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Optical multichannel correlators for high-speed targets detection and localization

Veacheslav Perju1, David Casasent2

1National Council for Accreditation and Attestation, 180, Stefan Mare Av.,

Chisinau, MD-2012, Republic of Moldova, Tel:(373)79431245; E-mail: [email protected]

2Carnegie Mellon University, Department of Electrical and Computer Engineering, Pittsburgh, PA 15213 USA, Tel: 412-268-2464, E-mail: [email protected]

ABSTRACT

In the article there are presented the mathematical and structural descriptions of the basic model of the optical correlator (BMOC), of the correlator using the matrixes of the lasers and filters (CMLF). In order to decrease the processing time in the correlators it is proposed to use the concept of the distribution of the operations of the targets detection and localization and to realize there in the different channels. At the stage of the targets detection it was proposed to use the filters generating the codified correlation functions consisting of a binary optical code which is analyzing in parallel with a high speed. There were elaborated new kinds of the correlators – with distributed targets detection and localization. There were given the analyses of the time expenditures and reliability in the different kinds of the correlators. Keywords: correlator, ditection, filter, localization, multichannel, optical, reliability, target

1. INTRODUCTION

In the signals and image analysis one of the basic operations represents the operation of correlation. In the optical correlator the function of correlation is calculated at a very high speed - of 1014 bytes/sec, which is compared with a super high computer’s capabilities. Unfortunately, the productivity of the optical correlator decrease at the targets detection and localization due to necessity to use a set of the filters. Also, the reliability of the targets detection can’t satisfy the standards at using of the filters which generate a single correlation maxima. The purpose of this article consists in the analysis of the basic model of the optical correlator, determination of the bottlenecks in the functioning, elaboration of the new structures of the optical correlators characterizing by enhanced processing time and reliability of the targets detection and localization. In section 2 there are presented the mathematical and structural descriptions of the basic model of the optical correlator. It is stipulated that a disadvantage of this correlator consists in the high processing time due to the necessity to input the different filters. In section 3 it is described the optical correlator based on the matrixes of the lasers and filters (CMLF) which permit to decrease the time of the targets detection.

Optical Pattern Recognition XXIII, edited by David P. Casasent, Tien-Hsin Chao, Proc. of SPIE Vol. 8398,83980C · © 2012 SPIE · CCC code: 0277-786X/12/$18 · doi: 10.1117/12.919142

Proc. of SPIE Vol. 8398 83980C-1

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It was observed that the disadvantage of the basic model of the optical correlator and the correlator CMLF consists in the necessity to spend high time at the stage of the optical correlation functions scanning at the targets detection and localization operations which are realized in one step. To avoid this problem there were elaborated the correlators in which the operations of the target detection and localization are realized in different channels (section 4). In section 5 it was given the analyses of the time expenditures in the different kinds of correlators. It was established that the correlator using matrixes of the lasers and filters with distributed detection and localization is characterized by a lowest processing time. In section 6 there were made the mathematical estimation and analyses of the targets detection and localization reliability. It was taken in to consideration that at the targets detection there are generated the correlation functions, which consists the binary optical codes. The probability of the correct and faulty targets detection was made at the occurrence of the errors of various orders.

2. THE BASIC MODEL OF THE OPTICAL CORRELATOR In signal and image analysis one of the basic operations represents the operation of correlation describing as: CRj(x,y) = P(x,y)*Hj(x,y), (1) where P(x,y) is the input signal (or image) and Hj(x,y) is the filter, j=1÷J. The operation of correlation can be described also as:

∞

CRj(ξ, η) = ∫∫ P(x,y)Hj*(x−ξ,y−η)dxdy, (2) −∞

or, using Fourier transformation: CRj(ξ,η)=F-1{F{P(x,y)}F*{Hj(x,y)}}=F-1{P(u,v)Hj*(u,v)}=F-1{Cj(u,v)} (3) where F and F-1 are the operations of the 2D Fourier transformation, direct and, respectively, inverse; the sign [*] is that of the complex conjugation; P(u,v), Hj(u,v) are the Fourier transformations of the functions P(x,y) and respectively, Hj(x,y); u, v – are the coordinates in the frequency space. The structure of the basic model of the optical correlator (BMOC) realizing formula (3) is presented on figure 1 [1]. The input image P(x,y) is placed on the Space Light Modulator 1 (SLM1) in the P1 focus plane of the Fourier Lens 1 (FL1). When the image is illuminated by a laser L, in the P2 focus plane of the lens FL1 it is formed the Fourier transformation of the function P(x,y): ∞

F{P(x,y)}=P(u,v)=∫∫P(x,y)exp[−i2π(xu+yv)]dxdy. (4) −∞

In the plane P2, on the SLM2 is introduced the filter Hj*(u,v). As a result, at the output of the plane P2 will be formed the product Cj(u,v)=P(u,v)Hj*(u,v). After the realization of the Fourier transformation of the function Cj(u,v) by the Fourier Lens FL2, in the P3 plane the optical distribution will be formed which will consists the correlation function CRj(ξ,η). This optical distribution will be scanned by the detector D and analyzed.



Figure 1. The structure of the basic model of the optical correlator

The time of the correlation functions calculation can be estimated by the formula: T1=TSLM1+ j(TSLM2 + TCO + TD), (5) where TSLM1, TSLM2 represent the time of the data input on SLM1 (the function P(x,y)) and on SLM2 (the function Hj*(u,v)) respectively; TCO – the time of the correlation operation performing; TD – the time of the optical correlation function scanning and analysis. If the initial image P(x,y) contains, for example, 106 pixels, and the time of the correlation function performing is determined by the light beam time passing from laser L to detector D and equal to TCO = 10-8 sec, the productivity of the processor at the correlation function calculation will be very high, equal to 1014

bytes/sec. Unfortunately, the productivity of the optical correlator BMOC decrease at the targets detection and localization due to necessity to use a set of the filters, to input the filters on the SLM2 (TSLM2 = 10ms), to scan the output optical correlation distribution by detector D (TD = 10ms). Also, the reliability of the targets detection can’t satisfy the standards at using of the filters which generate a single correlation maxima.

y

x

L

SLM1 SLM2

u v

D

ξ

η P1 P2

P3

FL1FL2



3. THE OPTICAL CORRELATOR BASED ON THE MATRIXES OF THE LASERS AND FILTERS To decrease the processing time by avoiding of the first factor from mentioned above can be used the optical correlator based on the matrixes of the lasers and filters (CMLF), presented in figure 2 [1].

Figure 2. Optical correlator based on the matrixes of the lasers and filters

The input image is fed to a modulator SLM1. The matrix of lasers ML represent the commutated matrix of the A1-GaAS diodes with radiation wave length 820nm, radiation capacity 5mW, width of radiation spectrum 4nm. The diode diameter is of 7 mm, emitter dimensions of 2x13μm, commutation time of 0.1mks. The matrix of filters MF is a set of the computer generated holograms calculated in advance or in real time mode and placed on a modulator SLM2. The advantage of this correlator consists in the significant reducing of the time at the filters input which in this case will be equal to the switching time Ts of a laser (Ts = 0.1mks). The processing time in the correlator CMLF can be estimated as: T2=TSLM1+d·TSLM2+j(Ts+TCO+TD) (6) were d=J/L; J is maximal number of the filters necessary to detect a target; L is the number of the lasers on the matrix ML.



4. THE CORRELATORS WITH SEPARATE CHANNELS FOR TARGETS’ DETECTION AND LOCALIZATION

The bottleneck of the correlators BMOC and CMLF consists in the necessity to spend high time at the stage of the optical correlation functions scanning by detector D at the targets detection and localization operations which are realized in one step. To avoid this problem we propose to divide the operations of the target detection and localization and to realize these operations in the different channels [2,3]. 4.1. The correlator CMLF with distributed functions of targets detection and localization To realize proposed approach there will be created two sets of the filters. The first set of the filters will be used for targets detection, and the second set – for target localization. The filters used for targets detection will generate the codified correlation functions, which will consists the binary optical code from a K “units” [1,2]: CRjd(x,y) = P(x,y)*Hjd(x,y). The filters response at the targets localization stage will consists of single correlation maxima, the coordinates’ of which will permit to localize the targets: CRjl(x,y) = P(x,y)*Hjl(x,y). The structure of the correlator CMLF will be modified in the next mode (Fig.3). The optical distribution after the lens FL2 is divided by the semitransparent mirror SM into two channels. In the first channel is realizing the operation of the targets detection with a high speed. For this purpose it is used the device for the operative correlation field analysis (DOCFA). This device consists of the matrix of the threshold optrons 1, optical collector 2, the line of the threshold detectors 3, the voting logical elements 4 and 5, the elements NOT 6 and OR 7. After the target will be detected, on the modulator SLM2 (Fig.2) will be input a filter for localization of the target, it will be switched on the laser from the matrix ML, which will permit to realize the operation of correlation between the input function P(x,y) and the filter Hjl

*(u,v) of the known target. This correlation function will consists a single correlation maxima which will be analyzed by detector D for determination of the correlation maxima coordinates which will permit to localize the target. The processing time will be described as (at TSLM1 = TSLM2): T21 = TSLM1 + d·TSLM2 + j(Ts + TCO + Td) + [TSLM2 + Ts + TCO + TD] = = TSLM (2+d) + j(Ts + TCO + Td) + (Ts + TCO + TD) (7) were Td is the time of the target detection, which is equal to the data time processing in the device DOCFA.



Figure 3. The channels for targets’ detection and localization. 4.2. The correlator BMOC with distributed detection and localization The structure of the correlator described in section 2, can be modified in the same way as was described in section 4.1 with the purpose to divide the stages of the target’s detection and localization and to decrease the processing time, to increase the reliability of the targets detection. In this case the filters used at the targets detection will be input on SLM2 (fig.1) consequently until the target will be detected by the device DOCFA (fig.3). After this on SLM2 will be input the filter for target localization. The processing time in this kind of the correlators will be estimated as: T11=TSLM1+j(TSLM2+TCO+Td)+(TSLM2+TCO+TD)=2TSLM1+j(TSLM2+TCO+Td)+(TCO+TD) (8)

5. THE ANALYSES OF THE TIME EXPENDITURES IN THE CORRELATORS There were calculated the processing time in the different types of the correlators using the formulas (5)-(8) for number of the filters j changed from 1 to 100, the time of the data input on SLM TSLM1 = TSLM2 = 10ms, the time of the correlation operation performing TCO = 10ns, the time of the optical correlation function scanning and analysis TD = 10ms, the number of the lasers in the matrix L=9, the switching time of a laser Ts = 0.1mks. The results of the calculations there are presented at the Fig.4 were T1 is the processing time in the correlator BMOC; T11 - in the correlator BMOC with distributed stages of the detection and localization of the targets (BMOC DDL); T2 – in the correlator CMLF; T21 – in the correlator CMLF with distributed detection and localization (CMLF DDL)

FL2

SM

DOCFAD

2 3 4

5

6

7

1



0,00

0,50

1,00

1,50

2,00

2,50

0 5 10 20 30 40 50 60 70 80 100

Nr Filters

Tim

e, s

ec

T1

T11

T2

T21

Figure 4. The processing time in the correlators

Also, there were calculated the time relations T1/T11, T2/T21, T1/T21 and T11/T21 presented on fig.5. The analyses of the data presented in fig.4 shows that the correlator using matrixes of the lasers and filters with distributed detection and localization is characterized by a lowest processing time changing from T21 = 0.04sec until 0.15sec at the changing of the number of the filters j from 1 to 100. The processing times in the correlator BMOC DDL - T11 and in the correlator CMLF - T2 are very closed and changed from 0.07sec to 1.13sec for the same diapason of the filters number j. The correlator BMOC is characterized by a highest processing time changing from T1=0.11sec up to 2.01sec. If the real time mode processing will be considered for the processing time less or equal to 0.5sec, this mode can be effectively realized in the correlator CMLF DDL - for j<=100, in the correlators CMLF and BMOC DDL - for j<=45, and in correlator BMOC - for j<=25. The proposed approach of the detection and localization operations distribution permit to decrease the processing time up to 2 times in the correlator BMOC DDL, and up to 8 times in the correlator CMLF DDL (Fig. 5). From the point of view of the processing time the correlator CMLF DDL is 7 times more effective than correlator BMOC DDL and 13 times than correlator BMOC.

6. THE RELIABILITY OF THE TARGETS DETECTION AND LOCALIZATION In the correlators BMOC and CMLF described in sections 2 and 3, there are used the filters which generate the correlation functions consisting of a single maxima. At the analysis of this maxima there are carried out the decisions regarding the targets detection and localization in one step. In the correlators BMOC DDL and CMLF DDL described in section 4, at the stage of the targets detection there are using the filters, which generate the correlation responses in the form of the optical binary codes, analyzed by the devices DOCFA.



0,00

2,00

4,00

6,00

8,00

10,00

12,00

14,00

16,00

0 5 10 20 30 40 50 60 70 80 100

Nr Filters

Tim

e R

elat

ions T1/T11

T2/T21T1/T21

T11/T21

Figure 5. The processing time relations Let’s appreciate the reliability of the targets detection in these two groups of the correlators. The general probability of the correct detection of a group of the targets in the correlators with a single correlation maxima (CSCM) can be described as the mathematical expectation of the probabilities Pi of the correct targets detection [4, 5]: M Ps = ΣaiPi = a1P1 + a2P2 + ….. + aMPM, (9) i=1

were iα is the a priory probability of the i-th target appearance; M is the number of the targets. In the correlators with codified correlation maxima (CCCM) there are generated the binary codes containing N digits, from which K - "units" and W=N-K - "zeroes". At the scanning of the optical codes and their transformation to electrical signals the errors are possible. Let consider the errors of the two types - first and second. At the error of the first type, in a binary code an "unit" will be detected as a "zero" or "zero" – as an "unit". At the error of the second type at one digit of the code an "unit" will be detected as a "zero" and simultaneously in other digit a "zero" - as an "unit". These errors may be of different order. In the optical correlators using devices DOCFA the errors of the first type will be detected and don’t influence on the detection reliability. The wrong targets’ detection can result only the errors of the second type. The probability of correct i-th target detection in the correlators CCCM at occurrence of the errors of various order k can be described as follows:

m Pci=1-Pcfi=1 - ∑ βikPcfik (10) k=1

where Pcfi - probability of the faulty detection of i-th target; m - maximum order of an error; βik - the probability of the second type errors occurrence; Pcfik - probability of faulty i-th target detection at the error of the order k.



Let A be the event, which consists in the fact that in a binary code an "unit" will be detected as a "zero" in one of the digits; B - event, concluded in the fact that in the same binary code a "zero" will be detected as an "unit " in other digit; C – the event, then in the binary code at one digit an "unit" will be detected as a "zero" and simultaneously in other digit a "zero" - as an "unit". It is obvious that the events A and B are independent. Under the theorem regarding the product of the probabilities of the independent events A and B we can determine the probability P(C) of the event C as simultaneous occurrence of events A and B. In this case P(C)=P(A)P(B). Let’s assum, that P(A) = P(B) = Pfi = (1-Pi), where Pfi, Pi are the probabilities of the faulty and correct i-th target detection at the forming of the single correlation maxima. Then the probability P(C) of occurrence of the event C is a probability of occurrence of an unitary error of the second type at the target detection, i.e. P(C)= βi1. The probability of occurrence of a k – order error will be equal to: βik=(1-Pi)2k. (11) Let’s define the probability Pcfik of the faulty detection of the i–th target as a result of occurrence of the errors of various order, taking into account the quantitative ratio in a code of "units" and "zeroes". Because on the wrong targets detection will result only errors of the second type, the probability of the faulty i-th target detection at occurrence of the errors of the second type of the order k will be defined as the relation of the non detected number of errors Qnk to the total number of possible errors Qtk: Pcfik =Qnk/Qtk (12) Let’s define the non detected number of errors Qnk of the second type of the order k. Let a binary code of the length N = 8 is describing as follows: Z=Z8,Z7,Z6,Z5,Z4,Z3,Z2,Z1. Let Zi.o and Zj,1 be the faulty detection of the signals in the digits i and j respectively (i.e. transitions of the signals 1→0 and 0→1). Let’s consider the number of "units" in the code is K=5, and number of “zeroes” W=3 and, for example Z=10110101. Then in this code the following combinations of the unitary errors of the second type are possible: Z8.0Z7.1;Z8.0Z4.1;Z8.0Z2.1;Z6.0Z7.1;Z6.0Z4.1; Z6.0Z2.1;Z5.0Z7.1;Z5.0Z4.1;Z5.0Z2.1;Z3.0Z7.1; Z3.0Z4.1;Z3.0Z2.1;Z1.0Z7.1;Z1.0Z4.1;Z1.0Z2.1 The number of such errors is equal to Qn1= C1

KC1W =15, where C1

K, C1J are the numbers of combinations

from K and J on one. In the similarly way it is possible to state, that Qn2=C2kC2

W; Qn3=C3kC3

W; Qn4=C4kC4

W etc. So, the number of the second type errors of the order k which will lead to the wrong target detection, can be defined as follows: Qnk=Ck

KCkW (13)

Taking into account the fact that the second type error of the order k is a consequence of the faulty detection of the signals simultaneously in n=2k digits, the total number of the errors can be defined as follows:



Qtk = CnN = C2k

N (14) On the basis of the expressions (13) and (14), the probability of the faulty target detection at occurrence of the errors of the order k will be defined as: Pcfik=Qnk/Qtk=Ck

KCkW/C2k

N. (15) The general probability of the faulty targets’ detection at occurrence of the errors of the second type of various order will be defined as: m m Pcfi=∑ βikPcfik = ∑{(1-Pi)2kCk

KCkW/C2k

N} (16) k=1 k=1

The probability of the correct target detection at occurrence of the errors of various order will be: m Pci = 1 – Pcfi= 1-∑{(1-Pi)2kCk

KCkW/C2k

N} (17) k=1

Then the expression for general probability of the correct targets detection in the correlators CCCM will be described as: M M m Pc= ∑αiPci = ∑{αi[1-∑{(1-Pi)2kCk

KCkW/C2k

N}]} (18) i=1 i=1 k=1

On the basis of expressions (9), (18) and (16) there have been calculated the probabilities of correct and faulty targets detection in the correlators with a single and codified correlation maxima at the following parameters: number of targets M=10, probability of the correct target detection Pi=0.95, number of “units” and “zeros” in the code K=W=4, second type error maximum order m = 4.

0

0,2

0,4

0,6

0,8

1

1,2

0,75 0,8 0,85 0,9 0,95

Ps

PcPfsPfc

Figure 6. The probabilities of targets detection: in the correlators CSCM - correct Ps

and faulty Pfs; in the correlators CCCM – correct Pc and faulty Pfc



The data presented in figure 6 show that the probability Pc of the correct targets detection in the correlators CCCM is higher and increase from the Pc = 0.962 until 0.998 at change of the Ps value from Ps = 0.75 to 0.95. The probabilities of the faulty targets detection in the correlators CCCM are smaller than in the correlators CSCM on 6.6 times at Ps = 0.75 and on 25 times at Ps = 0.95.

7. CONCLUSION The basic model of the optical correlator (BMOC) is characterized by a very high speed processor which can be compared with a super power computer system at the correlation function calculation stage. At the same time two factors influence negative on the optical correlator final productivity: necessity to input the filters and to scan the output optical correlation distribution at the stage of the targets detection and localization which are realized in one step. To decrease the processing time there were proposed new structures of the optical correlators, based on using of the matrixes of the lasers and filters, organizing of the separate channels for targets detection and localization and utilization of the filters with codified responses. The optical correlator based on the matrixes of the lasers and filters (CMLF) and correlator BMOC with distributed detection and localization operations (BMOC DDL) permit to increase (up to 2 times) the speed of the targets detection in comparison with a basic model of the optical correlator. Much more effective is the correlator CMLF with distributed detection and localization operations which permit to decrease the processing time until 13 times. It was established that the real time mode processing can be effectively realized in the correlator CMLF DDL at number of the filters j<=100, in the correlators CMLF and BMOC DDL for j<=45, and in the correlator BMOC for j<=25. The proposed approach of the operations of the detection and localization distribution permit to decrease the processing time up to 2 times in the correlator BMOC DDL and up to 8 times in the correlator CMLF DDL. From the point of the view of the processing time the correlator CMLF DDL is 7 times more effective than correlator BMOC DDL and on 13 times than correlator BMOC. An analytical estimation of targets detection reliability in the correlators has been performed, which has shown that the probability of faulty decisions in the systems with coded correlation responses is much lower (25 times lower) than in the systems with a single responses. The data presented in Figure 6 shows that the probability Pc of the correct targets detection in the correlators CCCM is higher and increase from the Pc = 0.962 until 0.998 at change of the Ps value from Ps = 0.75 to 0.95. The probabilities of the faulty targets detection in the correlators CCCM are smaller than in the CSCM correlators 6.6 times at Ps = 0.75 and 25 times - at Ps = 0.95.



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2. Katis G., Perju V., Rotari S. Methods and computing means for images processing. – Chisinau,

Stiinta, 1991.

3. Perju V. “Special-purpose optical-electronic computer systems, controlled by the images parameters” //In: Optical Pattern Recognition XI. - David P.Casasent, Tien-Hsin Chao, Eds. / Proc SPIE Vol.4043, pp.306-316 (2000).

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