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Performance of the tandem component filter for pattern recognition
Volume 65, number 5
OPTICS COMMUNICATIONS
l March 1988
P E R F O R M A N C E OF THE TANDEM C O M P O N E N T FILTER FOR PATTERN R E C O G N I T I O N K. CHALASINSKA-MACUKOW Ins#lute of Geophysics, Warsaw University, 02-093 Warsaw, Poland
and Henri H. ARSENAULT Laboratoire de recherches en optique et laser, Universit(, Laval Qu?bec, Canada, G IK 71"4 Received 22 July 1987
The performance of the tandem component filter for recogmzing one object from a set of objects is studied and compared with the performance of the classical matched spatial filter and the phase-only matched filter. The filler is found to have a performance comparable to the classical matched filler, wilh the advantage of a much higher light efficiency.
1. Introduction
Correlation methods play an important role in optical information processing and pattern recognition. For several applications, however, the light efficiency of the optical correlator is not sufficient [ 1,2]. The basic problems are the light losses by the holographic spatial filters, which are due to absorption and by diffraction into orders not used in the recognition process. Recent work has shown that this problem may be avoided by using one of several types of pure phase filters as matched spatial filters in optical correlator systems [ 3-6 ]. One type of phase-only filter (POF) was proposed by Horner and Gianino [3,4]. This type of filter uses only the phase information of the Fourier transform of the object for the correlation. An optical correlator with a pure phase correlation filter can have an optical efficiency of 100%. A comparison of phase-only filters with the classical matched spatial filter (MSF) using the criteria of discrimination, correlation peak value and optical efficiency [3] showed that the phase-only filter can discriminate between similar objects better than the classical matched filter. This is because of the domination of high spatial frequencies in the impulse response of the phase-only filter [7]. The high 334
frequencies make this filter more sensitive to scale and rotation changes as well as other intraclass variations [4]. These results initiated a series of studies analyzing the possibilities of more complicated phase-only filters, continuous as well as binary [ 8-13 ]. Another type of pure phase filter was proposed by Bartlet [5,6]. He proposed an optical subsystem consisting of two separated elements (tandem component) with a pure phase structure to modify a general wavefront. Such a tandem component allows a general wavefront modification with a theoretical efficiency of 100%, independently of the object function. One of the applications of the tandem component is a correlation-type measurement. With this component it is possible to transform an object function into an almost perfect peak in the output plane of the recognition system with 100% light efficiency. The tandem component filter (TCF) does not give an exact correlation similar to the one obtained with a conventional matched filter [14]. As in conventional matched filtering, all the information from the Fourier transform of an object is used in this recognition operation, but as opposed to a conventional correlator, this system is space-variant. Our goal was to study by computer the discrimi-
0 030-4018/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 65, number 5
OBJECT PLANE
OPTICS COMMUNICATIONS FOURIER TRANSFORM PLANE
OUTPUT PLANE
1 March 1988
can be multiplied by a phase function F~ (x, y) which is a solution to the equation ~ [ O ( x , y) F~ (x, y)] =A(u, v) exp[i~(u, v)] ,
Fig. I. Optical correlator employingthe tandem component filter composed of two pure phase fillers F~ and F2.
(1)
where ~ is the Fourier transform operator, and A (u, v) and O(u, v) are the amplitude and the phase of the Fourier transform of the product of the object function O(x, y) and of the phase-only function F~ (x, y); x, y and u, v are cartesian coordinates in the input plane and in the Fourier plane respectively. In solving eq. (1), the following condition must be satisfied
nation capability of the tandem component filter in the absence of noise, to compare the results with those obtained using the classical matched spatial filter and the phase-only filter, and to determine whether the tandem component filter has enhanced discrimination ability, like the phase-only filter.
A(u, v) =A= const(u, v) .
2. Tandem component filter
where ~b(x, y) is the phase distribution of the first phase-only element. The second phase-only element F2(u, v), whose purpose is to set the Fourier phase to a constant, takes the form
The tandem component filter is composed of two computer-generated phase-only elements. A single synthetic phase-only element is known as a kinoform [15]. There are many configurations of tandem component systems depending on the application [5]. The basic optical correlator employing a tandem component filter [ 10] is shown in fig. 1. It is a 4-f optical processing system with two phase-only elements F~ a n d / ~ located respectively in the object plane and in the Fourier plane. The first phase element F~ (x, y) placed in the object plane in contact with the object O(x, y) levels the Fourier amplitude of the object to a constant. The second phase element F2 (u, v) is placed in the Fourier transform plane and sets the phase of the Fourier transform to a constant. Therefore just after the second element both the amplitude and the phase of the Fourier transform of the object are constant and the wavefront is uniform. A Fourier transform of such a function will yield a perfect diffraction-limited peak in the output plane of the optical system shown in fig. 1. The main problem in the application of the tandem component to matched spatial filtering is the computer generation of two phase-only elements F~(x, y) and F2(u, v) satisfying given constraints. The tandem component filter has to be matched to the object O(x, y). To level the amplitude in the Fourier transform plane, the object function O(x, y)
(2)
The solution must be of the form
F~(x, y) = exp[i~(x, y)] ,
F2(u, v) = e x p [ - i O ( u , v)] .
(3)
(4)
In the output plane of the optical system with such a tandem component filter we obtain
R(xo, Yo)=~-~ {~[ O(x, y) F~(x,y)] F2(u, v)} , (5) which is equal to
R(xo, Yo)=[O(x,y) F,(x,y)]*Fz(x,y) ,
(6)
where Xo, Yo are cartesian coordinates in the output plane, and F2(x, y) is the Fourier transform of the second phase-only element F2(u, v) of the tandem component filter and the symbol • indicates a convolution. A tandem component filter composed of such phase-only elements matched to the object function O(x, y) yields a uniform wavefront after the second phase element and a perfect peak in the output plane of the system, but only if the input plane of the system contains the object O(x, y). For any other input function, the peak is degraded.
3. Digital analysis We now present, for different characters, a comparison of the recognition capability of a tandem 335
Volume 65, number 5
OPTICS COMMUNICATIONS
component filter with that of a conventional matched spatial filter and that of a phase-only filter.
3. I. Tandem component filter construction In the first step of this digital simulation, we generated a tandem component filter matched to the character A, composed of 64 × 64 pixels. Each phaseonly element was generated separately. The generation o f the element FI (x, y) is equivalent to solving a discrete version o f eq. (1) with the constraint A ( u, v) = const(u, v). A closed form solution cannot usually be found except for certain special cases [ 5]. This suggests the use of an iterative algorithm. The object-dependent phase elements can be calculated by means of several algorithms [ 16-19]. We chose a method which attempts to approach the o p t i m u m result by alternating between the object plane and the Fourier plane during each iteration, modifying the result after each iteration [ 10,17 ]. The iterative scheme used is represented in fig. 2. To begin the iteration, a uniform phase function is selected. Constraints are introduced in the Fourier plane and in the object plane. In the Fourier plane the amplitude A~(u, v) after each iteration is replaced by A,(u, v), where
A,(u,v)=l, A,(u,v)=l-e,
i f A ; ( u , u) ~< 1
ifA;(u,v)>l,
the value e = 0 . 9 was used to accelerate the convergence, and the index i indicates the iteration number. In the object plane, the amplitude a(x, y) of the
a{x'y)exp[ikOi(x'Y)]
~
I FFT
l March 1988
true object replaces the amplitude c(x, y) after each iteration. The convergence obtained by this method is shown in fig. 3. The error E is calculated after each iteration in the Fourier transform plane by the formula I/2
where A is the desired constant level of the amplitude and N z is the number of pixels. The tandem component filter matched to the letter A calculated by this method was obtained after five iterations.
3.2. Character recognition using the tandem component filter We used the tandem component filter matched to the letter A to recognize the reference character A among a set of tested characters A, X, D, O, U and Y, and compared the results with those obtained for the same set of characters using a conventional matched spatial filter and a phase-only filter. The results from this digital calculation are presented on figs. 4, 5, and 6 and on table 1. Figs. 4, 5 and 6 show the normalized output peaks for the different filters. All the filters are matched to the same character A. In the case of the matched spatial filter of fig. 4, the output fuction is o f course the autocorrelation function. For the phase-only filter of fig. 5, the output function is a quasi-autocorrelation function because this filter contains only phase information. In the case
---..-- A'i(u,v)exp[iq:)'
i (u,v)]
t c(x,y) = o (x,y) ~{x,y) = ~i(x,y)
c (x.y)exp[ikO' i (x,y)] ~
Ai(u,v) =1 or A i (u,v) =1 -f. cPi (u,v)=qb' i [u.v)
FFT -~ ~
Ai(u,v) exp [i qb i (u,v)]
Fig. 2. Diagram of the iteration scheme used to compute the first phase element FL of the tandem component filter. 336
Volume 65, number 5
OPTICS COMMUNICATIONS
I March 1988
E 2.10 -2,
{zI
0
1
E=
W
N2
iii ED D _J
10 -2
O_
<
I
1
I
L
I
I
2 3 4 5 NUMBER OF ITERATIONS
J
6
Fig. 3. Amplitude error calculated after each iteration in the Fourier plane. of the t a n d e m c o m p o n e n t filter of fig. 6, the output function is almost a delta function because of the u n i f o r m wavefront after the second phase element. The sharpness of the output peak is the major advantage of the t a n d e m c o m p o n e n t method. Table 1 gives the normalized output peak values
Fig. 4. Output signal tbr the conventional matched spatial filter.
Fig. 5. Output signal for the phase-onlyfilter.
Fig. 6. Output signal for the tandem component filler. 337
Volume 65, number 5
OPTICS COMMUNICATIONS
Table 1 Recognition of the character A with a matched spatial filter, a phase-onlyfilter and a tandem component filter. Object
A D O Y U X
Peak intensity MSF
POF
TCF
1.0 0.5 0.57 0.50 0.41 0.64
1.0 0.18 0.21 0.21 0.18 0.34
1.0 0.44 0.51 0.45 0.42 0.63
for all tested characters, using the three filters mentioned above designed for the letter A. The phase-only filter has the best d i s c r i m i n a t i o n ability. The performance of the t a n d e m c o m p o n e n t filter is almost identical to that of the c o n v e n t i o n a l matched spatial filter. This result is easy to u n d e r s t a n d when we rem e m b e r that the phase-only filter can be considered as a high-pass filter with enhanced d i s c r i m i n a t i o n ability. O n the other hand, the t a n d e m c o m p o n e n t filter's spatial frequency characteristic is more similar to that of a c o n v e n t i o n a l matched filter.
1 March 1988
a m o n g any set of objects whose spectra have identical phase distributions, which would be ambiguous to a phase-only filter. Also in the between-class disc r i m i n a t i o n problem, filters with enhanced discrimi n a t i o n capabilities might be too sensitive to small variations. In such cases, it might be c o n v e n i e n t to use a t a n d e m c o m p o n e n t filter with its high light efficiency and sharp output signal, instead of a conventional matched filter. The disadvantage of tandem c o m p o n e n t filters connected to the space-variant properties of the operation can be reduced by using a periodic filter F~(x, y) [5].
Acknowledgements We are grateful to Luc Leclerc for his assistance in the course of this work. This research was partially supported by research project CPBR 01.06, by the Natural Sciences and Engineering Research Council of Canada and by FCAR grants from Quebec.
References 4. Conclusion We have analyzed the ability of the t a n d e m comp o n e n t filter to recognize letters. The space-variant correlation operation involved yields a light efficiency of almost 100%, i n d e p e n d e n t l y of the object. A digital analysis has shown that the d i s c r i m i n a t i o n capability of the t a n d e m c o m p o n e n t filter is comparable to that of a c o n v e n t i o n a l matched spatial filter, which means that the filter does not belong to the class of filters with enhanced d i s c r i m i n a t i o n capability, like the phase-only filter. This is because both the c o n v e n t i o n a l matched spatial filter and the t a n d e m c o m p o n e n t filter use the whole Fourier spect r u m for the recognition process. This kind of filter can be useful for some recogn i t i o n problems, where complete i n f o r m a t i o n about the object is necessary. One example is recognition
338
[ 1] J.L. Homer, Appl. Optics 21 (1982) 4511. [2] H.J. Caulfield, Appl. Optics 21 (1982) 4391. [3] J.L. Horner and P.D. Gianino, Appl. Optics 23 (1984) 812. [4] P.D. Gianino and J.L. Horner, Opt. Eng. 23 (1984) 695. [5] H.O. Bartlet, Appl. Optics 23 (1984) 1499. [6] H.O. Bartlet, Appl. Optics 24 (1985) 3811. [ 7 ] K. Chalasinska-Macukowand T. Nitka, Proc. SPIE, Vol. 673 (1987), in press. [8 ] J.L. Horner and P.D. Gianino, Appl. Optics 24 ( 1985 ) 851. [9] J.L. Horner and H.O. Barlet, Appl. Optics 24 (1985) 2889. [ 10] H.O. Barlet and J.L. Horner, Appl. Optics 24 (1985) 2849. [ 11 ] J.L. Horner and J.L. Leger, Appl. Optics 24 (1985) 609. [ 12] D. Casasent and W.A. Rozzi, Appl. Optics 25 (1986) 3767. [ 13] R.R. Kallman, Appl. Optics 25 (1986) 4216. [ 14] A. Vander Lugt, Proc. IEEE, IT- 10 (1964) 139. [ 15 ] L.B. Lesem, P.M. Hirsch and J.A. Jordan, IBM J. Res. Dev. 13 (1969) 150. [16] P.M. Hirsch, J.A. Jordan and L.B. Lesem, US Patent 3.619.022, (1971). [ 17] J.R. Fienup, Appl. Optics 21 (1982) 2758. [ 18] N.C. Gallagher and B. Liu, Appl. Optics 12 (1973) 2328. [ 19] H. Akahori, Appl. Optics 12 (1973) 2336.
OPTICS COMMUNICATIONS
l March 1988
P E R F O R M A N C E OF THE TANDEM C O M P O N E N T FILTER FOR PATTERN R E C O G N I T I O N K. CHALASINSKA-MACUKOW Ins#lute of Geophysics, Warsaw University, 02-093 Warsaw, Poland
and Henri H. ARSENAULT Laboratoire de recherches en optique et laser, Universit(, Laval Qu?bec, Canada, G IK 71"4 Received 22 July 1987
The performance of the tandem component filter for recogmzing one object from a set of objects is studied and compared with the performance of the classical matched spatial filter and the phase-only matched filter. The filler is found to have a performance comparable to the classical matched filler, wilh the advantage of a much higher light efficiency.
1. Introduction
Correlation methods play an important role in optical information processing and pattern recognition. For several applications, however, the light efficiency of the optical correlator is not sufficient [ 1,2]. The basic problems are the light losses by the holographic spatial filters, which are due to absorption and by diffraction into orders not used in the recognition process. Recent work has shown that this problem may be avoided by using one of several types of pure phase filters as matched spatial filters in optical correlator systems [ 3-6 ]. One type of phase-only filter (POF) was proposed by Horner and Gianino [3,4]. This type of filter uses only the phase information of the Fourier transform of the object for the correlation. An optical correlator with a pure phase correlation filter can have an optical efficiency of 100%. A comparison of phase-only filters with the classical matched spatial filter (MSF) using the criteria of discrimination, correlation peak value and optical efficiency [3] showed that the phase-only filter can discriminate between similar objects better than the classical matched filter. This is because of the domination of high spatial frequencies in the impulse response of the phase-only filter [7]. The high 334
frequencies make this filter more sensitive to scale and rotation changes as well as other intraclass variations [4]. These results initiated a series of studies analyzing the possibilities of more complicated phase-only filters, continuous as well as binary [ 8-13 ]. Another type of pure phase filter was proposed by Bartlet [5,6]. He proposed an optical subsystem consisting of two separated elements (tandem component) with a pure phase structure to modify a general wavefront. Such a tandem component allows a general wavefront modification with a theoretical efficiency of 100%, independently of the object function. One of the applications of the tandem component is a correlation-type measurement. With this component it is possible to transform an object function into an almost perfect peak in the output plane of the recognition system with 100% light efficiency. The tandem component filter (TCF) does not give an exact correlation similar to the one obtained with a conventional matched filter [14]. As in conventional matched filtering, all the information from the Fourier transform of an object is used in this recognition operation, but as opposed to a conventional correlator, this system is space-variant. Our goal was to study by computer the discrimi-
0 030-4018/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 65, number 5
OBJECT PLANE
OPTICS COMMUNICATIONS FOURIER TRANSFORM PLANE
OUTPUT PLANE
1 March 1988
can be multiplied by a phase function F~ (x, y) which is a solution to the equation ~ [ O ( x , y) F~ (x, y)] =A(u, v) exp[i~(u, v)] ,
Fig. I. Optical correlator employingthe tandem component filter composed of two pure phase fillers F~ and F2.
(1)
where ~ is the Fourier transform operator, and A (u, v) and O(u, v) are the amplitude and the phase of the Fourier transform of the product of the object function O(x, y) and of the phase-only function F~ (x, y); x, y and u, v are cartesian coordinates in the input plane and in the Fourier plane respectively. In solving eq. (1), the following condition must be satisfied
nation capability of the tandem component filter in the absence of noise, to compare the results with those obtained using the classical matched spatial filter and the phase-only filter, and to determine whether the tandem component filter has enhanced discrimination ability, like the phase-only filter.
A(u, v) =A= const(u, v) .
2. Tandem component filter
where ~b(x, y) is the phase distribution of the first phase-only element. The second phase-only element F2(u, v), whose purpose is to set the Fourier phase to a constant, takes the form
The tandem component filter is composed of two computer-generated phase-only elements. A single synthetic phase-only element is known as a kinoform [15]. There are many configurations of tandem component systems depending on the application [5]. The basic optical correlator employing a tandem component filter [ 10] is shown in fig. 1. It is a 4-f optical processing system with two phase-only elements F~ a n d / ~ located respectively in the object plane and in the Fourier plane. The first phase element F~ (x, y) placed in the object plane in contact with the object O(x, y) levels the Fourier amplitude of the object to a constant. The second phase element F2 (u, v) is placed in the Fourier transform plane and sets the phase of the Fourier transform to a constant. Therefore just after the second element both the amplitude and the phase of the Fourier transform of the object are constant and the wavefront is uniform. A Fourier transform of such a function will yield a perfect diffraction-limited peak in the output plane of the optical system shown in fig. 1. The main problem in the application of the tandem component to matched spatial filtering is the computer generation of two phase-only elements F~(x, y) and F2(u, v) satisfying given constraints. The tandem component filter has to be matched to the object O(x, y). To level the amplitude in the Fourier transform plane, the object function O(x, y)
(2)
The solution must be of the form
F~(x, y) = exp[i~(x, y)] ,
F2(u, v) = e x p [ - i O ( u , v)] .
(3)
(4)
In the output plane of the optical system with such a tandem component filter we obtain
R(xo, Yo)=~-~ {~[ O(x, y) F~(x,y)] F2(u, v)} , (5) which is equal to
R(xo, Yo)=[O(x,y) F,(x,y)]*Fz(x,y) ,
(6)
where Xo, Yo are cartesian coordinates in the output plane, and F2(x, y) is the Fourier transform of the second phase-only element F2(u, v) of the tandem component filter and the symbol • indicates a convolution. A tandem component filter composed of such phase-only elements matched to the object function O(x, y) yields a uniform wavefront after the second phase element and a perfect peak in the output plane of the system, but only if the input plane of the system contains the object O(x, y). For any other input function, the peak is degraded.
3. Digital analysis We now present, for different characters, a comparison of the recognition capability of a tandem 335
Volume 65, number 5
OPTICS COMMUNICATIONS
component filter with that of a conventional matched spatial filter and that of a phase-only filter.
3. I. Tandem component filter construction In the first step of this digital simulation, we generated a tandem component filter matched to the character A, composed of 64 × 64 pixels. Each phaseonly element was generated separately. The generation o f the element FI (x, y) is equivalent to solving a discrete version o f eq. (1) with the constraint A ( u, v) = const(u, v). A closed form solution cannot usually be found except for certain special cases [ 5]. This suggests the use of an iterative algorithm. The object-dependent phase elements can be calculated by means of several algorithms [ 16-19]. We chose a method which attempts to approach the o p t i m u m result by alternating between the object plane and the Fourier plane during each iteration, modifying the result after each iteration [ 10,17 ]. The iterative scheme used is represented in fig. 2. To begin the iteration, a uniform phase function is selected. Constraints are introduced in the Fourier plane and in the object plane. In the Fourier plane the amplitude A~(u, v) after each iteration is replaced by A,(u, v), where
A,(u,v)=l, A,(u,v)=l-e,
i f A ; ( u , u) ~< 1
ifA;(u,v)>l,
the value e = 0 . 9 was used to accelerate the convergence, and the index i indicates the iteration number. In the object plane, the amplitude a(x, y) of the
a{x'y)exp[ikOi(x'Y)]
~
I FFT
l March 1988
true object replaces the amplitude c(x, y) after each iteration. The convergence obtained by this method is shown in fig. 3. The error E is calculated after each iteration in the Fourier transform plane by the formula I/2
where A is the desired constant level of the amplitude and N z is the number of pixels. The tandem component filter matched to the letter A calculated by this method was obtained after five iterations.
3.2. Character recognition using the tandem component filter We used the tandem component filter matched to the letter A to recognize the reference character A among a set of tested characters A, X, D, O, U and Y, and compared the results with those obtained for the same set of characters using a conventional matched spatial filter and a phase-only filter. The results from this digital calculation are presented on figs. 4, 5, and 6 and on table 1. Figs. 4, 5 and 6 show the normalized output peaks for the different filters. All the filters are matched to the same character A. In the case of the matched spatial filter of fig. 4, the output fuction is o f course the autocorrelation function. For the phase-only filter of fig. 5, the output function is a quasi-autocorrelation function because this filter contains only phase information. In the case
---..-- A'i(u,v)exp[iq:)'
i (u,v)]
t c(x,y) = o (x,y) ~{x,y) = ~i(x,y)
c (x.y)exp[ikO' i (x,y)] ~
Ai(u,v) =1 or A i (u,v) =1 -f. cPi (u,v)=qb' i [u.v)
FFT -~ ~
Ai(u,v) exp [i qb i (u,v)]
Fig. 2. Diagram of the iteration scheme used to compute the first phase element FL of the tandem component filter. 336
Volume 65, number 5
OPTICS COMMUNICATIONS
I March 1988
E 2.10 -2,
{zI
0
1
E=
W
N2
iii ED D _J
10 -2
O_
<
I
1
I
L
I
I
2 3 4 5 NUMBER OF ITERATIONS
J
6
Fig. 3. Amplitude error calculated after each iteration in the Fourier plane. of the t a n d e m c o m p o n e n t filter of fig. 6, the output function is almost a delta function because of the u n i f o r m wavefront after the second phase element. The sharpness of the output peak is the major advantage of the t a n d e m c o m p o n e n t method. Table 1 gives the normalized output peak values
Fig. 4. Output signal tbr the conventional matched spatial filter.
Fig. 5. Output signal for the phase-onlyfilter.
Fig. 6. Output signal for the tandem component filler. 337
Volume 65, number 5
OPTICS COMMUNICATIONS
Table 1 Recognition of the character A with a matched spatial filter, a phase-onlyfilter and a tandem component filter. Object
A D O Y U X
Peak intensity MSF
POF
TCF
1.0 0.5 0.57 0.50 0.41 0.64
1.0 0.18 0.21 0.21 0.18 0.34
1.0 0.44 0.51 0.45 0.42 0.63
for all tested characters, using the three filters mentioned above designed for the letter A. The phase-only filter has the best d i s c r i m i n a t i o n ability. The performance of the t a n d e m c o m p o n e n t filter is almost identical to that of the c o n v e n t i o n a l matched spatial filter. This result is easy to u n d e r s t a n d when we rem e m b e r that the phase-only filter can be considered as a high-pass filter with enhanced d i s c r i m i n a t i o n ability. O n the other hand, the t a n d e m c o m p o n e n t filter's spatial frequency characteristic is more similar to that of a c o n v e n t i o n a l matched filter.
1 March 1988
a m o n g any set of objects whose spectra have identical phase distributions, which would be ambiguous to a phase-only filter. Also in the between-class disc r i m i n a t i o n problem, filters with enhanced discrimi n a t i o n capabilities might be too sensitive to small variations. In such cases, it might be c o n v e n i e n t to use a t a n d e m c o m p o n e n t filter with its high light efficiency and sharp output signal, instead of a conventional matched filter. The disadvantage of tandem c o m p o n e n t filters connected to the space-variant properties of the operation can be reduced by using a periodic filter F~(x, y) [5].
Acknowledgements We are grateful to Luc Leclerc for his assistance in the course of this work. This research was partially supported by research project CPBR 01.06, by the Natural Sciences and Engineering Research Council of Canada and by FCAR grants from Quebec.
References 4. Conclusion We have analyzed the ability of the t a n d e m comp o n e n t filter to recognize letters. The space-variant correlation operation involved yields a light efficiency of almost 100%, i n d e p e n d e n t l y of the object. A digital analysis has shown that the d i s c r i m i n a t i o n capability of the t a n d e m c o m p o n e n t filter is comparable to that of a c o n v e n t i o n a l matched spatial filter, which means that the filter does not belong to the class of filters with enhanced d i s c r i m i n a t i o n capability, like the phase-only filter. This is because both the c o n v e n t i o n a l matched spatial filter and the t a n d e m c o m p o n e n t filter use the whole Fourier spect r u m for the recognition process. This kind of filter can be useful for some recogn i t i o n problems, where complete i n f o r m a t i o n about the object is necessary. One example is recognition
338
[ 1] J.L. Homer, Appl. Optics 21 (1982) 4511. [2] H.J. Caulfield, Appl. Optics 21 (1982) 4391. [3] J.L. Horner and P.D. Gianino, Appl. Optics 23 (1984) 812. [4] P.D. Gianino and J.L. Horner, Opt. Eng. 23 (1984) 695. [5] H.O. Bartlet, Appl. Optics 23 (1984) 1499. [6] H.O. Bartlet, Appl. Optics 24 (1985) 3811. [ 7 ] K. Chalasinska-Macukowand T. Nitka, Proc. SPIE, Vol. 673 (1987), in press. [8 ] J.L. Horner and P.D. Gianino, Appl. Optics 24 ( 1985 ) 851. [9] J.L. Horner and H.O. Barlet, Appl. Optics 24 (1985) 2889. [ 10] H.O. Barlet and J.L. Horner, Appl. Optics 24 (1985) 2849. [ 11 ] J.L. Horner and J.L. Leger, Appl. Optics 24 (1985) 609. [ 12] D. Casasent and W.A. Rozzi, Appl. Optics 25 (1986) 3767. [ 13] R.R. Kallman, Appl. Optics 25 (1986) 4216. [ 14] A. Vander Lugt, Proc. IEEE, IT- 10 (1964) 139. [ 15 ] L.B. Lesem, P.M. Hirsch and J.A. Jordan, IBM J. Res. Dev. 13 (1969) 150. [16] P.M. Hirsch, J.A. Jordan and L.B. Lesem, US Patent 3.619.022, (1971). [ 17] J.R. Fienup, Appl. Optics 21 (1982) 2758. [ 18] N.C. Gallagher and B. Liu, Appl. Optics 12 (1973) 2328. [ 19] H. Akahori, Appl. Optics 12 (1973) 2336.