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Rotational invariant pattern recognition using a composite circular harmonic and 2D isotropic Mexican-hat wavelet filter
a.__
i!!J I
1 November 1994
OPTICS COMMUNICATIONS
_
j--
._
ELSEVIER
Optics Communications 112 ( 1994) 9-l 5
Rotational invariant pattern recognition using a composite circular harmonic and 2D isotropic Mexican-hat wavelet filter Yih-Shyang Cheng, Tsair-Chun Liang Institute of Optical Sciences, National Central University, Chung-Li 320, Taiwan Received 11 January 1994
Abstract An improved shift- and rotational-invariant filter using wavelet transform and circular harmonic filtering at the same time is proposed. Computer simulation has shown that the filter is better than phase-onlycircular harmonic filter in peak sharpness,
discriminatingability,SNRand toleranceof positionerror of the filter.Furthermore,this filter possesseshighdiffractionefficiency.
1. Introduction
The first convenient and simple shift- and rotational-invariant correlation filter is the circular harmonic filter (CHF) proposed by Hsu and Arsenault in 1982 [ 1,2 1. This kind of filter is made from some circular harmonic component of the reference pattern. In late eighties, Yau and Chang, Rosen and Shamir, and Leclerc et al., proposed the phase-only circular harmonic filter (POCHF) [ 3-51. These kinds of filter are better than the CHF in light efficiency, correlation peak sharpness and discriminating ability, but it is noisy because of its high-pass nature. Although phase information is more important than amplitude information in pattern recognition in general [ 6 1, amplitude information can further improve the performance of a phase-only correlation filter if it is appropriately utilized. The amplitude compensated circular harmonic filter ( ACCHF) proposed by Sun et al. in 1990 [ 71, which is an example and this kind of filter can produce even sharper correlation peak than the POCHF does. Unfortunately, its light efficiency is ultra-low. This kind of filter is also noise-sensitive due to the tremendous
enhancement of some specific frequency components of the reference pattern. A good correlation filter should have sharp correlation peak, good discriminating ability, high noise discriminating ability in input plane, and high light efficiency. In this article we propose an improved shift- and rotational-invariant correlation filter which possesses all these merits. The basic scheme of this filter is a combination of an optical wavelet transform and a circular harmonic filtering. In principle, we first correlated the reference pattern with a 2D isotropic Mexican-hat wavelet. This kind of correlation is defined as wavelet transform. According to the theory of wavelet transform, the output intensity distribution will be an edgeenhanced reference pattern. The output amplitude distribution of this wavelet transform is then used as the input reference pattern for another 4fcorrelator. The filter in this correlator is a one-component circular harmonic filter of the output amplitude distribution of the preceding wavelet transform. The whole process is a double 4fcorrelation system. Here we will prove that this “double correlation” can be replaced by a single correlation and then it can be carried out
~30-4018/94/%07.00 0 1994 Elsevier Science B.V. All rights reserved XSDIOO30-4018(94)00416-l
10
Y.-S. Cheng, T.-C. Liang / Optics Communications 112 (1994) 9-15
with a single 4foptical correlator, thus simplifying the experimental set up. Results of computer simulation have shown that this new shift- and rotational-invariant filter not only yields sharp peaks, good discriminating ability but also tolerates, low signal-to-noise ratio (SNR) in the input plane and possesses high light efficiency. Another advantage is that this scheme can allow a relatively larger transverse mis-placement of the filter, thus relaxing the stringent requirement for the alignment when the experiment is carried out optically.
2. Basic theory
xexp[-i2nCf,x+f,y)]dxdy =ae(&
afy) ,
where Hcf,, f,) is Fourier transform of the mother wavelet h(x, y). The wavelet transform can also be written as +oO
Xexp[i2nCf,x+f,y)l
2.1. Wavelet transform Wavelet transforms have been proven effective in image analysis, data compression and feature extraction [ 8,9 1. The optical wavelet transforms are inherently shift-invariant [ IO]. In this paper the wavelet function we adopt is a 2D isotropic Mexican-hat wavelet function. Mexican-hat wavelet function is actually the secondary derivative of Gaussian function, which is widely used for zero-crossing multiresolution edge detection. This function can be written as h(x,y)=[l-(x2+y2)]
exp[-(x2+y2)/2].
(4)
dfxdf,,
where Scf,, f,) is the Fourier transform of the signal s(x, y) and the symbol “*” denotes the complex conjugate. So the final result of the wavelet transform depends upon the value of dilation factor a. The input reference pattern in this article is a 56 x 115 sized binary helicopter, embedded in a plane of 256~256 pixels, as shown in Fig. 1. In order to find the optimal dilation factor a for producing the best edge-enhanced reference pattern, we first obtain the intensity distribution of the input reference pattern after wavelet transform for various values of a, and then we calculate the autocorrelation of these in-
(1)
A set of 2D dilated wavelets hab(x, y) which has the same energy as that contained in the original mother wavelet h (x, y ) can be written as
(2) where a is the dilation factor and b is the amount of translation. In this paper, we chose the translation factor b = 0, and hence the subscript b is eliminated. Wavelet transform of an input pattern is the correlation operation between the input pattern s(x, y) and the wavelet function h, (x, y ), W,(a, 4 Y) =.%
~)*k(x,
Y) ,
(3)
here the symbol “*” denotes the correlation operation. In the following context, we denote the Fourier transform of the wavelet function h, (x, y ) as
(5)
Fig. 1. Input reference pattern.
Y.-S. Cheng, T.-C. Liang /Optics CommunicationsI12 (1994) 9-15
tensity distributions, respectively. Fig. 2 is the result of these autocorrelation peak intensities versus 1/a. The minimum peak intensity occurs at a= l/2.5. It is understandable that this value of a will result in the best edge enhancement, and thus this value of a is chosen to be used in the following analysis. Fig. 3 is the edge-enhanced reference pattern for a = 112.5. If the dilation factor a is slightly different from l/2.5, the edge becomes smeared out and indistinctive. If
106
v 0
1
2
3
4
I
5
11
we increase the factor a from its optimal value, the output helicopter pattern becomes distorted, dilated and unrecognizable. On the other hand, when the factor a is decreased, the output helicopter pattern approaches the original input helicopter. 2.2. Combination of optical wuvelet transform with circular harmonic filtering Let us first consider the operation of the schematic as shown in Fig. 4a. Let the input reference pattern be s( x, y) and the wavelet function h,(x, y ) be the Mexican-hat wavelet function. It is clear that the amplitude distribution W, [Eq. (3) ] appeared in the wavelet transform plane (WTP) is the correlation between the input reference pattern and the Mexican-hat wavelet function. According to the wavelet transform theory, the intensity distribution 1W, 1’will be an edge-enhanced reference pattern (Fig. 3). W, then serves as the input reference pattern for the second correlator in Fig. 4a. The spatial filter in this correlator is a computer generated hologram (CGH) in which is impressed the Fourier transform of a circular harmonic component, of some specific order m,
l/a
Fig. 2. Autocorrelation peak intensity for the wavelet transform of input reference pattern as a function of the dilation factor a.
I
L
Hat L
I
WTP
L
L (HaFr)*
L
I
I
I
Fr* L
0
0
‘f’f’f’f’ 04
Fig. 3. The edge-enhanced reference pattern image for a= l/2.5.
Fig. 4. I, input pattern; L, Fourier transform lens; Hz, wavelet filter; WTP, wavelet transform plane; F:, circular harmonic tilter; (Hzr)*, wavelet circular harmonic filter; 0, output plane;f; focal length. (a) The basic optical system employing a wavelet filter and a circular harmonic filter in tandem. (b) 4fsystem employing a wavelet circular harmonic filter.
Y.-S. Cheng, T.-C. Liang / Optics Communications 112 (1994) 9-15
12
of W, at the optimal a. We denote the mathematical expression of the CGH as F: dfx,f,). That is J’LLf,)
=%x(x,
v)} >
(6)
where 9 denotes the Fourier transform f,(x, y) =fm(r)
exp(im@
,
operation, (7)
and &(r)=
Ws(r, 0) exp(-ime)
&T
d6,
(8)
0
where (r, 6) are polar coordinates and Ws(r, 0) be the polar representation of the Ws(x, y). It is clear that the final output amplitude distribution in Fig. 4a will be K*L=
(s*ha)*J.
(9)
Based on correlation
theory
(S*-ha)*~=%-l{%{S*~n}(%-dfi})*} =%-‘{sy;,f,)H~Cf,,f)F~~,~)} = 8-‘{S(H,F,)*} =s_AY-'{H,F,.}
,
(10)
where the symbol “%-I” denotes the inverse Fourier transform operation. The above result implies that the double correlation operation performed in Fig. 4a can be carried with a simpler set up, as shown in Fig. 4b, with a filter function: (H,F,)*.
3. Computer simulation The computer simulation of the optical experiment as shown in Fig. 4b is described in this section. All the data and results involved in the computer simulation are expressed in 256x256 matrices. We have used the second order (m=2) circular harmonic component of the amplitude distribution appearing in the wavelet transform plane in Fig. 4a to make our filters. The calculation of the second order circular harmonic component and the finding of the proper center (optimal expansion center) [ 1 ] have been done with a VAX9320 computer. The other calculations are done with a personal computer using a PC matlab software. In fact we have done four sets of computer experiments with four different kinds of
filter. They are wavelet transform circular harmonic filter ( WTCHF), wavelet transform phase-only circular harmonic tilter (WTPOCHF), circular harmonic filter (CHF) and phase-only circular harmonic filter (POCHF). The third one and the fourth one are for comparison. Table 1 shows the output intensities I,, spot sizes Np and peak to sidelobe ratio (PSR) of correlation peaks for the four different kinds of filters. The Z, is in fact the relative correlation peak intensity with the correlation peak intensity produced by the CHF normalized to one. The spot size Np is defined as follows: it is the number of pixels in the autocorrelation distribution the intensity of which is larger than or equal to e-r times the peak value of the distribution. The PSR is the ratio of the correlation peak intensity to the largest sidelobe [ 111. The CHF and POCHF here are also made from the second order circular harmonic component of the reference pattern s(x, y). The WTPOCHF is a composite filter which combines together the wavelet transform filter and the phase-only circular harmonic filter. The process of making this filter is similar to that of making the WTCHF except that phase-only circular harmonic component is used instead of circular harmonic component. In Table 1 one can see that WTCHF and WTPOCHF have better PSR and Np which are measures of peak sharpness. One also finds that the peak intensities of WTCHF and WTPOCHF are close to or better than CHF. This implies that these two filters possess high diffraction efficiency. Fig. 5a is the input pattern for the computer experiments, in which there are two reference objects in different orientations in the upper half of the figure and two non-targets in the lower half of it. Fig. 5b is the correlation output with CHF. The two correlaTable 1 Autocorrelation results for input reference pattern Filter type WTCHF WTPOCHF CHF POCHF
1, 0.83 1.8
I 25.18
PSR
N,
I 1 786 3
6.22 4.24 1.8 2.64
I, is the normalized correlation peak intensity with the correlation peak intensity produced by the CHF normalized to 1. N, is the spot size and PSR is the ratio of correlation peak intensity to the largest sidelobe.
Y.-S. Cheng, T.-C. Liang /Optics Communications 112 (1994) 9-15
13
05
0 2 0
Fig. 5. (a) Input pattern consisting of two reference objects in different orientations in the upper half and two helicopters in different types in the lower half. (b) Correlation output with CHF. (c) Correlation output with POCHF. (d) Correlation output with WTCHF. (e ) Correlation output with WTPOCHF.
14
Y.-S. Cheng, T.-C. Liang / Optics Communications I12 (1994) 9-15
Table 2 The relative peak intensities corresponding to the four helicopters produced with WTCHF, WTPOCHF and POCHF Filter type
Upper left helicopter
Upper right helicopter
Lower left helicopter
Lower right helicopter
WTCHF WTPOCHF POCHF
1 1 1
1.02
0.22 0.26 0.24
0.16 0.08 0.40
1 0.79
(129,129)
(138,174)
(129,129)
(129,129)
I
-
-
(4
(b)
Fig. 6. Computer simulation correlation peaks yielded with different positioning of the CHF, POCHF, WTCHF and WTPOCHF from top to bottom. The number in the figure indicates the location of the peak. (a) The filters are positioned correctly. (b) The filters are m&positioned by 6-pixels in the x-direction, by 2-pixels in the y-direction.
Y.-S. Cheng, T.-C. Liang / Optics Communications 112 (1994) P-15
tion peaks are accompanied by high sidelobes, making them less visible. The prominence of the sidelobes makes it with this kind of filter impossible to recognize the reference objects when they are close to each other. Although the correlation peaks produced by POCHF are rather sharp (Fig. 5c), the correlation peaks produced by WTCHF (Fig. 5d) and WTPOCHF (Fig. 5e) are even sharper. The relative peak intensities corresponding to the four helicopters produced with WTCHF (Fig. 5d), WTPOCHF (Fig. 5e) and POCHF (Fig. 5c) are summarized in Table 2. These results show that our composite filters are better than the POCHF in discriminating the reference object from objects which closely resemble the reference object in shape. We also found that the WTCHF and WTPOCHF can tolerate larger transverse mis-placement than the CHF and POCHF do. In the computer simulation, if the CHF is shifted in the y-direction by one pixel, the correlation peak no longer occurs at its correct position. If it is shifted in the x-direction by one pixel, the correlation peak even becomes unrecognizable. For the case of POCHF, the correlation peak is still recognizable if the filter is shifted by one pixel in both the x- and the y-direction. However, if it is shifted by two pixels in y-direction, the position of correlation peak also shifts. Again, 2 pixels-shift in the x-direction makes the correlation peak unrecognizable. In the case of WTCHF and W’TPOCHF, the correlation peak is still recognizable even if the filter is shifted in x- (y-)direction by 6- (2-)pixels (see Fig. 6). Hence, the wavelet transform filter requires less position-accuracy than that with the CHF or with POCHF. The wavelet transform filter is a band-pass filter [9] while the phase-only filter is inherently a highpass filter. Hence, the wavelet transform filter should have better tolerance to additive random noises in the input plane. If random additive noises are added to the input pattern, computer simulation with CHF has shown that, when SNRc0.03, the correlation peak shifts from its correct position. For POCHF, the correlation peak shifts from its correct position and becomes unrecognizable when SNR < 0.15. For our WTCHF similar results took place only when SNR < 0.06 and when SNR < 0.09 for the case with our WTPOCHF.
15
4. Conclusion We have presented a new kind of composite wavelet circular harmonic filters for shift- and rotationalinvariant pattern recognition. Since h, is a circularlysymmetric Mexican-hat wavelet function, its Fourier transform H, should be a function of p only in the frequency domain. According to the property of circular harmonic, F, is a function of p multiplied by exp(im@), where p and @are the radial and angular coordinates, respectively, in the frequency plane, and m denotes the order of the circular harmonic component [ 5 1. So (H,F,)* is a function of p multiplied by exp( -im@). So this kind of correlation filtering is shift- and rotational-invariant according to the theory of circular harmonic filters. Computer simulations have shown that the correlation peaks are sharper than those yielded with POCHF’s. The correlation peaks obtained with these filters are less affected by the noises present in the input plane than those obtained with the phase-only filter. Furthermore, these filters possess high diffraction efftciency and they require less stringent positioning accuracy. These merits, we think, should make them superior to the CHF and POCHF especially when they are implemented in an optical system.
References [ 11 Y.N. Hsu, H.H. Arsenault and G. April, Appl. Optics 21 (1982) 4012. [2] Y.N. Hsu and H.H. Arsenault, Appl. Optics 21 (1982) 4016. [ 3] H.F. Yau and C.C. Chang, Appl. Optics 28 ( 1989) 2070. [4] J. Rosen and J. Shamir, Appl. Optics 27 (1988) 2895. [5] L. Leclerc, Y. Sheng and H.H. Arsenault, Appl. Optics 28 (1989) 1251. [ 61 A.V. Oppenheim and J.S. Lim, Proc. IEEE 69 ( 198 1) 529. [7] Y. Sun, Z.Q. Wang and G.G. Mu, Appl. Optics 29 (1990) 4779. [ 81 S.G. Mallat, IEEE Trans. Acoust. Speech Signal Process. 37 (1989) 2091. [ 9I X.J. Lu, A. Katz, E.G. Kanterakis, Y. Li, Y. Zhang and N.P. Caviris, Optics Comm. 92 (1992) 337. [ 101 Y. Sheng, D. Roberge, H. Szu and T. Lu, Optics Lett. 18 (1993) 299. [ 111 B. Javidi and Q. Tang, Appl. Optics 3 1 ( 1992) 4034.
i!!J I
1 November 1994
OPTICS COMMUNICATIONS
_
j--
._
ELSEVIER
Optics Communications 112 ( 1994) 9-l 5
Rotational invariant pattern recognition using a composite circular harmonic and 2D isotropic Mexican-hat wavelet filter Yih-Shyang Cheng, Tsair-Chun Liang Institute of Optical Sciences, National Central University, Chung-Li 320, Taiwan Received 11 January 1994
Abstract An improved shift- and rotational-invariant filter using wavelet transform and circular harmonic filtering at the same time is proposed. Computer simulation has shown that the filter is better than phase-onlycircular harmonic filter in peak sharpness,
discriminatingability,SNRand toleranceof positionerror of the filter.Furthermore,this filter possesseshighdiffractionefficiency.
1. Introduction
The first convenient and simple shift- and rotational-invariant correlation filter is the circular harmonic filter (CHF) proposed by Hsu and Arsenault in 1982 [ 1,2 1. This kind of filter is made from some circular harmonic component of the reference pattern. In late eighties, Yau and Chang, Rosen and Shamir, and Leclerc et al., proposed the phase-only circular harmonic filter (POCHF) [ 3-51. These kinds of filter are better than the CHF in light efficiency, correlation peak sharpness and discriminating ability, but it is noisy because of its high-pass nature. Although phase information is more important than amplitude information in pattern recognition in general [ 6 1, amplitude information can further improve the performance of a phase-only correlation filter if it is appropriately utilized. The amplitude compensated circular harmonic filter ( ACCHF) proposed by Sun et al. in 1990 [ 71, which is an example and this kind of filter can produce even sharper correlation peak than the POCHF does. Unfortunately, its light efficiency is ultra-low. This kind of filter is also noise-sensitive due to the tremendous
enhancement of some specific frequency components of the reference pattern. A good correlation filter should have sharp correlation peak, good discriminating ability, high noise discriminating ability in input plane, and high light efficiency. In this article we propose an improved shift- and rotational-invariant correlation filter which possesses all these merits. The basic scheme of this filter is a combination of an optical wavelet transform and a circular harmonic filtering. In principle, we first correlated the reference pattern with a 2D isotropic Mexican-hat wavelet. This kind of correlation is defined as wavelet transform. According to the theory of wavelet transform, the output intensity distribution will be an edgeenhanced reference pattern. The output amplitude distribution of this wavelet transform is then used as the input reference pattern for another 4fcorrelator. The filter in this correlator is a one-component circular harmonic filter of the output amplitude distribution of the preceding wavelet transform. The whole process is a double 4fcorrelation system. Here we will prove that this “double correlation” can be replaced by a single correlation and then it can be carried out
~30-4018/94/%07.00 0 1994 Elsevier Science B.V. All rights reserved XSDIOO30-4018(94)00416-l
10
Y.-S. Cheng, T.-C. Liang / Optics Communications 112 (1994) 9-15
with a single 4foptical correlator, thus simplifying the experimental set up. Results of computer simulation have shown that this new shift- and rotational-invariant filter not only yields sharp peaks, good discriminating ability but also tolerates, low signal-to-noise ratio (SNR) in the input plane and possesses high light efficiency. Another advantage is that this scheme can allow a relatively larger transverse mis-placement of the filter, thus relaxing the stringent requirement for the alignment when the experiment is carried out optically.
2. Basic theory
xexp[-i2nCf,x+f,y)]dxdy =ae(&
afy) ,
where Hcf,, f,) is Fourier transform of the mother wavelet h(x, y). The wavelet transform can also be written as +oO
Xexp[i2nCf,x+f,y)l
2.1. Wavelet transform Wavelet transforms have been proven effective in image analysis, data compression and feature extraction [ 8,9 1. The optical wavelet transforms are inherently shift-invariant [ IO]. In this paper the wavelet function we adopt is a 2D isotropic Mexican-hat wavelet function. Mexican-hat wavelet function is actually the secondary derivative of Gaussian function, which is widely used for zero-crossing multiresolution edge detection. This function can be written as h(x,y)=[l-(x2+y2)]
exp[-(x2+y2)/2].
(4)
dfxdf,,
where Scf,, f,) is the Fourier transform of the signal s(x, y) and the symbol “*” denotes the complex conjugate. So the final result of the wavelet transform depends upon the value of dilation factor a. The input reference pattern in this article is a 56 x 115 sized binary helicopter, embedded in a plane of 256~256 pixels, as shown in Fig. 1. In order to find the optimal dilation factor a for producing the best edge-enhanced reference pattern, we first obtain the intensity distribution of the input reference pattern after wavelet transform for various values of a, and then we calculate the autocorrelation of these in-
(1)
A set of 2D dilated wavelets hab(x, y) which has the same energy as that contained in the original mother wavelet h (x, y ) can be written as
(2) where a is the dilation factor and b is the amount of translation. In this paper, we chose the translation factor b = 0, and hence the subscript b is eliminated. Wavelet transform of an input pattern is the correlation operation between the input pattern s(x, y) and the wavelet function h, (x, y ), W,(a, 4 Y) =.%
~)*k(x,
Y) ,
(3)
here the symbol “*” denotes the correlation operation. In the following context, we denote the Fourier transform of the wavelet function h, (x, y ) as
(5)
Fig. 1. Input reference pattern.
Y.-S. Cheng, T.-C. Liang /Optics CommunicationsI12 (1994) 9-15
tensity distributions, respectively. Fig. 2 is the result of these autocorrelation peak intensities versus 1/a. The minimum peak intensity occurs at a= l/2.5. It is understandable that this value of a will result in the best edge enhancement, and thus this value of a is chosen to be used in the following analysis. Fig. 3 is the edge-enhanced reference pattern for a = 112.5. If the dilation factor a is slightly different from l/2.5, the edge becomes smeared out and indistinctive. If
106
v 0
1
2
3
4
I
5
11
we increase the factor a from its optimal value, the output helicopter pattern becomes distorted, dilated and unrecognizable. On the other hand, when the factor a is decreased, the output helicopter pattern approaches the original input helicopter. 2.2. Combination of optical wuvelet transform with circular harmonic filtering Let us first consider the operation of the schematic as shown in Fig. 4a. Let the input reference pattern be s( x, y) and the wavelet function h,(x, y ) be the Mexican-hat wavelet function. It is clear that the amplitude distribution W, [Eq. (3) ] appeared in the wavelet transform plane (WTP) is the correlation between the input reference pattern and the Mexican-hat wavelet function. According to the wavelet transform theory, the intensity distribution 1W, 1’will be an edge-enhanced reference pattern (Fig. 3). W, then serves as the input reference pattern for the second correlator in Fig. 4a. The spatial filter in this correlator is a computer generated hologram (CGH) in which is impressed the Fourier transform of a circular harmonic component, of some specific order m,
l/a
Fig. 2. Autocorrelation peak intensity for the wavelet transform of input reference pattern as a function of the dilation factor a.
I
L
Hat L
I
WTP
L
L (HaFr)*
L
I
I
I
Fr* L
0
0
‘f’f’f’f’ 04
Fig. 3. The edge-enhanced reference pattern image for a= l/2.5.
Fig. 4. I, input pattern; L, Fourier transform lens; Hz, wavelet filter; WTP, wavelet transform plane; F:, circular harmonic tilter; (Hzr)*, wavelet circular harmonic filter; 0, output plane;f; focal length. (a) The basic optical system employing a wavelet filter and a circular harmonic filter in tandem. (b) 4fsystem employing a wavelet circular harmonic filter.
Y.-S. Cheng, T.-C. Liang / Optics Communications 112 (1994) 9-15
12
of W, at the optimal a. We denote the mathematical expression of the CGH as F: dfx,f,). That is J’LLf,)
=%x(x,
v)} >
(6)
where 9 denotes the Fourier transform f,(x, y) =fm(r)
exp(im@
,
operation, (7)
and &(r)=
Ws(r, 0) exp(-ime)
&T
d6,
(8)
0
where (r, 6) are polar coordinates and Ws(r, 0) be the polar representation of the Ws(x, y). It is clear that the final output amplitude distribution in Fig. 4a will be K*L=
(s*ha)*J.
(9)
Based on correlation
theory
(S*-ha)*~=%-l{%{S*~n}(%-dfi})*} =%-‘{sy;,f,)H~Cf,,f)F~~,~)} = 8-‘{S(H,F,)*} =s_AY-'{H,F,.}
,
(10)
where the symbol “%-I” denotes the inverse Fourier transform operation. The above result implies that the double correlation operation performed in Fig. 4a can be carried with a simpler set up, as shown in Fig. 4b, with a filter function: (H,F,)*.
3. Computer simulation The computer simulation of the optical experiment as shown in Fig. 4b is described in this section. All the data and results involved in the computer simulation are expressed in 256x256 matrices. We have used the second order (m=2) circular harmonic component of the amplitude distribution appearing in the wavelet transform plane in Fig. 4a to make our filters. The calculation of the second order circular harmonic component and the finding of the proper center (optimal expansion center) [ 1 ] have been done with a VAX9320 computer. The other calculations are done with a personal computer using a PC matlab software. In fact we have done four sets of computer experiments with four different kinds of
filter. They are wavelet transform circular harmonic filter ( WTCHF), wavelet transform phase-only circular harmonic tilter (WTPOCHF), circular harmonic filter (CHF) and phase-only circular harmonic filter (POCHF). The third one and the fourth one are for comparison. Table 1 shows the output intensities I,, spot sizes Np and peak to sidelobe ratio (PSR) of correlation peaks for the four different kinds of filters. The Z, is in fact the relative correlation peak intensity with the correlation peak intensity produced by the CHF normalized to one. The spot size Np is defined as follows: it is the number of pixels in the autocorrelation distribution the intensity of which is larger than or equal to e-r times the peak value of the distribution. The PSR is the ratio of the correlation peak intensity to the largest sidelobe [ 111. The CHF and POCHF here are also made from the second order circular harmonic component of the reference pattern s(x, y). The WTPOCHF is a composite filter which combines together the wavelet transform filter and the phase-only circular harmonic filter. The process of making this filter is similar to that of making the WTCHF except that phase-only circular harmonic component is used instead of circular harmonic component. In Table 1 one can see that WTCHF and WTPOCHF have better PSR and Np which are measures of peak sharpness. One also finds that the peak intensities of WTCHF and WTPOCHF are close to or better than CHF. This implies that these two filters possess high diffraction efficiency. Fig. 5a is the input pattern for the computer experiments, in which there are two reference objects in different orientations in the upper half of the figure and two non-targets in the lower half of it. Fig. 5b is the correlation output with CHF. The two correlaTable 1 Autocorrelation results for input reference pattern Filter type WTCHF WTPOCHF CHF POCHF
1, 0.83 1.8
I 25.18
PSR
N,
I 1 786 3
6.22 4.24 1.8 2.64
I, is the normalized correlation peak intensity with the correlation peak intensity produced by the CHF normalized to 1. N, is the spot size and PSR is the ratio of correlation peak intensity to the largest sidelobe.
Y.-S. Cheng, T.-C. Liang /Optics Communications 112 (1994) 9-15
13
05
0 2 0
Fig. 5. (a) Input pattern consisting of two reference objects in different orientations in the upper half and two helicopters in different types in the lower half. (b) Correlation output with CHF. (c) Correlation output with POCHF. (d) Correlation output with WTCHF. (e ) Correlation output with WTPOCHF.
14
Y.-S. Cheng, T.-C. Liang / Optics Communications I12 (1994) 9-15
Table 2 The relative peak intensities corresponding to the four helicopters produced with WTCHF, WTPOCHF and POCHF Filter type
Upper left helicopter
Upper right helicopter
Lower left helicopter
Lower right helicopter
WTCHF WTPOCHF POCHF
1 1 1
1.02
0.22 0.26 0.24
0.16 0.08 0.40
1 0.79
(129,129)
(138,174)
(129,129)
(129,129)
I
-
-
(4
(b)
Fig. 6. Computer simulation correlation peaks yielded with different positioning of the CHF, POCHF, WTCHF and WTPOCHF from top to bottom. The number in the figure indicates the location of the peak. (a) The filters are positioned correctly. (b) The filters are m&positioned by 6-pixels in the x-direction, by 2-pixels in the y-direction.
Y.-S. Cheng, T.-C. Liang / Optics Communications 112 (1994) P-15
tion peaks are accompanied by high sidelobes, making them less visible. The prominence of the sidelobes makes it with this kind of filter impossible to recognize the reference objects when they are close to each other. Although the correlation peaks produced by POCHF are rather sharp (Fig. 5c), the correlation peaks produced by WTCHF (Fig. 5d) and WTPOCHF (Fig. 5e) are even sharper. The relative peak intensities corresponding to the four helicopters produced with WTCHF (Fig. 5d), WTPOCHF (Fig. 5e) and POCHF (Fig. 5c) are summarized in Table 2. These results show that our composite filters are better than the POCHF in discriminating the reference object from objects which closely resemble the reference object in shape. We also found that the WTCHF and WTPOCHF can tolerate larger transverse mis-placement than the CHF and POCHF do. In the computer simulation, if the CHF is shifted in the y-direction by one pixel, the correlation peak no longer occurs at its correct position. If it is shifted in the x-direction by one pixel, the correlation peak even becomes unrecognizable. For the case of POCHF, the correlation peak is still recognizable if the filter is shifted by one pixel in both the x- and the y-direction. However, if it is shifted by two pixels in y-direction, the position of correlation peak also shifts. Again, 2 pixels-shift in the x-direction makes the correlation peak unrecognizable. In the case of WTCHF and W’TPOCHF, the correlation peak is still recognizable even if the filter is shifted in x- (y-)direction by 6- (2-)pixels (see Fig. 6). Hence, the wavelet transform filter requires less position-accuracy than that with the CHF or with POCHF. The wavelet transform filter is a band-pass filter [9] while the phase-only filter is inherently a highpass filter. Hence, the wavelet transform filter should have better tolerance to additive random noises in the input plane. If random additive noises are added to the input pattern, computer simulation with CHF has shown that, when SNRc0.03, the correlation peak shifts from its correct position. For POCHF, the correlation peak shifts from its correct position and becomes unrecognizable when SNR < 0.15. For our WTCHF similar results took place only when SNR < 0.06 and when SNR < 0.09 for the case with our WTPOCHF.
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4. Conclusion We have presented a new kind of composite wavelet circular harmonic filters for shift- and rotationalinvariant pattern recognition. Since h, is a circularlysymmetric Mexican-hat wavelet function, its Fourier transform H, should be a function of p only in the frequency domain. According to the property of circular harmonic, F, is a function of p multiplied by exp(im@), where p and @are the radial and angular coordinates, respectively, in the frequency plane, and m denotes the order of the circular harmonic component [ 5 1. So (H,F,)* is a function of p multiplied by exp( -im@). So this kind of correlation filtering is shift- and rotational-invariant according to the theory of circular harmonic filters. Computer simulations have shown that the correlation peaks are sharper than those yielded with POCHF’s. The correlation peaks obtained with these filters are less affected by the noises present in the input plane than those obtained with the phase-only filter. Furthermore, these filters possess high diffraction efftciency and they require less stringent positioning accuracy. These merits, we think, should make them superior to the CHF and POCHF especially when they are implemented in an optical system.
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