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Influence of filter and signal types in optical matched filtering
Influence of filter and signal types in optical matched filtering N. DEMOLI The dependence of the output correlation signal on the filter parameters, filter types, and signal types in matched filtering is discussed. On the basis of the established filter criteria--the signal-to-noise ratio, peak-to-sidelobe ratio, and diffraction efficiency--a new measure is introduced giving an overall estimate of the feasibility of filters. According to that single measure, various filter types are compared and selected by numerical experiments. KEYWORDS: matched spatial filters, optical correlators
Introduction The classical matched filter (CMF) ~ provides the optimal signal-to-noise ratio in detecting a known signal in additive noise. To utilize this powerful tool more efficiently, many different types of matched spatial filters (MSFs) have been proposed. The most promising have proved to be the large family of phase-only filters (POFs) and inverse matched filters (IMFs) 2. However, to implement these in optical correlation problems one must use numerical experiments 3 or addressable spatial light modulators 4. Recently, the extended optical correlator (EOC) device 5, the quasi-phase-only 6-8 and the quasi-inverse 9 matched filtering techniques have been introduced as the experimental solutions for the corresponding concepts. New filter types are usually compared with the C M F to enhance one or a few favourable characteristics. The main performance criteria: the signal-to-noise ratio (SNR), peak-to-sidelobe ratio (PSR), diffraction efficiency (DE), and discrimination ability (DA), are the most often used measures to describe filters. It is not possible to obtain the maximum values for all measures at the same time and with a single filter type. The well-known trade-offs between SNR, PSR and DE are clearly given in Ref. 3. Briefly: to maximize the SNR we should use CMFs, to maximize the PSR we should use IMFs, while to maximize the DE we should use POFs. Which one is the optimal filter? It has been concluded 3, without proving it, that POFs may provide a good compromise between SNR and PSR while maximizing the light efficiency. On the The author is in the Institute of Physics of the University, PO Box 304, 41000 Zagreb, Croatia, Received 14 June 1993. Revised 8 September 1993.
other hand, some fundamental limitations of using POFs have been pointed oud °. The problem of filter type optimality has been also discussed l~ in terms of discrimination capability. In this paper, the influence of the filter parameters, filter types, and signal types on the output correlation signal is investigated by numerical experiments. A basic set of filter types is defined regarding the amplitude and phase information content. Then, on the basis of the SNR, PSR and DE, an overall estimate of the feasibility of filters is introduced. According to that single measure, called the feasibility estimate (FE), various filter types are compared and selected.
Preliminaries The optical matched filtering we are dealing with here, is based on correlating two complex signals by means of optical multiplication in the Fourier domain. We denote the coordinates of the input, filter, and ouput planes of such optical system with xi = (xi, y~), u = (u, v), and x0 = ()Co,Y0), respectively. We denote the input signal with f(xi), and the reference signal with h (xi). Upper case letters will be used to represent the Fourier transforms of spatial functions, while ~{.} will denote the Fourier transform operator. The reference spectrum H(u) can be described as a two-dimensional complex-valued function
n ( u ) = In(u)lexp{iO(u)}
(1)
having the amplitude [H(u)[ and the phase O(u). We shall refer to [H(u)[ as the reference amplitude, and to O(u) as the reference phase. The relative intensity of the correlation signal Ic(xo) is given by
Ic(xo) = I~{F(u).~(u)}l z
(2)
0030-3992/94/020119-08 © 1994 Butterworth-Heinemann Ltd Optics & Laser Technology Vol 26 No 2 1994
119
Filter and signal types in optical matched filtering: N. Demoli where o f ( u ) is the filter transfer function. Since filters in optical matched filtering systems are regularly passive elements, Iof(u)l <~ 1 holds. The o f ( u ) is related to the complex conjugated reference spectrum via the relation of(.) = z {/4.(.)}
(3)
where Z {.} is the filter operator. Z acts as a filter type selector as well as the preprocessor of the reference signal data. Numerous variations for the filter operator can be expected. For example, taking Z to be the unit operator, the classical MSF is simulated. The POF is obtained when g is a phase extractor; the edge enhancement is achieved when Z is a low-frequency cutter, etc. The approach in producing QPOFs has been 6 to level all maxima of the reference spectrum, which is attained by an attenuation mask placed in the first Fourier transform plane 5 of the EOC system. Thus, keeping the phase information unchanged and the beam balance ratio equal to unity over the spectrum maxima, the high-efficiency and the high-discrimination-capability filters have been synthesized. However, the original idea underlying this technique was in producing filters having the transfer function.
1, IH(u)l > H0} of(u) =
0, IH(u)l <~Ho exp{-tO(u)}
(4)
where/4o is the constant. The filter operator is now the phase extractor and the binary low amplitude cutter. The level H0 should be chosen such as to preserve the spectrum energy (like POFs do), while keeping some of the main characteristics of the reference amplitude (such as the amplitude minima). It will be shown later that the filter parameter H0 is of great importance when dealing with signals that have a real and positive Fourier transform. The next step is to determine a basic set of MSFs according to the amplitude and phase information content.
Basic set of matched filter types According to the well-known importance ~2 of the phase information of signals, the phase part of the filter should be: • matched to the phase of the reference spectrum; or • binary matched to the phase of the reference spectrum. Otherwise, for example, by setting the phase information equal to unity, or inversely matched to the phase of the reference spectrum, the resulting matched filtering would be very poor. Although the amplitude part of the filter could be arbitrarily designed, we consider the special cases in which the filter amplitude is: • matched to the amplitude of the reference spectrum; • equal to unity; • binary matched to the amplitude of the reference spectrum; • modified according to the QPOF model; or • inversely matched to the amplitude of the reference spectrum.
120
Combining all proposed amplitude and phase filter information contents, we define the basic set of matched filter types, see Appendix A. Roughly speaking, the main filter groups are: • CMFs (APF and ABPF); • POFs (POF, BPOF, BHAPF, BHABPF, M A P F and MABPF); and • IMFs (IAPF, IABPF, BLAPF and BLABPF). In most applications, for example when the reference signal has no symmetry, the filters within one group will show similar behaviour. However, in some other cases the output correlation results will be significantly different.
Performance quantities The signal-to-noise ratio (SNR) is defined assuming that the power spectral density of the noise is equal to unity SNR =
IFI 2
(5)
F
where <> indicates the ensemble average over the filter area F. The peak-to-sidelobe ratio (PSR) is defined as the ratio of the correlation peak intensity and the average intensity around the peak
-
PSR
Ic. . . .
(6)
c
where Ic,max is the correlation peak intensity. The denominator indicates the average value over the correlation area C in all points except I c.... • The diffraction efficiency (DE) of the filter is commonly ~3 defined as DE=
F F
(7)
The discrimination ability (DA) is defined as the difference between the auto-correlation and cross-correlation peak intensities, as a fraction of the auto-correlation peak intensity DA
-
Ia ....
-
Ic ....
(8)
a,max
where Ia.... and I c. . . . are the auto-correlation and cross-correlation peak intensities, respectively.
Feasibility estimate We consider the ouput correlation signal to be entirely described by the ability of the matched filter to tolerate input noise, by the sharpness of the correlation peak, and by the matched filter efficiency. These three measures (SNR, PSR and DE), as seen from the definitions (5)-(7), are mutually independent. On the other hand, the ability of the matched filter to discriminate various inputs (or the DA), described by (8), provides information pertaining to the sharpness Optics & Laser Technology Vol 26 No 2 1994
Filter and signal types in optical matched filtering: N. Demoli difference of various correlation signals (see Ref. 14), i.e. DA = 1 (I'~(xo))cPSRc (9)
(I'~(Xo))cPSRA
where PSRc is the PSR corresponding to the cross-correlation output and PSR A is that corresponding to the auto-correlation output. Evidently, the higher PSR sensitivity means the better Ler DA. It is convenient to normalize the main performance quantities and calculate in decibels (dB), i.e. Q~(da) = 10 log~o(Q,/Qo,)
(10)
where Q~ (i = 1, 2, 3) are the SNR, PSR, and DE quantities and Q0~ (i = 1, 2, 3) are the normalization71 values. Thus, the main quantities now describe eachh filter in comparison with the reference one. We introduce a new parameter, the feasibility estimate FE FE(dB) = ~ SNR(dB) + flPSR(dB) + ~,DE(dB)
(11)
which gives an overall estimate of the feasibility of filters. According to (11), an optimal filter ought to possess various properties in such a balance to ensure the highest FE. However, the feasibility estimate may include more terms than shown on the right-hand side of (11). Also, the coefficients ~, fl and 7 could be arbitrarily determined. We choose ~t = fl = 7 -- 1 for all our calculations, thus giving equal weights to SNR, PSR and DE measures.
b
Computer experiments We use numerical methods to investigate the correlation dependence on the filter type, the filter parameter, and the signal used. The aim of these experiments is also to demonstrate the usefulness of the FE quantity. The training set of real signals is shown in Fig. 1, where (a) represents a general signal; (b) a symmetrical signal; and (c) a symmetrical signal with the real and positive Fourier spectrum. The 30 x 30 pixel images are placed in a 64 x 64 array and zero padded. A 64 × 64 point twodimensional F F T routine is used to simulate the desired filter type and to compute the SNR, PSR, DE and FE values. The SNR, PSR and DE are normalized (set to unity for the APF) and calculated in decibels. The filter parameter influence on the auto-correlation signal for the BHAPF, MAPF, IAPF and BLAPF is shown in Fig. 2. From Fig. 2(a) we can see that quantities describing the BHAPF are equal to those describing the POF for the low threshold parameter H0. Increasing H0, the PSR and DE slowly decrease while the SNR firstly increases and then decreases. Since the FE has a shallow maximum, it defines the representative parameter value H 0-- 0.01 (relative units) for describing BHAPFs. The characteristic curves for the M A P F result in a broad maximum in the FE from z equal to 1 to 1000 (relative units), see Fig. 2(b). MAPFs, formerly known as QPOFs, have Optics & Laser Technology Vol 26 No 2 1994
Fig. 1 Images used in numerical experiments. (a) 'F104'; (b) "FORT'; (c) 'PYR'
been introduced 7 using the numerical model that well describes the experimental conditions for obtaining phase-only filters. Therefore, the choice z = 60 is determined by comparing the SNR = - 5 . 2 dB and PSR = 12.9 dB values with the corresponding POF values: SNR = - 6 . 7 dB and PSR = 12.7 dB. From Fig. 2(c) we can see that by increasing the threshold parameter H1 of the IAPF, the PSR decreases while the DE and the SNR increase. For HI = 1 (relative units) all performance quantities saturate to the POF values. We choose H~--0.05 as the representative value for describing IAPFs since, at that value, the FE remains high enough, while the other quantities considerably differ from the corresponding POF values. Figure 2(d) shows that quantities describing BLAPFs saturate to those describing the POF for the threshold parameter H2 = 1 (relative units). We choose H2 = 0.1 to characterize BLAPFs. Comparing the behaviour of the quantity values near the selected parameters, it is obvious that IAPFs and BLAPFs belong to the same group of filters. We made calculations (not shown here) for other general-type
1 21
Filter and signal types in optical.matched filtering: N. Demoli 4o.~
n-t "1o
a
i
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.... .
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___ SNR .... PSR . . . . . . DE __FE - ~ . 0 ~
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H0 (rel. units) 4o.oo
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, , , , .
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SNR PSR DE
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,,,,,,,i
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,,,,,,,
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(rel. units)
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Fig. 2 Filter auto-correlation quantities as a function of filter parameters for 'F104' image. (a) BHAPF; (b) M A P F ; (c) IAPF; (d) BLAPF
signals such as a number, a character, etc. The calculations confirmed that our choice has been satisfactory. We have also calculated quantities for T a b l e 1. Filter type APF ABPF
122
the binary-phase versions of the upper described filters. The resulting curves (not shown here) showed very similar behaviour to those presented, but also
Filter a u t o - c o r r e l a t i o n DE (dB)
q u a n t i t i e s f o r t h e 'F104' i m a g e
SNR (dB)
PSR (dB)
FE (dB)
0.00 0.00
0.00 -1.86
0.00 -1.89
0.00 -3.75
POF BPOF BHAPF BHABPF MAPF MABPF
5.64 5.64 5.63 5.63 -1.75 -1.75
-6.71 -9.86 -5.69 -8.83 -5.20 -8.54
12.68 8.95 12.00 8.37 12.89 8.93
11.61 4.73 11.94 5.17 5.94 -1.36
IAPF lAB PF BLAPF BLAB PF
-2.32 -2.32 -2.18 -2.18
-9.59 - 13.17 -10.27 -14.02
20.84 14.26 19.07 13.05
8.93 - 1.23 6.62 -3.15
Optics & Laser Technology Vol 26 No 2 1 9 9 4
Filter and signal types in optical matched fihering: N. Demoli Table 2.
Filter auto-correlation quantities for the 'FORT" image
Filter type
DE (dB)
APF, ABPF
-2.59
-2.68
5.43
0.16
POF, BPOF BHAPF, BHABPF MAPF, MABPF
5.64 5.64 -0.10
-7.66 -6.69 -6.98
15.03 14.70 15.81
13.01 13.65 8.73
IAPF, IABPF BLAPF, BLABPF
2.59 -2.55
-11.61 -13.22
22.05 18.39
7.85 2.58
Table 3.
SNR (dB)
DE (dB)
APF, ABPF POF, BPOF BHAPF, BHABPF MAPF, MABPF
SNR (dB)
FE (dB)
2.26
-12.80
-7.09
5.64 5.64 -7.46
-24.05 -2.71 -9.96
-7.96 -8.76 -7.42
-26.37 -5.83 -24.84
-11.46 -1 5.53
-34.75 -38.61
-1.52 -1.34
-47.73 -55.48
The quantities that describe the auto-correlations using the signal shown in Fig. l(a) were calculated as a function of the filter type and listed in Table 1. Note the feasibility estimate for those filters. For example, the POF shows 11.6 dB gain in FE relative to the APF. We can see that the best FE possesses the BHAPF (11.9 dB). The IAPF also shows the high FE (8.9 dB). The results (not shown here) performed with other general type signals were similar to those presented in Table 1. The typical auto-correlation quantities for the class of symmetrical signals, such as is shown in Fig. l(b), are presented in Table 2. Since the Fourier transform of an even and real signal is itself even and real, the binaryphase versions of filters and the corresponding filters possess equal quantity values. Again the BHAPF (BHABPF) have the best FE (13.7 dB). The auto-correlation and cross-correlation quantity results for the signal of Fig. l(c) are shown in Tables 3 and 4, respectively. We also provide the three-dimensional plots of correlation outputs, see Fig. 3. Obviously, in the case of the symmetrical
signal with its Fourier transform positive and real, only the BHAPF (BHABPF) and the CMFs are the useful ones. Only they can successfully discriminate different inputs. Note that the discrimination ability was unable to calculate for the MAPF (MABPF) and the IMF group of filters.
S u m m a r y and concl usi ons Numerous filter design variations can be found in the optical correlation literature. We have defined the basic set of matched filter types in accordance with the importance of the amplitude and phase information contents of signals. It is considered that the phase part of the filter is allowed to be matched or binary matched to the phase of the reference signal, while the amplitude part could be designed arbitrarily. A mathematical formulation for each filter type has been reviewed and its performance without noise has been quantified. The results presented in this paper are strongly dependent on the filter parameters, filter types and signal types. To describe filters, and taking into consideration the major filter performance criteria, we have introduced an overall single measure, the feasibility estimate, defined as a sum of logarithmic relative values of SNR, PSR and DE (in decibels). The classical filter characteristics served for the reference values.
Filter cross-correlation quantities for the 'PYR" and 'F104' signals
Filter type
DE (dB)
APF, ABPF
-4.48
POF, BPOF BHAPF, BHABPF MAPF, MABPF IAPF, IABPF BLAPF, BLABPF Optics & LaserTechnology Vol 26 No 2 1994
PSR (dB)
3.45
5-10 dB lower values in SNR, PSR and FE, while the DEs remain unchanged. Thus, by defining the representative parameter values the basic set of filters was also defined.
Table 4.
FE (dB)
Filter auto-correlation quantities for the "PYR' image
Filter type
IAPF, IABPF BLAPF, BLABPF
PSR (dB)
SNR (dB)
PSR (dB)
FE (dB)
DA (%)
-5.27
-10.10
-19.85
82.2
5.64 1.84 -4.59
-24.05 -6.20 -8.51
-5.59 -5.46 -5.37
-24.00 -9.82 -18.46
0.0 47.2 --
3.92 3.85
-27.91 -28.92
-5.54 -5.71
-29.53 -30.78
---
1 23
Filter and signal types in optical matched filtering: N. Demoli We have shown the dependence of the correlation results on the filter parameters. Then, by choosing the representative parameters for various filter types, the basic set of filter types has been defined. The optimality of the filter types has been discussed in relation to the signal types. Generally,
according to the FE measure, the POF group represents optimal filters, in particular BHAPFs (also called theoretical QPOFs), which preserve the spectrum energy (like POFs do) and keep some characteristics of the reference amplitude (POFs do not possess such an attribute)• For the class of
1
a
b
%
%
d Fig. 3 The correlations of 'PYR" as a reference function with"PYR" (left-hand side, the autocorrelation case) and 'F104' (right-hand side, cross-correlation case). (a) APF; (b) POF; (c) BHAPF; (d) MAPF
124
Optics & Laser Technology Vol 26 No 2 1994
Filter and signal types in optical matched filtering: N. Demoli
t
61
"
tg
I Fig. 3 (continued) The correlations of "PYR" as a reference function with 'PYR' (left-hand side, t h e auto-correlation case) and 'F104' ( r i g h t - h a n d side, cross-correlation case). (e) IAPF; (f) BLAPF.
symmetrical signals, the binary-phase versions of filters and the corresponding filters possess equal quantity values. BHAPFs and POFs show very similar behaviour in most applications. However, the superior performance of BHAPFs was demonstrated in the case of symmetrical signals with the Fourier transform real and positive. In such cases, only BHAPFs (BHABPFs) and CMFs are able to discriminate the input signal. Although not addressed in these simulations, BHAPFs should be analysed for more complex applications.
Acknowledgements This work was supported by the Ministry for Science of the Republic of Croatia. The author would like to thank the reviewers for their helpful comments.
• Phase-Only Filter (POF) ZPOF= exp{ -- i0 (u) } •
{ •BPOF = •
•
{1,Im.)l> 0, In(u)l ~ / % 3
Binary-High-Amplitude Binary-Phase Filter (BHABPF) ~BHABPF ~---
The basic set of matched spatial filters • Amplitude Phase Filter (APF) or CMF
• Amplitude Binary-Phase Filter (ABPF)
os0u,00}
[H(u)lmax -- , cos O(u) <
0, IH(u)l ~
• Modified-Amplitude Phase Filter (MAPF) or experimental QPOF
IH(u)] exp{-i0(u)} ZAPF--iH(u)lma x
Optics & Laser Technology Vol 26 No 2 1994
1' c°s 0(u) ~>00} - - l , COS 0 ( U ) <
Binary-High-Amplitude Phase Filter (BHAPF) or theoretical QPOF ~(BHAPF
Appendix A
J(ABPF =
Binary-Phase-Only Filter (BPOF)
ZMAPF
•
--
IH(u)l "--i0(u)} 1 -t--z2--~u)l 4exp/
Modified-Amplitude Binary-Phase Filter (MABPF) IH(u), 5 1, cos0(u)>~00} ZMABPF= 1 + CIH(u)I 4 [ - - 1, cos O(u) <
125
Filter and signal types in optical matched filtering: N. Demoli •
Inverse-Amplitude Phase Filter (IAPF) or amplitude-compensated matched filtert5 XIAPF =
t
-IH(u)l - ,
L 1, •
[H(u)[
H1
exp{ -i0(u)}
IH(u)l ~
Inverse-Amplitude Binary-Phase Filter (IABPF) Z,ABeF =
t
1, cos O(u) >~0"~
IH----~' IH(u)l > H,
(
1,
IH(u)l ~< H i
- 1, cos O(u) < OJ
• Binary-Low-Amplitude Phase Filter (BLAPF) {0, [H(u)] > H2}exp{_iO(u) } ZBLAPZ= 1, IH(u)I ~
Binary-Low-Amplitude Binary-Phase Filter (BLABPF) XBLABPF=
1,
IH(u)l ~
-- 1, cos
O(u) <
References 1 Vander Lugt, A. Signal detection by complex spatial filtering, IEEE Trans Inf Theory, ITI0 (1964) 139-145
2 Pernick, B.J. Phase-only and binary phase-only spatial filters for optical correlators: a survey, Opt Laser Technol, 23 (1991) 273-282 3 Vijaya Kumar, B.V.K., Hassebrook, L. Performance measures for correlation filters, Appl Opt, 29 (1990) 2997-3006 4 Chalasinska-Macukow, K., Goreeki, C. Optoelectronic implementation of the quasi-phase correlator, Opt Commun, 93 (1992) 11-18 5 Demoli, N., Vuki~evi~, D. Discrimination sensitivity of the extended optical correlator, Opt Commun, 64 (1987) 417-420 6 Demoli, N. Quasiphase-only matched filtering, Appl Opt, 26 (1987) 2058-2061 7 Demoli, N. Properties of quasi phase-only matched spatial filters, Opt Eng, 31 (1992) 275-279 8 Janowska-Dmoch, B., Chalasinska-Macukow, K., Piliszek, J. Matched filters and quasiphase-only filters recorded in silver halide (sensitized) gelatin, Opt Laser Technol, 24 (1992) 279~84 9 Demoli, N. Quasi phase-only filters as an inverse matched spatial filter, Optik, 91 (1992) 11 15 10 Wunsch II, D.C., Marks II, R.J., Caudeil, T.P., Capps, C.D. Limitations of a class of binary phase-only filters, Appl Opt, 31 (1992) 5681 5687 11 Yaroslavsky, L.P. Is the phase-only filter and its modifications optimal in terms of the discrimination capability in pattern recognition? Appl Opt, 31 (1992) 1677-1679 12 Oppenheim, A.V., Lira, J.S. The importance of phase in signals. Proc IEEE, 69 (1981) 529-541 13 Demoli, N. Optical power efficiency in coherent optical correlater systems, Opt and Quant Electron, 21 (1989) 179 182 14 Zheng, S.H., Karim, M.A., Gao, M.L. Amplitude-modulated phase-only filter performance in presence of noise, Opt Commun, 89 (1992) 296-305 15 Mu, G., Wang, X., Wang, Z. Amplitude-compensated matched filtering, Appl Opt, 27 (1988) 3461-3463
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Optics & Laser Technology Vol 26 No 2 1994
Introduction The classical matched filter (CMF) ~ provides the optimal signal-to-noise ratio in detecting a known signal in additive noise. To utilize this powerful tool more efficiently, many different types of matched spatial filters (MSFs) have been proposed. The most promising have proved to be the large family of phase-only filters (POFs) and inverse matched filters (IMFs) 2. However, to implement these in optical correlation problems one must use numerical experiments 3 or addressable spatial light modulators 4. Recently, the extended optical correlator (EOC) device 5, the quasi-phase-only 6-8 and the quasi-inverse 9 matched filtering techniques have been introduced as the experimental solutions for the corresponding concepts. New filter types are usually compared with the C M F to enhance one or a few favourable characteristics. The main performance criteria: the signal-to-noise ratio (SNR), peak-to-sidelobe ratio (PSR), diffraction efficiency (DE), and discrimination ability (DA), are the most often used measures to describe filters. It is not possible to obtain the maximum values for all measures at the same time and with a single filter type. The well-known trade-offs between SNR, PSR and DE are clearly given in Ref. 3. Briefly: to maximize the SNR we should use CMFs, to maximize the PSR we should use IMFs, while to maximize the DE we should use POFs. Which one is the optimal filter? It has been concluded 3, without proving it, that POFs may provide a good compromise between SNR and PSR while maximizing the light efficiency. On the The author is in the Institute of Physics of the University, PO Box 304, 41000 Zagreb, Croatia, Received 14 June 1993. Revised 8 September 1993.
other hand, some fundamental limitations of using POFs have been pointed oud °. The problem of filter type optimality has been also discussed l~ in terms of discrimination capability. In this paper, the influence of the filter parameters, filter types, and signal types on the output correlation signal is investigated by numerical experiments. A basic set of filter types is defined regarding the amplitude and phase information content. Then, on the basis of the SNR, PSR and DE, an overall estimate of the feasibility of filters is introduced. According to that single measure, called the feasibility estimate (FE), various filter types are compared and selected.
Preliminaries The optical matched filtering we are dealing with here, is based on correlating two complex signals by means of optical multiplication in the Fourier domain. We denote the coordinates of the input, filter, and ouput planes of such optical system with xi = (xi, y~), u = (u, v), and x0 = ()Co,Y0), respectively. We denote the input signal with f(xi), and the reference signal with h (xi). Upper case letters will be used to represent the Fourier transforms of spatial functions, while ~{.} will denote the Fourier transform operator. The reference spectrum H(u) can be described as a two-dimensional complex-valued function
n ( u ) = In(u)lexp{iO(u)}
(1)
having the amplitude [H(u)[ and the phase O(u). We shall refer to [H(u)[ as the reference amplitude, and to O(u) as the reference phase. The relative intensity of the correlation signal Ic(xo) is given by
Ic(xo) = I~{F(u).~(u)}l z
(2)
0030-3992/94/020119-08 © 1994 Butterworth-Heinemann Ltd Optics & Laser Technology Vol 26 No 2 1994
119
Filter and signal types in optical matched filtering: N. Demoli where o f ( u ) is the filter transfer function. Since filters in optical matched filtering systems are regularly passive elements, Iof(u)l <~ 1 holds. The o f ( u ) is related to the complex conjugated reference spectrum via the relation of(.) = z {/4.(.)}
(3)
where Z {.} is the filter operator. Z acts as a filter type selector as well as the preprocessor of the reference signal data. Numerous variations for the filter operator can be expected. For example, taking Z to be the unit operator, the classical MSF is simulated. The POF is obtained when g is a phase extractor; the edge enhancement is achieved when Z is a low-frequency cutter, etc. The approach in producing QPOFs has been 6 to level all maxima of the reference spectrum, which is attained by an attenuation mask placed in the first Fourier transform plane 5 of the EOC system. Thus, keeping the phase information unchanged and the beam balance ratio equal to unity over the spectrum maxima, the high-efficiency and the high-discrimination-capability filters have been synthesized. However, the original idea underlying this technique was in producing filters having the transfer function.
1, IH(u)l > H0} of(u) =
0, IH(u)l <~Ho exp{-tO(u)}
(4)
where/4o is the constant. The filter operator is now the phase extractor and the binary low amplitude cutter. The level H0 should be chosen such as to preserve the spectrum energy (like POFs do), while keeping some of the main characteristics of the reference amplitude (such as the amplitude minima). It will be shown later that the filter parameter H0 is of great importance when dealing with signals that have a real and positive Fourier transform. The next step is to determine a basic set of MSFs according to the amplitude and phase information content.
Basic set of matched filter types According to the well-known importance ~2 of the phase information of signals, the phase part of the filter should be: • matched to the phase of the reference spectrum; or • binary matched to the phase of the reference spectrum. Otherwise, for example, by setting the phase information equal to unity, or inversely matched to the phase of the reference spectrum, the resulting matched filtering would be very poor. Although the amplitude part of the filter could be arbitrarily designed, we consider the special cases in which the filter amplitude is: • matched to the amplitude of the reference spectrum; • equal to unity; • binary matched to the amplitude of the reference spectrum; • modified according to the QPOF model; or • inversely matched to the amplitude of the reference spectrum.
120
Combining all proposed amplitude and phase filter information contents, we define the basic set of matched filter types, see Appendix A. Roughly speaking, the main filter groups are: • CMFs (APF and ABPF); • POFs (POF, BPOF, BHAPF, BHABPF, M A P F and MABPF); and • IMFs (IAPF, IABPF, BLAPF and BLABPF). In most applications, for example when the reference signal has no symmetry, the filters within one group will show similar behaviour. However, in some other cases the output correlation results will be significantly different.
Performance quantities The signal-to-noise ratio (SNR) is defined assuming that the power spectral density of the noise is equal to unity SNR =
I
(5)
where <> indicates the ensemble average over the filter area F. The peak-to-sidelobe ratio (PSR) is defined as the ratio of the correlation peak intensity and the average intensity around the peak
-
PSR
Ic. . . .
(6)
where Ic,max is the correlation peak intensity. The denominator indicates the average value over the correlation area C in all points except I c.... • The diffraction efficiency (DE) of the filter is commonly ~3 defined as DE=
(7)
The discrimination ability (DA) is defined as the difference between the auto-correlation and cross-correlation peak intensities, as a fraction of the auto-correlation peak intensity DA
-
Ia ....
-
Ic ....
(8)
a,max
where Ia.... and I c. . . . are the auto-correlation and cross-correlation peak intensities, respectively.
Feasibility estimate We consider the ouput correlation signal to be entirely described by the ability of the matched filter to tolerate input noise, by the sharpness of the correlation peak, and by the matched filter efficiency. These three measures (SNR, PSR and DE), as seen from the definitions (5)-(7), are mutually independent. On the other hand, the ability of the matched filter to discriminate various inputs (or the DA), described by (8), provides information pertaining to the sharpness Optics & Laser Technology Vol 26 No 2 1994
Filter and signal types in optical matched filtering: N. Demoli difference of various correlation signals (see Ref. 14), i.e. DA = 1 (I'~(xo))cPSRc (9)
(I'~(Xo))cPSRA
where PSRc is the PSR corresponding to the cross-correlation output and PSR A is that corresponding to the auto-correlation output. Evidently, the higher PSR sensitivity means the better Ler DA. It is convenient to normalize the main performance quantities and calculate in decibels (dB), i.e. Q~(da) = 10 log~o(Q,/Qo,)
(10)
where Q~ (i = 1, 2, 3) are the SNR, PSR, and DE quantities and Q0~ (i = 1, 2, 3) are the normalization71 values. Thus, the main quantities now describe eachh filter in comparison with the reference one. We introduce a new parameter, the feasibility estimate FE FE(dB) = ~ SNR(dB) + flPSR(dB) + ~,DE(dB)
(11)
which gives an overall estimate of the feasibility of filters. According to (11), an optimal filter ought to possess various properties in such a balance to ensure the highest FE. However, the feasibility estimate may include more terms than shown on the right-hand side of (11). Also, the coefficients ~, fl and 7 could be arbitrarily determined. We choose ~t = fl = 7 -- 1 for all our calculations, thus giving equal weights to SNR, PSR and DE measures.
b
Computer experiments We use numerical methods to investigate the correlation dependence on the filter type, the filter parameter, and the signal used. The aim of these experiments is also to demonstrate the usefulness of the FE quantity. The training set of real signals is shown in Fig. 1, where (a) represents a general signal; (b) a symmetrical signal; and (c) a symmetrical signal with the real and positive Fourier spectrum. The 30 x 30 pixel images are placed in a 64 x 64 array and zero padded. A 64 × 64 point twodimensional F F T routine is used to simulate the desired filter type and to compute the SNR, PSR, DE and FE values. The SNR, PSR and DE are normalized (set to unity for the APF) and calculated in decibels. The filter parameter influence on the auto-correlation signal for the BHAPF, MAPF, IAPF and BLAPF is shown in Fig. 2. From Fig. 2(a) we can see that quantities describing the BHAPF are equal to those describing the POF for the low threshold parameter H0. Increasing H0, the PSR and DE slowly decrease while the SNR firstly increases and then decreases. Since the FE has a shallow maximum, it defines the representative parameter value H 0-- 0.01 (relative units) for describing BHAPFs. The characteristic curves for the M A P F result in a broad maximum in the FE from z equal to 1 to 1000 (relative units), see Fig. 2(b). MAPFs, formerly known as QPOFs, have Optics & Laser Technology Vol 26 No 2 1994
Fig. 1 Images used in numerical experiments. (a) 'F104'; (b) "FORT'; (c) 'PYR'
been introduced 7 using the numerical model that well describes the experimental conditions for obtaining phase-only filters. Therefore, the choice z = 60 is determined by comparing the SNR = - 5 . 2 dB and PSR = 12.9 dB values with the corresponding POF values: SNR = - 6 . 7 dB and PSR = 12.7 dB. From Fig. 2(c) we can see that by increasing the threshold parameter H1 of the IAPF, the PSR decreases while the DE and the SNR increase. For HI = 1 (relative units) all performance quantities saturate to the POF values. We choose H~--0.05 as the representative value for describing IAPFs since, at that value, the FE remains high enough, while the other quantities considerably differ from the corresponding POF values. Figure 2(d) shows that quantities describing BLAPFs saturate to those describing the POF for the threshold parameter H2 = 1 (relative units). We choose H2 = 0.1 to characterize BLAPFs. Comparing the behaviour of the quantity values near the selected parameters, it is obvious that IAPFs and BLAPFs belong to the same group of filters. We made calculations (not shown here) for other general-type
1 21
Filter and signal types in optical.matched filtering: N. Demoli 4o.~
n-t "1o
a
i
I
4o.oo
2o.~.
nn
I
I
I
1
i
20.00-
v ¢-
/
d
O0
o.~
.... .
-
-
k:k:
-
0.00
>
-$ n-
-20.~-
-20.00
I
___ SNR .... PSR . . . . . . DE __FE - ~ . 0 ~
I
,
,
,
, , , , , t
0 . 001
,
,
,
, , , , , I
0.01
a
,
I
__ ....
I ,
,
...... _ _ F E
, ,,,,,,i
.
,,,,,,,i
,
J
m
,
,,,,,,,!
10
T 40.00
" - .
,,,,,,,i
1
0.1
b
i
20.00
,
0.01
H0 (rel. units) 4o.oo
-40. gg
, , , , .
0.1
SNR PSR DE
,
100
,,,,,,,i
,
1000
,,,,,,,
10000
(rel. units)
i
,
20.00
v t00
= >
0.~
.
I-
"
/
t,
/ /
0.00:
-$
CE
(3E -20.~
~ ~ ~ 7"7
-20.00
/" ____
,'"
......
SNR PSR
,/
DE
,
,
,
l i i l ,
0.1t01
t
,
I
,
, l , , l
//
C
H, (tel.
.......
-40.00
0.'1
0.01
I . . . . psR I I . . . . . . DE I
.,,/
__FE -40,~
/ ,'/ -'~ /
,
0.001
units)
L=
< I, ,,I . . . . . . . . . . g.01
d
Hz (rel.
, g.1
F'E
........
units)
Fig. 2 Filter auto-correlation quantities as a function of filter parameters for 'F104' image. (a) BHAPF; (b) M A P F ; (c) IAPF; (d) BLAPF
signals such as a number, a character, etc. The calculations confirmed that our choice has been satisfactory. We have also calculated quantities for T a b l e 1. Filter type APF ABPF
122
the binary-phase versions of the upper described filters. The resulting curves (not shown here) showed very similar behaviour to those presented, but also
Filter a u t o - c o r r e l a t i o n DE (dB)
q u a n t i t i e s f o r t h e 'F104' i m a g e
SNR (dB)
PSR (dB)
FE (dB)
0.00 0.00
0.00 -1.86
0.00 -1.89
0.00 -3.75
POF BPOF BHAPF BHABPF MAPF MABPF
5.64 5.64 5.63 5.63 -1.75 -1.75
-6.71 -9.86 -5.69 -8.83 -5.20 -8.54
12.68 8.95 12.00 8.37 12.89 8.93
11.61 4.73 11.94 5.17 5.94 -1.36
IAPF lAB PF BLAPF BLAB PF
-2.32 -2.32 -2.18 -2.18
-9.59 - 13.17 -10.27 -14.02
20.84 14.26 19.07 13.05
8.93 - 1.23 6.62 -3.15
Optics & Laser Technology Vol 26 No 2 1 9 9 4
Filter and signal types in optical matched fihering: N. Demoli Table 2.
Filter auto-correlation quantities for the 'FORT" image
Filter type
DE (dB)
APF, ABPF
-2.59
-2.68
5.43
0.16
POF, BPOF BHAPF, BHABPF MAPF, MABPF
5.64 5.64 -0.10
-7.66 -6.69 -6.98
15.03 14.70 15.81
13.01 13.65 8.73
IAPF, IABPF BLAPF, BLABPF
2.59 -2.55
-11.61 -13.22
22.05 18.39
7.85 2.58
Table 3.
SNR (dB)
DE (dB)
APF, ABPF POF, BPOF BHAPF, BHABPF MAPF, MABPF
SNR (dB)
FE (dB)
2.26
-12.80
-7.09
5.64 5.64 -7.46
-24.05 -2.71 -9.96
-7.96 -8.76 -7.42
-26.37 -5.83 -24.84
-11.46 -1 5.53
-34.75 -38.61
-1.52 -1.34
-47.73 -55.48
The quantities that describe the auto-correlations using the signal shown in Fig. l(a) were calculated as a function of the filter type and listed in Table 1. Note the feasibility estimate for those filters. For example, the POF shows 11.6 dB gain in FE relative to the APF. We can see that the best FE possesses the BHAPF (11.9 dB). The IAPF also shows the high FE (8.9 dB). The results (not shown here) performed with other general type signals were similar to those presented in Table 1. The typical auto-correlation quantities for the class of symmetrical signals, such as is shown in Fig. l(b), are presented in Table 2. Since the Fourier transform of an even and real signal is itself even and real, the binaryphase versions of filters and the corresponding filters possess equal quantity values. Again the BHAPF (BHABPF) have the best FE (13.7 dB). The auto-correlation and cross-correlation quantity results for the signal of Fig. l(c) are shown in Tables 3 and 4, respectively. We also provide the three-dimensional plots of correlation outputs, see Fig. 3. Obviously, in the case of the symmetrical
signal with its Fourier transform positive and real, only the BHAPF (BHABPF) and the CMFs are the useful ones. Only they can successfully discriminate different inputs. Note that the discrimination ability was unable to calculate for the MAPF (MABPF) and the IMF group of filters.
S u m m a r y and concl usi ons Numerous filter design variations can be found in the optical correlation literature. We have defined the basic set of matched filter types in accordance with the importance of the amplitude and phase information contents of signals. It is considered that the phase part of the filter is allowed to be matched or binary matched to the phase of the reference signal, while the amplitude part could be designed arbitrarily. A mathematical formulation for each filter type has been reviewed and its performance without noise has been quantified. The results presented in this paper are strongly dependent on the filter parameters, filter types and signal types. To describe filters, and taking into consideration the major filter performance criteria, we have introduced an overall single measure, the feasibility estimate, defined as a sum of logarithmic relative values of SNR, PSR and DE (in decibels). The classical filter characteristics served for the reference values.
Filter cross-correlation quantities for the 'PYR" and 'F104' signals
Filter type
DE (dB)
APF, ABPF
-4.48
POF, BPOF BHAPF, BHABPF MAPF, MABPF IAPF, IABPF BLAPF, BLABPF Optics & LaserTechnology Vol 26 No 2 1994
PSR (dB)
3.45
5-10 dB lower values in SNR, PSR and FE, while the DEs remain unchanged. Thus, by defining the representative parameter values the basic set of filters was also defined.
Table 4.
FE (dB)
Filter auto-correlation quantities for the "PYR' image
Filter type
IAPF, IABPF BLAPF, BLABPF
PSR (dB)
SNR (dB)
PSR (dB)
FE (dB)
DA (%)
-5.27
-10.10
-19.85
82.2
5.64 1.84 -4.59
-24.05 -6.20 -8.51
-5.59 -5.46 -5.37
-24.00 -9.82 -18.46
0.0 47.2 --
3.92 3.85
-27.91 -28.92
-5.54 -5.71
-29.53 -30.78
---
1 23
Filter and signal types in optical matched filtering: N. Demoli We have shown the dependence of the correlation results on the filter parameters. Then, by choosing the representative parameters for various filter types, the basic set of filter types has been defined. The optimality of the filter types has been discussed in relation to the signal types. Generally,
according to the FE measure, the POF group represents optimal filters, in particular BHAPFs (also called theoretical QPOFs), which preserve the spectrum energy (like POFs do) and keep some characteristics of the reference amplitude (POFs do not possess such an attribute)• For the class of
1
a
b
%
%
d Fig. 3 The correlations of 'PYR" as a reference function with"PYR" (left-hand side, the autocorrelation case) and 'F104' (right-hand side, cross-correlation case). (a) APF; (b) POF; (c) BHAPF; (d) MAPF
124
Optics & Laser Technology Vol 26 No 2 1994
Filter and signal types in optical matched filtering: N. Demoli
t
61
"
tg
I Fig. 3 (continued) The correlations of "PYR" as a reference function with 'PYR' (left-hand side, t h e auto-correlation case) and 'F104' ( r i g h t - h a n d side, cross-correlation case). (e) IAPF; (f) BLAPF.
symmetrical signals, the binary-phase versions of filters and the corresponding filters possess equal quantity values. BHAPFs and POFs show very similar behaviour in most applications. However, the superior performance of BHAPFs was demonstrated in the case of symmetrical signals with the Fourier transform real and positive. In such cases, only BHAPFs (BHABPFs) and CMFs are able to discriminate the input signal. Although not addressed in these simulations, BHAPFs should be analysed for more complex applications.
Acknowledgements This work was supported by the Ministry for Science of the Republic of Croatia. The author would like to thank the reviewers for their helpful comments.
• Phase-Only Filter (POF) ZPOF= exp{ -- i0 (u) } •
{ •BPOF = •
•
{1,Im.)l> 0, In(u)l ~ / % 3
Binary-High-Amplitude Binary-Phase Filter (BHABPF) ~BHABPF ~---
The basic set of matched spatial filters • Amplitude Phase Filter (APF) or CMF
• Amplitude Binary-Phase Filter (ABPF)
os0u,00}
[H(u)lmax -- , cos O(u) <
0, IH(u)l ~
• Modified-Amplitude Phase Filter (MAPF) or experimental QPOF
IH(u)] exp{-i0(u)} ZAPF--iH(u)lma x
Optics & Laser Technology Vol 26 No 2 1994
1' c°s 0(u) ~>00} - - l , COS 0 ( U ) <
Binary-High-Amplitude Phase Filter (BHAPF) or theoretical QPOF ~(BHAPF
Appendix A
J(ABPF =
Binary-Phase-Only Filter (BPOF)
ZMAPF
•
--
IH(u)l "--i0(u)} 1 -t--z2--~u)l 4exp/
Modified-Amplitude Binary-Phase Filter (MABPF) IH(u), 5 1, cos0(u)>~00} ZMABPF= 1 + CIH(u)I 4 [ - - 1, cos O(u) <
125
Filter and signal types in optical matched filtering: N. Demoli •
Inverse-Amplitude Phase Filter (IAPF) or amplitude-compensated matched filtert5 XIAPF =
t
-IH(u)l - ,
L 1, •
[H(u)[
H1
exp{ -i0(u)}
IH(u)l ~
Inverse-Amplitude Binary-Phase Filter (IABPF) Z,ABeF =
t
1, cos O(u) >~0"~
IH----~' IH(u)l > H,
(
1,
IH(u)l ~< H i
- 1, cos O(u) < OJ
• Binary-Low-Amplitude Phase Filter (BLAPF) {0, [H(u)] > H2}exp{_iO(u) } ZBLAPZ= 1, IH(u)I ~
Binary-Low-Amplitude Binary-Phase Filter (BLABPF) XBLABPF=
1,
IH(u)l ~
-- 1, cos
O(u) <
References 1 Vander Lugt, A. Signal detection by complex spatial filtering, IEEE Trans Inf Theory, ITI0 (1964) 139-145
2 Pernick, B.J. Phase-only and binary phase-only spatial filters for optical correlators: a survey, Opt Laser Technol, 23 (1991) 273-282 3 Vijaya Kumar, B.V.K., Hassebrook, L. Performance measures for correlation filters, Appl Opt, 29 (1990) 2997-3006 4 Chalasinska-Macukow, K., Goreeki, C. Optoelectronic implementation of the quasi-phase correlator, Opt Commun, 93 (1992) 11-18 5 Demoli, N., Vuki~evi~, D. Discrimination sensitivity of the extended optical correlator, Opt Commun, 64 (1987) 417-420 6 Demoli, N. Quasiphase-only matched filtering, Appl Opt, 26 (1987) 2058-2061 7 Demoli, N. Properties of quasi phase-only matched spatial filters, Opt Eng, 31 (1992) 275-279 8 Janowska-Dmoch, B., Chalasinska-Macukow, K., Piliszek, J. Matched filters and quasiphase-only filters recorded in silver halide (sensitized) gelatin, Opt Laser Technol, 24 (1992) 279~84 9 Demoli, N. Quasi phase-only filters as an inverse matched spatial filter, Optik, 91 (1992) 11 15 10 Wunsch II, D.C., Marks II, R.J., Caudeil, T.P., Capps, C.D. Limitations of a class of binary phase-only filters, Appl Opt, 31 (1992) 5681 5687 11 Yaroslavsky, L.P. Is the phase-only filter and its modifications optimal in terms of the discrimination capability in pattern recognition? Appl Opt, 31 (1992) 1677-1679 12 Oppenheim, A.V., Lira, J.S. The importance of phase in signals. Proc IEEE, 69 (1981) 529-541 13 Demoli, N. Optical power efficiency in coherent optical correlater systems, Opt and Quant Electron, 21 (1989) 179 182 14 Zheng, S.H., Karim, M.A., Gao, M.L. Amplitude-modulated phase-only filter performance in presence of noise, Opt Commun, 89 (1992) 296-305 15 Mu, G., Wang, X., Wang, Z. Amplitude-compensated matched filtering, Appl Opt, 27 (1988) 3461-3463
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