Upper bound on false alarm rate for landmine detection and classification using syntactic pattern recognition

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1 Upper bound on false alar rate for landine detection and classification using syntactic pattern recognition Ahed O. Nasif, Brian L. Mark, Kenneth J. Hintz, and Nathalia Peixoto Dept. of Electrical and Coputer Engineering George Mason University Fairfax, VA 22030, U.S.A. ABSTRACT Recently, there has been considerable interest in the developent of robust, cost-effective and high perforance non-etallic landine detection systes using ground penetrating radar (GPR). Many of the available solutions try to discriinate landines fro clutter by extracting soe for of statistical or geoetrical inforation fro the raw GPR data, and oftenties, it is difficult to assess the perforance of such systes without perforing extensive field experients. In our approach, a landine is characterized by a binary-valued string corresponding to its ipedance discontinuity profile in the depth direction. This profile can be detected very quickly utilizing syntactic pattern recognition. Such an approach is expected to be very robust in ters of probability of detection (P d ) and low false alar rates (FAR), since it exploits the inner structure of a landine. In this paper, we develop a ethod to calculate an upper bound on the FAR, which is the probability of false alar per unit area. First, we paraeterize the nuber of possible ine patterns in ters of the nuber of ipedance discontinuities, dither and noise. Then, a cobinatorial enueration technique is used to quantify the nuber of adissible strings. The upper bound on FAR is given as the ratio of an upper bound on the nuber of possible ine pattern strings to the nuber of adissible strings per unit area. The nuerical results show that the upper bound is saller than the FAR reported in the literature for a wide range of paraeter choices. Keywords: Landine detection, Ground penetrating radar, False alar rate 1. INTRODUCTION There is a continued interest in the developent of a robust, cost-effective and high perforance landine detection and classification syste. A variety of approaches have been discussed in the research literature. 1, 2 Most approaches ake a distinction between two ajor classes of landines based on their etal content. There are those which contain a sufficient aount of etal to be detected by electroagnetic induction (EMI) detectors and those which are essentially non-etallic. For non-etallic landines, the ost coon sensor used for detection is ground penetrating radar (GPR) 3, 4 and the ost coon detection ethodology is anoaly detection, soeties colloquially called blob detection. This ethodology is typically ipleented utilizing a pre-screener followed by a ore detailed analysis of the GPR data for detection leading to a two-pass operational approach. Iproveents to the anoaly detector have recently been ade which fuse EMI and GPR data as well as take into account contextual inforation by alternative ethods of analyzing the GPR data before selecting the algorith appropriate to the soil type and environent. Anoaly detection typically depends on coputing running statistics which characterize the clutter background and select for further analysis those regions whose statistics differ fro the background statistics by soe threshold value. An alternative approach to anoaly detection and its variants is the application of syntactic pattern recognition (SPR) 5. In SPR, single coluns of range data fro the GPR (A-Scan) are processed so as to convert the signal into a binary-valued string of the sae length as the original data. This binary-valued string is coprised of zeros except for where there are changes in ipedance. The binary-valued string is then presented to a finite state achine (FSM) language recognizer for detection. This is a single-pass operation which does not require a pre-screener. It is also extreely fast in execution since the FSM operates

2 at one clock cycle per state transition, naely one clock cycle per range bin, independent of the nuber of states in the FSM. The binarization of the A-scan approach was chosen because it is well known that the radar cross section (RCS) easured by a radar looking at a coplex target is highly variable with aspect angle. We have the distinct advantage of being able to assue that ost landines are oriented approxiately parallel to the earth yielding a repeatable and recognizable pattern. While this slight variation in either the orientation of the landine or the relative geoetry between the GPR and the landine contributes to aplitude variations in the easured RCS with range, it has been deonstrated that the locations of ipedance discontinuities can be reliably detected through preprocessing followed by a binarization procedure. That is, the binary-valued string representing the location of ipedance discontinuities of a prototypical buried landine vary only a slight aount which is well within the capabilities of being recognized as a language by an appropriately designed FSM. The key etrics of interest for a landine detection syste are: (1) probability of detection (P d ), and (2) false alar rate (FAR). The FAR is defined as the probability of false alar per unit area, which for landine detection, is a ore eaningful perforance easure than siply the probability of false alar. Given that we have a suitable ethod for the preprocessing and production of the binary-valued A-scan, it reains to recognize the string associated with a landine and discriinate it fro other strings produced by clutter and noise. We call the process of producing the binary-valued A-scans a binarization schee. It is a fundaental requireent of language recognizers that they not only recognize all the words in the language but no others. In the coputation of the false alar rate (FAR) of an SPR language recognizer, one needs to know two nubers. The first is the nuber of strings that represent the language of the landine. While it ay see that one can have a language consisting of a single string, operationally this is not possible since the statistical variations in the A-scan of a landine present theselves as sall oveents of ipedance discontinuities about their noinal locations. These variations are sall and enuerable and it is not difficult to copute the nuber of strings representing variations on a landine s characteristic string and this coputation is discussed in Section 3. The second nuber required for FAR estiation is the total nuber of all adissible strings. Adissible strings are those which can be produced by the binarization process. It ay first appear that for a landine string of length n, that there would be 2 n possible strings produced by the binarization process, soe of which would be as a result of the landine and soe as a result of clutter. A siple exaple shows that this is not true. If we assue that the binarization process cannot produce adjacent ipedance discontinuities, then strings of the for 0110 are not possible. This liitation, aong others, eans that the nuber of adissible strings is less than the 2 n possible strings in a string of length n. It is for this reason that a ethodology based on cobinatorial enueration has been developed in this paper, naely to be able to copute the nuber of adissible strings as a first attept to put a bound on the FAR of SPR as well as provide guidance in the developent of the binarization process. We note that in the SPR approach, such a bound can serve as a rough estiate of the expected perforance in ters of FAR, whereas in ost traditional approaches one needs to perfor detailed field tests to evaluate the syste s FAR. The reainder of the paper is organized as follows. In Section 2, we characterize the binary sequence space due to an arbitrary binarization schee using a cobinatorial enueration technique. In particular, we copute the total nuber of adissible strings for a fixed string length. In Section 3, we copute an upper bound on the FAR. Section 4 presents soe nuerical results showing the expected FAR perforance of the SPR approach. We then use the results fro Section 2 and contrast the, in Section 5, against statistics of binary strings obtained fro real landine data. In Section 6 we discuss the significance of the experiental and theoretical results to our future work in detecting landines with the syntactic pattern recognition ethod. 2. CHARACTERIZATION OF THE BINARY SEQUENCE SPACE Suppose we are given a rando sequence, {X i R:1 i n}, wheren denotes the length of the sequence. The sequence {X i } could represent, for exaple, a sequence of easureents coprising an A-scan obtained

3 using GPR to detect the presence of landines. It could also represent a sequence which is generated as a result of perforing soe processing on the raw data of a GPR A-scan. To perfor further processing of the easureents using the SPR approach, the rando sequence {X i } is converted into a binary sequence {B i {0, 1}}. There are various ways to convert a given sequence {x i } to {b i }. As entioned before, we refer to such a conversion process as binarization. For exaple, we can have the following siple binarization rule: b 0 0, (1) { 1, if xi >x b i = i 1 and x i >x i+1, (2) 0, otherwise, for i 1. In words, b i is set to one if x i is a local axiu with respect to x i 1 and x i+1.otherwise,b i is set to 0. Another schee ay be to consider all local extrea of {x i }. Obviously, we can have ore sophisticated binarization schees. In this section, we study the characteristics of the deterinistic properties of the binary sequence space that results fro application of an arbitrary binarization rule, subject to the constraint that no consecutive 1 s are allowed. The sequence {b i } ay be viewed as a binary string with certain properties. Let B denote the set of binary strings that could possibly correspond to a sequence {b i } obtained by applying a binarization rule to a finite or infinite sequence of values {x i }.Inotherwords, {b i } B, for all binary sequences {b i } generated by the binarization ethod. Furtherore, any sequence b B corresponds to a valid binary sequence {b i } obtained fro the binarization process. A given sequence v Bay be characterized as a binary string that begins and ends with 0 and does not contain the string 11. We refer to such a sequence as an adissible string. We can copactly represent the set B by the regular expression (see Appendix A) B =(00 1) 00. (3) Let Φ(x) denote the generating function of the set B with respect to sequence length (see Appendix B). In particular, define the weight function w(σ) length of σ, σ B. Then Φ(x) = σ B x w(σ). Fro (3), we can obtain the following expression for Φ(x): Φ(x) = Let B n denote the set of strings of length n that belong to B: [ 1 x ] 1 1 x x x 1 x = x 1 x x 2. (4) B n {b B: b = n}. (5) The following proposition gives a closed-for expression for the nuber of adissible strings of length n in B. A proof is given in Appendix C. We will follow the convention that lowercase letters represent deterinistic variables, whereas uppercase letters represent rando variables.

4 Proposition 2.1 (Adissible strings). The nuber of adissible strings of length n in B is given by B n = n 1 j= n 2 ( ) j. (6) n j 1 Hence, the fraction of binary strings of length n that belong to the set B is given by f n = B n 2 n =2 n n 1 j= n 2 ( ) j. (7) n j 1 Fig.1showsaplotoff n vs. n for n = 1 to 20. One sees that f n decreases rapidly as a function of n. In particular, f , which eans that out of the 2 20 binary strings of length 20, less than 0.65% of the are in the space B. Notethatf 2 = 1 2 and f 2 3 = It ay also be of interest to consider the nuber of 1 s that occur in a string belonging to the set B. Let w 1 (σ) denote the nuber of 1 s in string σ B. To address this issue, we consider a bivariate generating function Φ(x, y), where the coefficient of ter x n y represents the nuber of strings of length n, containing 1 s. In particular, define the weight function w 1 (σ) nuber of 1 s in σ, σ B. and Φ(x, y) σ B x w(σ) y w1(σ). (8) Fro (3), we can then obtain the following expression for Φ(x, y): Φ(x, y) = [ 1 xy ] 1 1 x x x 1 x = x 1 x(1 + xy). (9) Let B n, denote the set of strings in B of length n and containing 1 s. The nuber of strings in the set B n, is given by following proposition. A proof is given in Appendix D. Proposition 2.2 (Adissible strings with 1 s). The nuber of adissible strings of length n containing 1 s in B is given by ( n 1 B n, = ), 0 n 2 1. (10) Let f n, denote the fraction of strings of length n in the set B, containing 1 s. Fro (10) and (6), we obtain f n, = B ( n 1 ) n, n = ), 0 1. (11) B n 2 n 1 j= n 2 ( j n j 1 Fig. 2 shows a bar graph of f n, vs. when n = 20. Note that the distribution has a ode at =5and has a relatively syetric shape about this ode.

5 3. UPPER BOUND ON FALSE ALARM RATE As entioned in Section 1, due to statistical variations it ay not be possible to have an unique binary string corresponding to a particular landine. However, we can introduce soe paraeters that capture statistical uncertainties. We assue that all valid ine patterns have a known prefix 100. We denote the locations of the 1 s in a valid ine pattern of length n by a vector q. For exaple, if q =(q(1), q(2),, q(k),, q(k)) = (1, 10,, 79,, L), where K < L n, iteansthat the location of the k th 1 of the ine pattern is 79. In the SPR approach, it is possible to represent a particular landine by q and its variation according to soe paraeterization. We propose to characterize the binary-valued string corresponding to a landine pattern (of length n) using the following three paraeters: (i) the total nuber of 1 s in the binary string sequence (K, the length of q), (ii) dither or by how uch the eleents of q ay get shifted (d), and (iii) the nuber of noise pulses (J), that are not contained in q but ay appear erroneously. A ine pattern with dither value d is characterized by a set Q d = {q ± 1, q ± 2,, q ± d}. By a noise pulse we ean the erroneous change of a 0 bit to 1. We assue that a noise pulse does not occur in any of the positions contained in q, Q d,the top of the ine, and at ine prefix ( 100 ) locations. The nuber of valid ine patterns of length n using the paraeters (K, d, J) is upper bounded by N (u) ine =(2d +1)K J ( ) n (2d +1)K 3. (12) j j=0 Note that the (2d + 1) refers to the nuber of variations in the string pattern caused by each 1 present in the ine string, and the 3 present in the top arguent of the choose function denotes the length of the ine prefix. Clearly, it is possible to have soe other criteria or paraeters to select the valid ine patterns. But as long as we can evaluate, either analytically or epirically, an upper bound on the nuber of such valid ine patterns, N (u) ine, our approach of coputing an upper bound on FAR would be valid. Assuing that each string is equally likely to occur, an upper bound on the probability of false alar is given by the ratio of the nuber of valid ine pattern strings to the nuber of adissible strings, P (u) FA = N (u) ine B n. Suppose the footprint area of the GPR corresponding to each A-scan is denoted by D in units of 2. Since the FAR is defined as the probability of false alar per unit area we have the following upper bound on FAR: FAR = P (u) FA D = N (u) ine B n D. (13) This upper bound on FAR can serve as a theoretical perforance easure of the SPR approach for landine detection. It can also guide us towards an appropriate binarization schee such that acceptable FAR values can be achieved. 4. NUMERICAL EXAMPLES In this section, we provide two nuerical exaples to illustrate the dependence of the ine pattern paraeters on the FAR upper bound. We consider the footprint of a typical GPR, which has area D = and a ine prefix of length 3. In Fig. 3, we plot the upper bound on FAR as a function of string length (n), and the nuber of 1 s in the ine pattern (), for different values of noise (J =0, 1, 2) and a dither value of d = 2. The surface plot shows that for a fixed string length n, as the nuber of 1 s in the ine pattern increases, the FAR also increases. This is because the nuerator of the bound (N (u) ine )increaseswith increasing, while the denoinator ( B n ) reains unchanged (cf. (6),(12), and (13)). On the other hand, for fixed values of, the FAR decreases as the string length n increases. This is because B n increases faster than N (u) ine as a function of n, causing the bound to decrease. As expected, the bound uniforly increases with increasing noise. In Fig. 4, the bound is plotted as a function of for fixed values dither and noise, (d, J) =(2, 2), with various values of n. We observe that there is roughly a two orders of agnitude decrease in the bound with every increase in the string length of 10 bits.

6 5. COMPARISON OF EXPERIMENTAL AND THEORETICAL STRING DISTRIBUTIONS Once the theoretical approach has been established and argued for, one interesting question reains: to what extent do the experiental results atch the nuerical data? The iplications of such a question for the SPR ethod, or for any ethod applied to the identification of a pattern, are crucial: atching distributions ay guarantee low false alar rate, and that guarantee is one ain goal of binarization algoriths. We have used ultiple ethods for binarization of raw and processed data. As representative saples, here we inspect results obtained with two independent binarization schees, labeled schees b1 and b2. Every dataset is exained for strings of length n [20, 140], and for strings containing 1 s, with [2, 13]. For each cobination of (n, ), entire sets of data were surveyed, containing approxiately 133 illion datapoints (i.e., pixels). For b1, over 2 illion strings are found to coply with the (n, ) ranges. For b2, over 20 illion strings are retrieved, indicating a first difference between the two schees. When graphing these results, the dichotoy is ore obvious: the peaks in b2 (see Fig. 6) are absent in b1 (see Fig. 5). Furtherore, in b1 the position of the single peak on the length of the string (n), as if reiniscent of a Rayleigh distribution, is directly proportional to the nuber of 1 s in the string (). While this is intuitively acceptable, the spiky distributions of b2 are not. As stated above, the task of coparing theoretical to experiental results would not be coplete without a side-by-side plot. Aong the any possibilities of realizing such exercise, we have chosen to fix n =80 (strings of length 80 pixels) and vary (nuber of 1 s), siilarly to the histogra presented in Fig. 2. Figs. 7 and 8 show the resulting distributions of the experiental (blue) datasets for binarization schees b1 and b2, respectively. Both are copared with the theoretical counterpart (green) (cf. (11)). The experiental data clearly has a different ean copared to the theoretical one, and this is not surprising since the theoretical distribution assues a unifor distribution of all adissible strings. Such plots can be extreely useful in extracting the ine pattern strings and also assessing the perforance of different binarization schees. 6. CONCLUSION In this paper, we have studied the perforance of the SPR approach to landine detection and classification in ters of the FAR. In particular, an upper bound on the FAR is proposed, where the bound is given as a fraction of the possible ine pattern strings with respect to the total nuber of adissible strings that can result fro an arbitrary binarization schee. It is deonstrated that the tool of cobinatorial enueration is very useful in studying the string statistics for syntactic landine detection. In this initial study we have ade the siplifying assuption that all strings are equally likely, which ay not be the case in real landine data. Therefore, it would be of great interest to relax this assuption in the adissible set, and derive a new upper bound on the FAR. It is also of interest to incorporate the ine pattern paraeters in our string enueration technique. These developents can potentially guide us to a ethod of extracting landine string patterns that have very attractive detection and false alar perforances. ACKNOWLEDGMENTS The authors would like to thank the Office of Naval Research for supporting this work under Grant No. N The authors would also like to express their gratitude to the Night Vision and Electronic Sensors Directorate (NVESD), Ft. Belvoir, VA, for providing sensor data.

7 APPENDIX A. NOTATION FOR REGULAR EXPRESSIONS If a = 0101 and b = 1110 are binary strings, then ab is the string fored by concatenating a and b: If A and B are sets of binary strings ab = AB {ab : a A, b B}, is the set of strings fored by concatenating any string in A with any string in B. We also define A 2 AA, and in general, A n AA }{{ A }. n ties Let ɛ denote the epty string. By convention, we define A 0 {ɛ} as the set consisting of just the epty string. Next we define A ɛ A A 2 A 3 = i 0 A i. For brevity of notation, we define 0 {0} and 1 {1}. It is iportant to note that AB is not, in general, equivalent to the Cartesian product A B. However, if each of the eleents of AB is uniquely generated, i.e., if we can deterine uniquely which part of the eleent cae fro A and which part fro B, thenab is in fact equivalent to A B. In this case, the Product Lea applies to generating functions defined on the set AB (see Appendix B). APPENDIX B. GENERATING FUNCTIONS A generating function of a real-valued sequence {a i } is a foral power series 6 Φ(x) = n 0 a i x i. (14) We will use [x i ]Φ(x) to denote the coefficient of x i in Φ(x). Hence, if Φ(x) is given by (14), a i =[x i ]Φ(x). Given a set S, we associated with each eleent a nonnegative integer w(σ) called the weight of σ. Let us define the generating function on S with respect to the weight function w( ) as follows: Φ S (x) = σ S x w(σ). (15) If we write Φ S (x) in the for (14), i.e., Φ S (x) = i 0 a i x i, we see that a i = nuber of eleents of S of weight i. Two useful rules for anipulating generating functions defined on sets according to (15) are the so-called Su and Product Leas, which are straightforward to prove. Lea B.1 (Su Lea). Suppose S = A B and A B =. Then Φ S (x) =Φ A (x)+φ B (x).

8 Lea B.2 (Product Lea). Suppose S = A B and for each σ =(a, b) S, w(σ) =w(a)+w(b). Then Φ S (x) =Φ A (x)φ B (x). As an exaple, let us represent the set of all binary strings using the following regular expression (see Appendix A): S = A, where A = {0, 1}. Letw(σ) denote the length of a binary string σ A. The generating function of the set {0} with respect to w( ) is siply x. Likewise, the generating function of {1} is x. Therefore, the generating function of A with respect to w( ) is given by 2x, i.e., Φ A (x) =2x. BytheProduct Lea, the generating function of A i,fori 0, is given by Φ A i(x) =(2x) i.bythesu Lea, Φ S (x) = i 0 Φ A i(x) = i 0 Therefore, the nuber of binary strings of length n is given by [x n ]Φ S (x) =2 n, (2x) i = 1 1 2x. as we would expect. Though this exaple is rather trivial, it illustrates how the su and product leas can be used to derive an expression for the generating function, and finally, how the coefficient [x n ]Φ S (x) can be obtained. The results (4) and (6) are obtained using this approach. APPENDIX C. PROOF OF PROPOSITION ON ADMISSIBLE STRINGS The nuber of adissible strings of length n in B is given by B n =[x n ]Φ(x) =[x n x ] 1 x(1 + x) =[xn ] x[x(1 + x)] j (16) j 0 = [x n j 1 ](1 + x) j = ( ) n 1 j ( ) j =. (17) n j 1 n j 1 j 0 j 0 j= n 2 APPENDIX D. PROOF OF PROPOSITION ON ADMISSIBLE STRINGS CONTAINING M 1 S The nuber of adissible strings of length n containing 1 s in B is given by B n, =[x n y ]Φ(x, y) =[x n y x ] (18) 1 xy(1 + x) =[y ] [x n ]x[x(1 + xy)] j =[y ] j 0 j 0[x n j 1 ](1 + xy) j (19) =[y ] ( ) ( ) j n 1 n y n j 1 =, 0 1. (20) n j 1 2 j 0 This result can also be derived using a ore direct arguent.

9 REFERENCES [1] D. J. Daniels, A review of landine detection using GPR, Proc. European Radar Conf. (EuRAD 08), pp , Oct [2] J. N. Wilson, P. Gader, L. Wen-Hsiung, H. Frigui, and K. C. Ho, A Large-Scale Systeatic Evaluation of Algoriths Using Ground-Penetrating Radar for Landine Detection and Discriination, IEEE Trans. Geosci. Reote Sens. 45, pp , Aug [3] H.M.Jol,Ground Penetrating Radar Theory and Applications, Elsevier, [4] D. J. Daniels, Ground Penetrating Radar, MPG Books Ltd., reprint of 2004 edition, UK, [5] K. J. Hintz, N. Peixoto, and D. Hwang, Syntactic landine detection and classification, in Detection and Sensing of Mines, Explosive Objects, and Obscured Targets XIV, R. S. Haron, J. T. Broach, J. H. Holloway, Jr., ed., Proc. SPIE 7303, May [6] I. P. Goulden and D. M. Jackson, Cobinatorial Enueration, Dover Publications, reprint of 1983 edition, 2004.

10 Figure 1: Fraction of strings, f n, in the binary sequence space B, as a function of string length n. Figure 2: Fraction, f n, of strings of length n = 20 containing ones in the binary sequence space B, asa function of. dither=2, ine prefix length= noise=0 noise=1 noise=2 FAR upper bound (per 2 ) string length, n Figure 3: FAR upper bound as a function of string length and the nuber of 1 s in the ine pattern, (n, )

11 10 2 dither=2, noise=2, ine prefix length= FAR upper bound (per 2 ) n= n=80 n=90 n= Figure 4: FAR upper bound as a function of the nuber of 1 s in the ine pattern,, for different string lengths. 2 x 10 5 nuber of strings Figure 5: String distribution due to binarization schee b1. n 4 x 10 5 Nuber of strings Figure 6: String distribution due to binarization schee b2. n

12 theoretical binarization schee b1 Fraction of strings of lenght Figure 7: Fraction of strings of length 80 using binarization schee b theoretical binarization schee b2 0.2 Fraction of strings of lenght Figure 8: Fraction of strings of length 80 using binarization schee b2.