- Email: [email protected]
Second-order interpattern neural networks for optical pattern recognition
1 July 1997 OPTICS COMMUNICATIONS ELSEWIER
Optics Communications 139 (1997) 182-186
Second-order interpattern neural networks for optical pattern recognition Huiquan Cheng *, Liren Liu, Guoqiang Li, Lan Shao, Changhe Zhou Irzformation Optics Laboratory, Slwznghai hzstitute of Optics and Fine Mechanics, Academia Sinica, P.O. Box 800-211 Shanghai 201800, China
Received 4 July 1996; accepted 18 December 1996
Abstract By combining the autoassociative model with the heteroassociative model, we propose a second-order semi-complex IPA neural network. In this neural network, the real part or the imaginary part of a complete or partial complex vector not only addresses the corresponding part of the complex vector prestored in the neural network by autoassociation, but also addresses the corresponding imaginary or real part by heteroassociation. Owing to the introduction of the heteroassociation, the correct addressing probability is increased. Based on this neural network, a scheme for optical pattern recognition (OPR) is proposed. This scheme is composed of two cascaded optical systems; each performs a relative simple function. Compared with the matched spatial filter OPR system, the addressing results are direct and our scheme is feasible and promising.
1. Introduction Neural networks have received extensive investigations and have been applied to different fields for their potential superior capabilities [ 1,2]. Pattern recognition, which involves many factors and is difficult to handle, has been closely connected with neural networks for smart decision. Optical pattern recognition (OPR) based on optical correlation has long been investigated since VandeLugt proposed the matched filter concept [3]. There are several methods such as synthetic discriminant function [4], symbolic substitution [5], circular harmonic expansion [6], etc. Recently owing to the advantage of windowed Fourier transform, a wavelet matched filter was applied for OPR [7]. Commonly, they abide by the so-called “constant constraint criterion” (CCC). Tao analyzed CCC in detail and drew the conclusion that the CCC is ill-posed and proposed the use of filters banks [S]. In this paper, we studied the optical pattern recognition from the viewpoint of second-order interpattern association neural network and came to a
XCorresponding author. E-mail: [email protected]
similar conclusion. Moreover, we proposed the concept of an interconnection matrices bank for optical association pattern recognition. High-order neural networks have more interconnections between neurons than the first-order neural networks. But for a long time high-order neural networks were neglected by most researchers owing to the restriction of perceptron [9]. Until 1986, they were revalued and received plenty appreciation for their impressive storage and learning capabilities [lo]. Therefore particular problems, such as geometric invariance, can be encoded in high-order neural networks to eliminate the redundant freedoms and these neural networks are efficient. The interpattern association (IPA) model is a new method to constitute the interconnection matrix [I I]. The second order IPA, which is developed from the IPA model, has been investigated [12-141. On the basis of the second-order IPA, we set up a neural network (we caIled it the second-order semi-complex IPA model) for optical pattern recognition in this paper. This network employs both autoassociation and heteroassociation. The vectors in the network take the complex form, but we assume that the real part of one vector does not affect the imaginary part of the other vector. Similar cases can be seen in optical pattern recognition. Section 2 deals
0030.4018/97/$17,00 Copyright 0 1997 Published by Elsevier Science B.V. All rights reserved. PII SOO30-4018(97)00023-O
183
H. Cheng et al. / Optics Communications 139 (1997) 182-186 with the basic theory of second-order autoassociative and heteroassociative IPA models and the second-order semicomplex IPA model with the consideration of mapping these models to aspects of optical distortion invariant pattern recognition. In Section 3, we gave a pattern addressing example and proposed a possible optical implementation based on vector matrix product (VMP) structure [20] and redundant elimination arrangement. At last, we made an overall analysis of the presented second-order semi-complex IPA model.
2. Basic theory of second order model and computer simulations
semi-complex
IPA
The interpattern association (IPA) can be straightforwardly extended to the second order [12-141. Once one of the stored vectors (patterns) was fed into the association system, the second order autoassociative memories can be addressed by the following expression:
j=l
(1)
k=l
and fi (next state) = 1, = 0,
if fi > 0, (2)
if& < 0,
where {f,}{fk} is the input vector; N is the length of the vectors; {fi> is the recalled vector; Wijk is the second order interconnection matrix. As for the heteroassociation model, its basic idea is using different pairs to constitute the interconnection matrices. Within a single pair we can use a to address b and vice versa. But in most cases the interconnection matrices are different, the reason why will be discussed in the following sections. Let {f,gX be heteroassociation pairs, if we use the quadratic f to address g, we have
gi =
5 Ii K>k.iyk
j=l
(3)
k=l
and gi (next state) = 1, = 0,
ifg,>O, (4)
if gjIO.
The constitute rules of Wijk and W& are different. This can be seen from the following expressions in Eqs. (5) and (6X wiir: = 1 ,
if f
= 0,
=
m=l (6)
-1, =
0,
if E fThm(l-gm)= in= 1
E fjmJrkm#=O m=l
otherwise,
where M is the number of stored patterns. The case Wijk (or W&) = 1 occurs when both the jth pixel and the kth pixel are excited and produce an excitatory signal to the ith pixel activity (fi for the autoassociation, gj for the heteroassociation). The case Wijk (or W/j,> = - 1 occurs when any of the stored patterns that have included both the jth pixel and kth pixel do not have the ith pixel i.e. fj = fk = 1 and fj (or gi> = 0. This case embodies an inhibitory effect. The case Wiik (or Wi’j,) = 0 means that both the jth pixel’s and the kth pixel’s excitation have no relationship with the ith pixel’s excitation. Now we consider the following complex second-order interpattern association case. As we have stated in Section 1, the stored vectors have real part and imaginary part. Both parts take values from {O,l>. For each vector, its real part and imaginary part become a heteroassociation pair {fi,gi}; and for these vectors, they contribute to Wijk with their real or imaginary part in Eq. (5). In order to clarify this point, we choose two such vectors in the following diagram (Fig. 1) to see their relationships. When one complete complex model is input, through the addressing equation both the real part and the imaginary part memorize themselves (autoassociation). If only the real part or the imaginary part is input, it recalls itself by autoassociation and recalls the corresponding imaginary part (or real part> by heteroassociation. Such constraints make the neural network in this paper applicable for pattern recognition. We can map these characteristics described above to optical pattern recognition, for example, let the scale distort the real part and the rotation distort the imaginary part. The scale distortion is along the radius while the rotation distortion is vertical to the radius, therefore we can assume that the scale distortion does not affect the rotation distortion. For a specific object, scale distortion and rotation distortion usually exist simultaneously. We can describe its distortion status with scale distortion and rotation distortion. According to the distortion condition and degree, we can classify scale distortions and rotation distortions into different classes. This work can be done before training neural networks using different samples to form the interconnection matrices. When using optical correlation methods, before the fabrication of
fk"f,fT=2 s,'"fk"+ 0
m=l
= -1,
m=l
if 5 fk”f,?“(l -fim) m=l
m=l
=
5 fjmfrfO ??l=l otherwise,
(5) Fig. 1. The association relationships between two complex vectors.
184
H. Cheng et al. /Optics samplesof
Communications 139 (1997) 182-186
samples of rotation distortion
scale distortion
wijk(s)
Fig. 2. The mapping of auto- and hetero-association to optical pattern recognition, where double arrows mean s has quadratic order weight in Wi;, while Y has order one.
matched filters, this classification is necessary. For optical pattern recognition as discussed above (if we assume having only scale and rotation distortions), the relationship of the Wijk and W;lik can be expressed as in Fig. 2. We can see from Fig. 2 that Wjjk( s> and Wjj,( u) make two autoassociations to represent scale distortion and rotation distortion interconnection matrices respectively. As for the heteroassociative interconnection matrix W/jk in Eq. (6), we can use quadratic order fj for addressing gi. And we can also use quadratic order gi for addressing fi. In the latter case, the interconnection matrix takes the following form: M
1,
w;k=
if C
M
g;g,Ffiim =
,?z=1 =
-1
if t
,
C
gygp
# 0
m=l
grg]y(
1 -AT’) =
m=l
E
3. A pattern addressing example and the architecture for optical implementation
gJFgT # 0
m=l
0.
otherwise.
(7) The corresponding N
N
C
C
j=l
k=l
fi=
example f2 represents fifj. If more factors are needed to be taken into consideration, a bank of such heteroassociation interconnection matrices should be introduced in this neural network to determine different classes. Since the above neural network combines autoassociation and heteroassociation together but it does not abide by the complex multiplication rules, we called it the second-order semi-complex IPA model. A comparison between this second-order semi-complex IPA model and optical correlation matched filter system can be made. For a matched filter correlation optical pattern recognition system, the attempt to synthesize different factors into one matched filter to obtain a constant correlation output in the correlation plane is usually unsuccessful. The reason why this happens has been analyzed by Tao in Ref. 181. Tao’s conclusion is to use a bank of filters instead of a single synthetic filter. Here we put forward the heteroassociation interconnection matrices bank concept to solve heteroassociation problem. The essence of the two banks is the same. In the following section, we propose a scheme for optical pattern recognition based on the second-order semi-complex IPA model we have discussed above.
addressing
In this paper, three complex vectors are used to see what happens when we introduce heteroassociation be-
equation becomes complex
(8)
IY$kgjgk
vectors:
A
realpart:
l,l,l,O
imaginary part:
LO, 1 ,O
B
C
input
l,l,OJ
l,O,Ll
O,O,1 ,O
O,l,O,l
O,O,Ll
LO,l,O
and (8)
fj (next state) = 1 , = 0,
iffj>O,
(9
iffiSOo.
From Eqs. (7), (8) and (9), we can see that for a second order heteroassociative IPA model if we want to address gi we use Eqs. (31, (4) and (6). If we want to address fi we use Eqs. (7), (8) and (9). This difference shows that we cannot add Wiik and WGk together to form a new heteroassociative matrix to addressing both fi and gi for the reason that the heteroassociative interconnection of f&g, will inevitably become a noise source to the heteroassociative interconnection of gigjfk. If the heteroassociative IPA matrix takes symmetric interconnection i.e. W$,, or Wi$tmn etc., where both f and g take the same interconnection weight in W$,, or W$tmn etc., it can be used to address both f and g. In the other words, use f to address g and vice versa with the same interconnection matrices. In most cases the interconnection matrices are not symmetric; so for a single interconnection matrix if gj = CW”f” is a correct
association
Notice
that
then & = CW”g”
the notation
f”
means
is a false association. n-order
weights,
for
autoassociative interconnection matrix Wijk(f, and the addressed real part 1 1 1 1
0100
0010
0001
1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 0 I o-1 01-10
0 0 l-l 1 1 1 1 O-110
00-I 1 o-1 0 1 1 1 1 1
1
0
1
0
hetemassociative interconnection matrix W{k and the addressed real part I-1 l-1 -1 l-l 1 l-1 11
1-l -:.;
-1
-1
1 1 1
l-l ;_; l-1
0
l-l -;;;
1-l -; ;l
-1-l
1 0
-1-l -1 -I -1 l-1 1 -1-l 0 1 -1
11
10
1 1 1
autoassociative interconnection matrix Wfik(g) and the addressed imaginary part l-l l-1 -1 -1 -1 -1 I-1 O-l -I -1-l -1
-1 -I -1-l -1 l-1 1 -1 -1 -I -1 -1 l-l 0
1-l l-l -1 -1 -1-1 l-l 11 -1-1 1 0
-1-l-l-l -1 l-1 1 -1-l 0 1 -1 1 1 1
1
0
10
autoassociative interconnection matix W'ijk and the addressed imagine 0 0 0 -1
0 o-1 0 l-l 1 O-l -1-l -1
lll~‘1’: -1 -1-l 01-10
;I;_; -1
1 1 1 1 O-110
0001 00-l 1 o-1 0 Y 1 1 1 1
0
0
part
10
(b)
Fig. 3. The prestored complex vectors and the second order interconnection matrices; (a) the prestored vectors and the input vector for addressing; (b) the interconnection matrices and corresponding addressed result.
185
H. Cheng et al. / Optics Communications 139 (1997) 182-186
i Wlk
Wbk
(b) Fig. 4. The optical architecture for the second order IPA model with the unipolar (0,l) vectors as the input vectors; (a) the least combination number; (b) the interconnection matrix W,jk corresponding to f( j)f(k) in (a).
tween the real part and the imaginary part along with the autoassociation of the real parts and the imaginary parts. One partial complex vector is used as the input vector and the corresponding address result is given. The result along with the second-order IPA interconnection matrices and the stored vectors is shown in Fig. 3. From Fig. 3 we see that for such an input complex vector {O,O,l,O} + j{ l,O,l,O} the addressed real part by autoassociation and the addressed imaginary part by heteroassociation are incorrect, but the addressed imaginary part by autoassociation and the addressed real part by heteroassociation are correct. Similarly the correct real part input addresses the correct imaginary part by heteroassociation. We also discovered that if the deviation between the input vector and one prestored vector falls in the attraction radius [ 151, the prestored pattern can be correctly addressed. These features are more helpful if we have some prior knowledge about the prestored patterns (for example in Fig. 3 the number of “1” in the real parts is 3 and the number of “1” in the imaginary part is 2), the addressed vector in Fig. 3 is vector A. Owing to the introduction of heteroassociation between the real part and the imaginary part, the pattern recognition probability is greatly increased. Optical implementation of both auto- and hetero-associations have been reported by Yu’s group [11,12]. Several optical architectures have been put forward for implementing high-order neural networks [15-191. The number of
nonredundant weights in the second-order interconnection matrix is mentioned in Ref. [15]. A compact optical architecture for implementing second-order neural networks considering the symmetric characteristic of the interconnection matrix is suggested in Ref. [ 181. Later this architecture is improved by considering redundance elimination in Ref. [13]. For each recalling process we use the improved architecture suggested in Ref. [13] as shown in Fig. 4. For all the four addressing processes (if all the four interconnection matrices W&f), IVJg), IV&, IV&, are taken into consideration) we set up an optical experiment based on the above discussion, the diagram is depicted in Fig. 5. The input vector is put on the front focus plane of the lenslet in Fig. 5 (it comprises three lenses). In this figure,
mask lens 1 ’
mask 2 lenslet
Iphotodetector
source -fl
-t---f2
1
L3p_ Fig. 5. The optical arrangement of the semi-complex second order IPA model; the input vector is encoded on mask 1, the real part and the imaginary part are symmetric; the four interconnection matrices are encoded in the four blocks on mask 2.
186
H. Cheng et al./ Optics Communications
the electronic device for postprocessing and feedback is omitted (in a practical system, it exists). In this section, we proposed one scheme for optical pattern recognition. This scheme takes the following steps: (i> extracting the edges of the input object as the feature to be fed into the optical second-order IPA neural network; (ii) using the above second-order semi-complex IPA neural network for pattern addressing (recalling). The direct result we get is the addressed pattern (a prestored pattern>. Step i can be performed by an optical differential system, Recently, using an optical wavelet system to perform optical differentiation arose great interest and can also be employed in step i. The continuous result image of step i can be discretized as the digital model stored in a intermediate computer. Then the discrete model is fed into the input plane of step ii by the control instructions from the computer. The feedback of the correlation results in that step ii from the correlation plane to the input plane is also realized by this computer. The interconnection matrices are generated based on the pattern classification we have stated. We should have precollected distortion sample object images. These images can be discretized optically or electronically to form the prestored patterns. These patterns are stored in the interconnection matrices. These matrices can be encoded on mask 2 in Fig. 5. The optical system proposed here gives a new idea for optical pattern recognition. It is the cascade of two optical systems corresponding to the two steps in the above discussions, each performs a relative simple task, so that the two-step optical system proposed here is feasible and direct. While in a matched spatial filter optical pattern recognition system, the fabrication of a matched filter synthesizing more factors is time consuming and the recognition result is usually unsatisfactory.
4. Conclusion In this paper, we combined the autoassociation neural network and the heteroassociation neural network for pattern recognition. We proposed a second-order semi-complex interpattern association neural network based on this combination. We also proposed the interconnection matrices bank concept for heteroassociative multi-factor pattern recognition. For a single factor pattern recognition we adopted the autoassociation to address the prestored pattern. The main work aims at second order auto- and hetero-associative interpattern neural networks and can be
139 (1997) 182-186
extended to higher order neural networks. Symmetric interconnection can be taken into consideration. After introducing heteroassociation between the real part and the imaginary part, a correct (or partial within the attraction radius) real or imaginary part can address the complete complex vector correctly. In addition, we studied optical pattern recognition from the viewpoint of the second-order semicomplex IPA neural network and put forward a scheme for optical pattern recognition. An optical architecture for auto- and hetero-association is suggested, and is promising for optical association pattern recognition.
Acknowledgements This work is supported by the National Natural Science Foundation of China under the contract 69477001.
References [ll J.J. Hopfield, Proc. Nat. Acad. Sci. USA 79 (1982) 2554. El D.E. Rumelhart and J.L. McClelland, Parallel distributed processing, Vol. 1 and 2 (MIT Press, Cambridge, 1986). [31 A. VandeLugt, IEEE Trans. Inf. Theory IT-10 (1964) 139. [41 C.F. Hester and D. Casasent, Appl. Optics 19 (1980) 1758. [51 A. Huang, IEEE Tenth Intern. Opt. Comput. Conf. (1983) p. 13. 161 Y.N. Hsu and H.H. Arsennault, Appl. Optics 21 (1982) 4016. [71 Y. Sheng, D. Roberge, H. Szu and T. Lu, Optics Lett. 18 (1993) 299. @I C. Tao, J. Opt. Sot. Am. A 12 (1995) 2114. 191 M.L. Minsky and S. Papert, Perceptrons (MIT Press, Cambridge, 1969). DO1C.L. Giles and T. Maxwell, Appl. Optics 26 (1987) 4972. [111 T. Lu, X. Xu, S. Wu and F.T.S. Yu, Appl. Optics 29 (1990) 284. 1121 F.T.S. Yu, T. Lu and X. Yang, Int. J. Opt. Computing 1 (1990) 129. [131 C. Zhou, L. Liu and Z. Wang, Optics Comm. 94 (1992) 25. 1141 C. Zhou, L. Liu and Z. Wang, Chinese J. Lasers B 1 (1992) 6. [151 D. Psaltis and C.H. Park, AR’ Conf. Proc. 151 (1986) 370. [161 S. Lin and L. Liu, Optics Comm. 73 (1989) 268. [171 J. Jang, S.Y. Shin and S. Lee, Optics Lett. 14 (1989) 838. [HI S. Lin, J. Wu and L. Liu, Micro. Opt. Tech. Lett. 2 (1989) 304. [191 L. Zhang, M.G. Robinson and K.M. Johnson, Optics Lett. 16 (1991) 45. 1201 C. Zhou, L. Liu and Z. Wang, Optics Lett. 17 (1992) 1800.
Optics Communications 139 (1997) 182-186
Second-order interpattern neural networks for optical pattern recognition Huiquan Cheng *, Liren Liu, Guoqiang Li, Lan Shao, Changhe Zhou Irzformation Optics Laboratory, Slwznghai hzstitute of Optics and Fine Mechanics, Academia Sinica, P.O. Box 800-211 Shanghai 201800, China
Received 4 July 1996; accepted 18 December 1996
Abstract By combining the autoassociative model with the heteroassociative model, we propose a second-order semi-complex IPA neural network. In this neural network, the real part or the imaginary part of a complete or partial complex vector not only addresses the corresponding part of the complex vector prestored in the neural network by autoassociation, but also addresses the corresponding imaginary or real part by heteroassociation. Owing to the introduction of the heteroassociation, the correct addressing probability is increased. Based on this neural network, a scheme for optical pattern recognition (OPR) is proposed. This scheme is composed of two cascaded optical systems; each performs a relative simple function. Compared with the matched spatial filter OPR system, the addressing results are direct and our scheme is feasible and promising.
1. Introduction Neural networks have received extensive investigations and have been applied to different fields for their potential superior capabilities [ 1,2]. Pattern recognition, which involves many factors and is difficult to handle, has been closely connected with neural networks for smart decision. Optical pattern recognition (OPR) based on optical correlation has long been investigated since VandeLugt proposed the matched filter concept [3]. There are several methods such as synthetic discriminant function [4], symbolic substitution [5], circular harmonic expansion [6], etc. Recently owing to the advantage of windowed Fourier transform, a wavelet matched filter was applied for OPR [7]. Commonly, they abide by the so-called “constant constraint criterion” (CCC). Tao analyzed CCC in detail and drew the conclusion that the CCC is ill-posed and proposed the use of filters banks [S]. In this paper, we studied the optical pattern recognition from the viewpoint of second-order interpattern association neural network and came to a
XCorresponding author. E-mail: [email protected]
similar conclusion. Moreover, we proposed the concept of an interconnection matrices bank for optical association pattern recognition. High-order neural networks have more interconnections between neurons than the first-order neural networks. But for a long time high-order neural networks were neglected by most researchers owing to the restriction of perceptron [9]. Until 1986, they were revalued and received plenty appreciation for their impressive storage and learning capabilities [lo]. Therefore particular problems, such as geometric invariance, can be encoded in high-order neural networks to eliminate the redundant freedoms and these neural networks are efficient. The interpattern association (IPA) model is a new method to constitute the interconnection matrix [I I]. The second order IPA, which is developed from the IPA model, has been investigated [12-141. On the basis of the second-order IPA, we set up a neural network (we caIled it the second-order semi-complex IPA model) for optical pattern recognition in this paper. This network employs both autoassociation and heteroassociation. The vectors in the network take the complex form, but we assume that the real part of one vector does not affect the imaginary part of the other vector. Similar cases can be seen in optical pattern recognition. Section 2 deals
0030.4018/97/$17,00 Copyright 0 1997 Published by Elsevier Science B.V. All rights reserved. PII SOO30-4018(97)00023-O
183
H. Cheng et al. / Optics Communications 139 (1997) 182-186 with the basic theory of second-order autoassociative and heteroassociative IPA models and the second-order semicomplex IPA model with the consideration of mapping these models to aspects of optical distortion invariant pattern recognition. In Section 3, we gave a pattern addressing example and proposed a possible optical implementation based on vector matrix product (VMP) structure [20] and redundant elimination arrangement. At last, we made an overall analysis of the presented second-order semi-complex IPA model.
2. Basic theory of second order model and computer simulations
semi-complex
IPA
The interpattern association (IPA) can be straightforwardly extended to the second order [12-141. Once one of the stored vectors (patterns) was fed into the association system, the second order autoassociative memories can be addressed by the following expression:
j=l
(1)
k=l
and fi (next state) = 1, = 0,
if fi > 0, (2)
if& < 0,
where {f,}{fk} is the input vector; N is the length of the vectors; {fi> is the recalled vector; Wijk is the second order interconnection matrix. As for the heteroassociation model, its basic idea is using different pairs to constitute the interconnection matrices. Within a single pair we can use a to address b and vice versa. But in most cases the interconnection matrices are different, the reason why will be discussed in the following sections. Let {f,gX be heteroassociation pairs, if we use the quadratic f to address g, we have
gi =
5 Ii K>k.iyk
j=l
(3)
k=l
and gi (next state) = 1, = 0,
ifg,>O, (4)
if gjIO.
The constitute rules of Wijk and W& are different. This can be seen from the following expressions in Eqs. (5) and (6X wiir: = 1 ,
if f
= 0,
=
m=l (6)
-1, =
0,
if E fThm(l-gm)= in= 1
E fjmJrkm#=O m=l
otherwise,
where M is the number of stored patterns. The case Wijk (or W&) = 1 occurs when both the jth pixel and the kth pixel are excited and produce an excitatory signal to the ith pixel activity (fi for the autoassociation, gj for the heteroassociation). The case Wijk (or W/j,> = - 1 occurs when any of the stored patterns that have included both the jth pixel and kth pixel do not have the ith pixel i.e. fj = fk = 1 and fj (or gi> = 0. This case embodies an inhibitory effect. The case Wiik (or Wi’j,) = 0 means that both the jth pixel’s and the kth pixel’s excitation have no relationship with the ith pixel’s excitation. Now we consider the following complex second-order interpattern association case. As we have stated in Section 1, the stored vectors have real part and imaginary part. Both parts take values from {O,l>. For each vector, its real part and imaginary part become a heteroassociation pair {fi,gi}; and for these vectors, they contribute to Wijk with their real or imaginary part in Eq. (5). In order to clarify this point, we choose two such vectors in the following diagram (Fig. 1) to see their relationships. When one complete complex model is input, through the addressing equation both the real part and the imaginary part memorize themselves (autoassociation). If only the real part or the imaginary part is input, it recalls itself by autoassociation and recalls the corresponding imaginary part (or real part> by heteroassociation. Such constraints make the neural network in this paper applicable for pattern recognition. We can map these characteristics described above to optical pattern recognition, for example, let the scale distort the real part and the rotation distort the imaginary part. The scale distortion is along the radius while the rotation distortion is vertical to the radius, therefore we can assume that the scale distortion does not affect the rotation distortion. For a specific object, scale distortion and rotation distortion usually exist simultaneously. We can describe its distortion status with scale distortion and rotation distortion. According to the distortion condition and degree, we can classify scale distortions and rotation distortions into different classes. This work can be done before training neural networks using different samples to form the interconnection matrices. When using optical correlation methods, before the fabrication of
fk"f,fT=2 s,'"fk"+ 0
m=l
= -1,
m=l
if 5 fk”f,?“(l -fim) m=l
m=l
=
5 fjmfrfO ??l=l otherwise,
(5) Fig. 1. The association relationships between two complex vectors.
184
H. Cheng et al. /Optics samplesof
Communications 139 (1997) 182-186
samples of rotation distortion
scale distortion
wijk(s)
Fig. 2. The mapping of auto- and hetero-association to optical pattern recognition, where double arrows mean s has quadratic order weight in Wi;, while Y has order one.
matched filters, this classification is necessary. For optical pattern recognition as discussed above (if we assume having only scale and rotation distortions), the relationship of the Wijk and W;lik can be expressed as in Fig. 2. We can see from Fig. 2 that Wjjk( s> and Wjj,( u) make two autoassociations to represent scale distortion and rotation distortion interconnection matrices respectively. As for the heteroassociative interconnection matrix W/jk in Eq. (6), we can use quadratic order fj for addressing gi. And we can also use quadratic order gi for addressing fi. In the latter case, the interconnection matrix takes the following form: M
1,
w;k=
if C
M
g;g,Ffiim =
,?z=1 =
-1
if t
,
C
gygp
# 0
m=l
grg]y(
1 -AT’) =
m=l
E
3. A pattern addressing example and the architecture for optical implementation
gJFgT # 0
m=l
0.
otherwise.
(7) The corresponding N
N
C
C
j=l
k=l
fi=
example f2 represents fifj. If more factors are needed to be taken into consideration, a bank of such heteroassociation interconnection matrices should be introduced in this neural network to determine different classes. Since the above neural network combines autoassociation and heteroassociation together but it does not abide by the complex multiplication rules, we called it the second-order semi-complex IPA model. A comparison between this second-order semi-complex IPA model and optical correlation matched filter system can be made. For a matched filter correlation optical pattern recognition system, the attempt to synthesize different factors into one matched filter to obtain a constant correlation output in the correlation plane is usually unsuccessful. The reason why this happens has been analyzed by Tao in Ref. 181. Tao’s conclusion is to use a bank of filters instead of a single synthetic filter. Here we put forward the heteroassociation interconnection matrices bank concept to solve heteroassociation problem. The essence of the two banks is the same. In the following section, we propose a scheme for optical pattern recognition based on the second-order semi-complex IPA model we have discussed above.
addressing
In this paper, three complex vectors are used to see what happens when we introduce heteroassociation be-
equation becomes complex
(8)
IY$kgjgk
vectors:
A
realpart:
l,l,l,O
imaginary part:
LO, 1 ,O
B
C
input
l,l,OJ
l,O,Ll
O,O,1 ,O
O,l,O,l
O,O,Ll
LO,l,O
and (8)
fj (next state) = 1 , = 0,
iffj>O,
(9
iffiSOo.
From Eqs. (7), (8) and (9), we can see that for a second order heteroassociative IPA model if we want to address gi we use Eqs. (31, (4) and (6). If we want to address fi we use Eqs. (7), (8) and (9). This difference shows that we cannot add Wiik and WGk together to form a new heteroassociative matrix to addressing both fi and gi for the reason that the heteroassociative interconnection of f&g, will inevitably become a noise source to the heteroassociative interconnection of gigjfk. If the heteroassociative IPA matrix takes symmetric interconnection i.e. W$,, or Wi$tmn etc., where both f and g take the same interconnection weight in W$,, or W$tmn etc., it can be used to address both f and g. In the other words, use f to address g and vice versa with the same interconnection matrices. In most cases the interconnection matrices are not symmetric; so for a single interconnection matrix if gj = CW”f” is a correct
association
Notice
that
then & = CW”g”
the notation
f”
means
is a false association. n-order
weights,
for
autoassociative interconnection matrix Wijk(f, and the addressed real part 1 1 1 1
0100
0010
0001
1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 0 I o-1 01-10
0 0 l-l 1 1 1 1 O-110
00-I 1 o-1 0 1 1 1 1 1
1
0
1
0
hetemassociative interconnection matrix W{k and the addressed real part I-1 l-1 -1 l-l 1 l-1 11
1-l -:.;
-1
-1
1 1 1
l-l ;_; l-1
0
l-l -;;;
1-l -; ;l
-1-l
1 0
-1-l -1 -I -1 l-1 1 -1-l 0 1 -1
11
10
1 1 1
autoassociative interconnection matrix Wfik(g) and the addressed imaginary part l-l l-1 -1 -1 -1 -1 I-1 O-l -I -1-l -1
-1 -I -1-l -1 l-1 1 -1 -1 -I -1 -1 l-l 0
1-l l-l -1 -1 -1-1 l-l 11 -1-1 1 0
-1-l-l-l -1 l-1 1 -1-l 0 1 -1 1 1 1
1
0
10
autoassociative interconnection matix W'ijk and the addressed imagine 0 0 0 -1
0 o-1 0 l-l 1 O-l -1-l -1
lll~‘1’: -1 -1-l 01-10
;I;_; -1
1 1 1 1 O-110
0001 00-l 1 o-1 0 Y 1 1 1 1
0
0
part
10
(b)
Fig. 3. The prestored complex vectors and the second order interconnection matrices; (a) the prestored vectors and the input vector for addressing; (b) the interconnection matrices and corresponding addressed result.
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H. Cheng et al. / Optics Communications 139 (1997) 182-186
i Wlk
Wbk
(b) Fig. 4. The optical architecture for the second order IPA model with the unipolar (0,l) vectors as the input vectors; (a) the least combination number; (b) the interconnection matrix W,jk corresponding to f( j)f(k) in (a).
tween the real part and the imaginary part along with the autoassociation of the real parts and the imaginary parts. One partial complex vector is used as the input vector and the corresponding address result is given. The result along with the second-order IPA interconnection matrices and the stored vectors is shown in Fig. 3. From Fig. 3 we see that for such an input complex vector {O,O,l,O} + j{ l,O,l,O} the addressed real part by autoassociation and the addressed imaginary part by heteroassociation are incorrect, but the addressed imaginary part by autoassociation and the addressed real part by heteroassociation are correct. Similarly the correct real part input addresses the correct imaginary part by heteroassociation. We also discovered that if the deviation between the input vector and one prestored vector falls in the attraction radius [ 151, the prestored pattern can be correctly addressed. These features are more helpful if we have some prior knowledge about the prestored patterns (for example in Fig. 3 the number of “1” in the real parts is 3 and the number of “1” in the imaginary part is 2), the addressed vector in Fig. 3 is vector A. Owing to the introduction of heteroassociation between the real part and the imaginary part, the pattern recognition probability is greatly increased. Optical implementation of both auto- and hetero-associations have been reported by Yu’s group [11,12]. Several optical architectures have been put forward for implementing high-order neural networks [15-191. The number of
nonredundant weights in the second-order interconnection matrix is mentioned in Ref. [15]. A compact optical architecture for implementing second-order neural networks considering the symmetric characteristic of the interconnection matrix is suggested in Ref. [ 181. Later this architecture is improved by considering redundance elimination in Ref. [13]. For each recalling process we use the improved architecture suggested in Ref. [13] as shown in Fig. 4. For all the four addressing processes (if all the four interconnection matrices W&f), IVJg), IV&, IV&, are taken into consideration) we set up an optical experiment based on the above discussion, the diagram is depicted in Fig. 5. The input vector is put on the front focus plane of the lenslet in Fig. 5 (it comprises three lenses). In this figure,
mask lens 1 ’
mask 2 lenslet
Iphotodetector
source -fl
-t---f2
1
L3p_ Fig. 5. The optical arrangement of the semi-complex second order IPA model; the input vector is encoded on mask 1, the real part and the imaginary part are symmetric; the four interconnection matrices are encoded in the four blocks on mask 2.
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H. Cheng et al./ Optics Communications
the electronic device for postprocessing and feedback is omitted (in a practical system, it exists). In this section, we proposed one scheme for optical pattern recognition. This scheme takes the following steps: (i> extracting the edges of the input object as the feature to be fed into the optical second-order IPA neural network; (ii) using the above second-order semi-complex IPA neural network for pattern addressing (recalling). The direct result we get is the addressed pattern (a prestored pattern>. Step i can be performed by an optical differential system, Recently, using an optical wavelet system to perform optical differentiation arose great interest and can also be employed in step i. The continuous result image of step i can be discretized as the digital model stored in a intermediate computer. Then the discrete model is fed into the input plane of step ii by the control instructions from the computer. The feedback of the correlation results in that step ii from the correlation plane to the input plane is also realized by this computer. The interconnection matrices are generated based on the pattern classification we have stated. We should have precollected distortion sample object images. These images can be discretized optically or electronically to form the prestored patterns. These patterns are stored in the interconnection matrices. These matrices can be encoded on mask 2 in Fig. 5. The optical system proposed here gives a new idea for optical pattern recognition. It is the cascade of two optical systems corresponding to the two steps in the above discussions, each performs a relative simple task, so that the two-step optical system proposed here is feasible and direct. While in a matched spatial filter optical pattern recognition system, the fabrication of a matched filter synthesizing more factors is time consuming and the recognition result is usually unsatisfactory.
4. Conclusion In this paper, we combined the autoassociation neural network and the heteroassociation neural network for pattern recognition. We proposed a second-order semi-complex interpattern association neural network based on this combination. We also proposed the interconnection matrices bank concept for heteroassociative multi-factor pattern recognition. For a single factor pattern recognition we adopted the autoassociation to address the prestored pattern. The main work aims at second order auto- and hetero-associative interpattern neural networks and can be
139 (1997) 182-186
extended to higher order neural networks. Symmetric interconnection can be taken into consideration. After introducing heteroassociation between the real part and the imaginary part, a correct (or partial within the attraction radius) real or imaginary part can address the complete complex vector correctly. In addition, we studied optical pattern recognition from the viewpoint of the second-order semicomplex IPA neural network and put forward a scheme for optical pattern recognition. An optical architecture for auto- and hetero-association is suggested, and is promising for optical association pattern recognition.
Acknowledgements This work is supported by the National Natural Science Foundation of China under the contract 69477001.
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