. 2, 90–92 (2010). 11. M. A. Mitrokhin and G. K. Chistova, “Algorithm of Classification of Equipment in Terms of the Seismic Signal,” in: Modern Monitoring Technologies and Means for Provision of Comprehensive Security of Objects Proc. VII All-Russian Conf., Izd. Penz. Gos. Univ., Penza, 2008, pp. 151–153. 12. Z. Liang, J. Wei, J. Zhao, et al., “The Statistical Meaning of Kurtosis and its New Application to Identification of Persons Based on Seismic Signals,” Sensors, No. 8, 5106–5119 (2008). 13. B. R. Levin, Theoretical Fundamentals of Statistical Radio Engineering (Sov. Radio, Moscow, 1969) [in Russian]. 14. Ya. A. Fomin and G. R. Tarlovskii, Statistical Theory of Pattern Recognition (Radio i Svyaz, Moscow, 1986) [in Russian].
OPTOELECTRONICS, INSTRUMENTATION AND DATA PROCESSING
Vol. 48
No. 5
2012
