the differences between cellular neural network based and fuzzy cellular nneural network based...

Correspondence to: T. Yang, Electronics Research Laboratory and Department of Electrical Engineering and Computer Sciences,University of California at Berkeley, Berkeley, CA 94720, U.S.A.

CCC 0098—9886/98/010013—13$17.50 Received 14 June 1996( 1998 John Wiley & Sons, Ltd. Revised 3 February 1997

INTERNATIONAL JOURNAL OF CIRCUIT THEORY AND APPLICATIONSInt. J. Circ. ¹heor. Appl., 26, 13—25 (1998)

THE DIFFERENCES BETWEEN CELLULAR NEURAL NETWORKBASED AND FUZZY CELLULAR NEURAL NETWORK BASED

MATHEMATICAL MORPHOLOGICAL OPERATIONS

TAO YANG1,*, CHUN-MEI YANG2, LIN-BAO YANG3

1 Electronics Research Laboratory and Department of Electrical Engineering and Computer Sciences,University of California at Berkeley, Berkeley, CA 94720, U.S.A.

2 China Construction Bank Songjiang Sub-branch, Shanghai 201611, P.R. China3 University of E-Zhou, E-Zhou, Hubei 436000, P.R. China

SUMMARY

In this paper, the differences between cellular neural network (CNN)-based and fuzzy CNN (FCNN)-based grey-scalemathematical morphological operations are presented. The performances of the CNN-based mathematical morphologi-cal operations are analyzed. The stability and the basin of attraction of conventional CNN-based grey-scale erosion andgrey-scale dilation are studied. We find that the conventional CNN-based grey-scale erosion and grey-scale dilation canintroduce some time varying and ‘random’ errors in their outputs. For comparison, the performances of the FCNN-based grey-scale mathematical morphological operations are also presented. We find that the FCNN-based erosion anddilation can give error-free results. Simulation results are given. ( 1998 John Wiley & Sons, Ltd.

KEY WORDS: CNN; fuzzy CNN; FCNN; erosion; dilation; morphology

1. INTRODUCTION

Mathematical morphology1,2 is a very effective tool for image processing and pattern recognition. Onedisadvantage of mathematical morphology is the high computational complexity when the images or/and thestructuring elements are of large sizes. One approach to overcome this disadvantage is to map themorphological operations into some parallel computational arrays. Cellular neural networks (CNN)3 arevery ideal for implementing the mathematical morphological operations because the local connectedness ofcells.

To implement binary mathematical morphological operations, there are three kinds of CNN structurescan be used: namely, discrete-time CNN (DTCNN),4 conventional CNN3 and fuzzy CNN(FCNN).5~10 TheDTCNN can be used because we can decompose the binary erosion and dilation with a threshold in interval(0, 1).11 The conventional CNN can be used because the positive and the negative saturation operationregions in the standard cell12 can be used to represent binary output. The FCNN5~10 can be used becausethe equivalence between OR(AND) and FUZZY OR (FUZZY AND) in boolean set.5,6 In a DTCNNstructure, a single template can be used to implement both erosion and dilation while the input function andthe output function are switched between two configurations.11 This structure is very ideal for VLSIimplementation with fixed parameters. On the other hand, the conventional CNN structure needs two sets oftemplates to implement erosion and dilation, respectively. Since an analog CNN chip whose templates can bedynamically programmed is far away from practical applications, the DTCNN may be the only CNNimplementation of binary mathematical morphological operations in the near future.

To implement the grey-scale mathematical morphological operations, both conventional CNN andFCNN can be used. When the conventional CNN-based methods12 are used, each synaptic law isa threshold-type non-linear function whose argument is the difference between the output and the input. Thethreshold, which represents the grey value of an entry of the structuring element, is programmable. When theFCNN-based methods5,6 are used, each cell only needs a local min (or a local max) operation. The FCNNimplementation is much simpler than that of the conventional CNN. For example, if a 3]3 structuringelement is used, then a cell in a conventional CNN-based erosion (dilation) has nine threshold-typenon-linear synaptic laws and other nine minus operations. Since the conventional CNN structure proposedin Reference 12 is structurally unstable, there exist some problems in its VLSI implementation. Furthermore,this structure is very sensitive to initial conditions and it usually provides unstable and error results. TheFCNN implementation is globally and asymptotically stable and error-free. In this paper, we show that theFCNN-based mathematical morphologic operations are superior to the conventional CNN-based ones.

The organization of this paper is as follows. In Section 2, the stability and the basin of attraction of theconventional CNN-based erosion and dilation are given. In Section 3, the performances of the FCNN-basederosion and dilation are presented. In Section 4, the conclusions are contained.

2. PERFORMANCES OF THE CONVENTIONAL CNN-BASEDMORPHOLOGICAL OPERATIONS

There are two basic operations in mathematical morphology:1,2 erosion and dilation. The basic idea ofmathematical morphology is to probe an image with a structuring element, which is often very small (typicallyof 3]3 or 5]5 sizes), and to quantify the manner in which the structuring element fits (or does not fit) withinthe image.

Let f : X>E denote a grey-scale image and s :S>E denote a grey-scale structuring element, where E isthe range of grey values. X is the domain of the grey-scale image, S is the domain of the grey-scale structuringelement. In a 2-D digital image, the set X and S are subsets of Z2. Then the basic morphological operations oferosion and dilation for grey-scale images are given by

Grey-scale erosion:

X>S"minM f (x#z)!s(z)N (1)

for all z3S and x#z3X. Since z and x are two points on Z2 lattice, we can also denote them by two vectorsin Z2. In this sense, x#z denotes a translation of the point x by the vector z. For example, let x"(i, j) andz"(k, l), then we have x#z"(i#k, j#l ), where i, j, k, l3Z.

Grey-scale dilation:

X=S"maxM f (x!z)#s (z)N (2)

for all z3S and x!z3X. Similarly, x!z denotes a translation of the point x by the vector !z. If we alsochoose x"(i, j ) and z"(k, l), where i, j, k, l3Z, then we have (x!z)"(i!k, j!l ).

The elementary unit in a CNN is called cell. In an M]N 2-D CNN, we use Cij

to denote the cell located inthe ith row and the jth column. Then, an elementary definition is used to describe the local connectedness ofCNN.

Definition 1 (r-neighbourhood3). The r-neighbourhood of a cell Cij, in an M]N CNN is defined by

Nr(i, j )"MC

klDmax( Dk!i D, D l!j D ))r, 1)k)M, 1)l)NN

where r is a positive integer.

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14 T. YANG E¹ A¸.

Int. J. Circ. ¹heor. Appl., 26, 13—25 (1998) ( 1998 John Wiley & Sons, Ltd.

In Reference 12, the authors proposed two conventional CNN structures to implement the basic grey-scalemorphological operations. The conventional CNN-based erosion proposed in Reference 12 is given by

State equation:

xRij"!x

ij#y

ij# +

Ckl|Nr (i, j)

DEkl(u

kl!x

ij)#1, x

ij(0)"!1 (3)

where uij, x

ijand y

ijare input, state variable and output of C

ij, respectively. x

ij(0) is the initial condition of C

ij,

and the function DEkl( · ) is given by

DEkl(x)"G

0, x'Sij(kl)

!1, x)Sij(kl)

0, Sij(kl) undefined

(4)

where Sij(kl) is defined by

Sij(kl)"s(k!i, l!j), (k!i, l!j )3S (5)

The output yij

in Equation (3) is given by the following output equation:

yij"1

2( Dx

ij#1 D!Dx

ij!1 D ) (6)

The conventional CNN-based dilation is given by the following state equation:

xRij"!x

ij#y

ij# +

Ckl|Nr (i, j)

DDkl(u

kl!x

ij)!1, x

ij(0)"1 (7)

where the function DDkl( · ) is given by

DDkl(x)"G

1, x*S*ij(kl)

0, x(S*ij(kl)

0, S*ij(kl) undefined

(8)

where S*ij(kl) is defined by

S*ij(kl)"s (i!k, j!l), (i!k, j!l)3S (9)

The output equation is the same as that in equation (6). Without loss of generality, in this paper we only study0 height flat structuring element of size 3]3, i.e.

0 0 0

0 0 0

0 0 0

(10)

then the erosion CNN in Equation (3) can be rewritten as

xRij"!x

ij#y

ij# +

Ckl|N1(i,j)

dE(u

kl!x

ij)#1 (11)

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CNN AND FCNN-BASED MATHEMATICAL MORPHOLOGICAL OPERATIONS 15

( 1998 John Wiley & Sons, Ltd. Int. J. Circ. ¹heor. Appl., 26, 13—25 (1998)

where the function dE( · ) is given by

dE(x)"G

0,

!1,

x'0

x)0(12)

Suppose that the input uij

is normalized in the interval (!1, 1), then the equilibrium point of equation (11)should satisfy

+Ckl|N1 (i,j)

dE(u

kl!x

ij)"!1 (13)

We sort all of the inputs ukl3N

1(i, j) in a non-increasing order as Mu

1, u

2,2, u

8, u

9N, and suppose that

u1Ou

2is satisfied. To study the stability of the equilibrium point of equation (11), we have the following

theorem.

¹heorem 1. Suppose that u1Ou

2is satisfied, then u

1is an asymptotically stable equilibrium point in the basin

of attraction (!R, u1).

Proof. We can see that if u1Ou

2, then x*

ij"u

1is a solution of equation (13), namely, u

1is an equilibrium

point of equation (11). We construct the following Lyapunov function:

»ij(t)"1

2(x

ij!u

1)2'0 (14)

where xij3 (!R, u

1). Taking the derivatives of »

ij(t) along the solutions of equation (11), we have

d»ij(t)

dt K%26!5*0/ (11)

"(xij!u

1)xR

ij

"(xij!u

1)A!x

ij#y

ij# +

Ckl|N1(i,j)

dE(u

kl!x

ij)#1B

"(xij!u

1) (!x

ij#y

ij#1)(0 (15)

The last inequality is in view of xij(u

1. Since x

ij(u

1(1, we have (!x

ij#y

ij#1)'0. K

When u1Ou

2, any x

ij3[u

1, u

2) is an equilibrium point of equation (11). If u

2!u

1is very big, then any

noise in a cell circuit can make a cell output any value in [u1, u

2). The performance of this circuit is not good.

Furthermore, this circuit is structurally unstable. In a VLSI implementation, a very small positive error in thefeedback weight can eventually blow up the state.

When u1"u

2"2"u

i, i)9, the cell in equation (11) has no equilibrium point. In fact, the cell will

fluctuate around an m-equilibrium point,13 which will stay at u1

with a probability of zero measurement.When the structuring element in equation (10) is used, the dilation CNN in equation (7) can be rewritten as

xRij"!x

ij#y

ij# +

Ckl|N1(i,j)

dD(u

kl!x

ij)!1 (16)

where the function dD( ·) is given by

dD(x)"G

1,

0,

x*0

x(0(17)

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16 T. YANG E¹ A¸.


Suppose that the input uij

is normalized in (!1, 1), then the equilibrium point of equation (16) should satisfy

+Ckl|N1 (i,j)

dD(u

kl!x

ij)"1 (18)

To study the stability of the equilibrium point of equation (16), we have the following theorem.

¹heorem 2. Suppose that u8Ou

9is satisfied, then u

9is an asymptotically stable equilibrium point of equation

(16) in the basin of attraction (u9, #R).

Proof. We can see that if u8Ou

9, then x*

ij"u

9is a solution of equation (18), namely, u

9is an equilibrium

point of equation (16). We construct the following Lyapunov function:

»ij(t)"1

2(x

ij!u

9)2'0 (19)

where xij3 (u

9, #R). Taking the derivatives of »

ij(t) along the solutions of equation (16), we have

d»ij(t)

dt K%26!5*0/ (16)

"(xij!u

9)xR

ij

"(xij!u

9)A!x

ij#y

ij# +

Ckl|N1(i,j)

dD(u

kl!x

ij)!1B

"(xij!u

9) (!x

ij#y

ij!1)(0 (20)

The last inequality is in view of xij'u

9. Since x

ij'u

9'!1, we have (!x

ij#y

ij#1)(0. K

If u8Ou

9, then any x

ij3 (u

8, u

9] is an equilibrium point of equation (16). When u

9"u

8"2"u

j, j*1,

the cell in equation (16) has no equilibrium point.To show the output errors of the above conventional CNN-based erosion and dilation, we provide some

simulation results, Figure 1(a) shows an artificial image of size 20]20 with 256 grey levels. In this simulation,we normalize the input image in interval (!1, 1). The 0 V voltage corresponds to the 128th grey level, and thespan between two consecutive grey-levels is represented by a voltage of about 8 mV. We can see that with thestructuring element in equation (10), the erosion makes all the pixels have the 128th grey level. The dilationwill make all the pixels in the centre region have the 140th grey level and make all the pixels in the outerregion have the 128th grey level. Figure 1(b) shows the output result of CNN in equation (3) at instant t"4 s.The initial state is !1. We can see that this output is totally different from the standard erosion result whichshould be a flat surface with height of 128. Since the CNN in equation (3) is unstable, it will oscillate for ever.To demonstrate this, we show the snapshot at instant t"4·3 s in Figure 1(c). We can see that Figure 1(c) istotally different from Figure 1(b) and there exist more errors in Figure 1(c). From above we known that theerosion results given by the CNN in equation (3) are ‘random’ images with uncontrollable and unpredictableerrors which are decided by the input images and the stop moments.

The same can be addressed to the dilation CNN in equation (7). Figure 1(d) shows the output of CNN inequation (7) at instant t"4 s. The initial state is 1. Figure 1(e) shows the output result at instant t"4·5 s. Wecan find that it is hard to believe the output results of the CNN in equation (7) because the results in Figure1(d) and 1(e) share no similarity. In the above simulations, the fourth-order Runge—Kuta method with stepsize 0·01 is used.

When the input image is a real image, it is very hard to predict and control the errors in the output resultsof the CNNs in equations (3) and (7). The simulation results are shown in Figure 2. Figure 2(a) shows a faceimage of a Chinese girl of size 63]63 with 256 grey levels. The structuring element in equation (10) is used.

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Fig. 1. The output results of the conventional CNN-based mathematical morphological operations proposed in Reference 12: (a) anartificial image; (b) the output of the erosion CNN in equation (3) at instant t"4 s; (c) the output of the erosion CNN in equation (3) atinstant t"4·3 s; (d) the output of the dilation CNN in equation (7) at instant t"4 s; (e) the output of the dilation CNN in equation (7) at

instant t"4·5 s

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18 T. YANG E¹ A¸.


Figure 1 (Continued)

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*An FCNN is called multiplicative if it has multiplicative fuzzy synaptic laws. An FCNN is called additive if it has additive fuzzysynaptic laws.

The difference between the output of the CNN in equation (3) and the result of the standard erosion at instantt"4 s is shown in Figure 2(b). We can see that there exists lots of errors. The largest error is equal to !5grey levels. The initial state is !1. The difference between the output of the CNN in equation (7) and theresult of standard dilation at instant t"4 s is shown in Figure 2(c). We can see that there exists lots of errors.The largest error is equal to 5 grey levels. The initial state is 1. In the above simulations, the fourth-orderRunge—Kutta method with step size 0·01 is used.

3. PERFORMANCES OF THE FCNN-BASED MORPHOLOGICAL OPERATIONS

In References 5 and 6, a unified framework for implementing mathematical morphological operations usingFCNN is proposed. The following multiplicative FCNN* is used to implement a morphological operatorwith a flat structuring element:

xRij"!x

ij# +

Ckl|Nr (i,j)

A(i, j ; k, l)ykl# +

Ckl|Nr (i,j)

B (i, j ; k, l )ukl#I (21)

#

&

§Ckl|Nr (i,j)

Af.*/

(i, j ; k, l)ykl#

&

¨Ckl|Nr (i,j)

Af.!9

(i, j ; k, l )ykl

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20 T. YANG E¹ A¸.


Fig. 2. The output results of conventional CNN-based mathematical morphological operations proposed in Reference 12: (a) thegrey-scale image of a Chinese girl (the second author); (b) the difference between the output of the CNN in equation (3) and the result ofthe standard erosion at instant t"4 s. (c) the difference between the output of CNN in equation (7) and the result of the standard

dilation at instant t"4 s

#

&

§Ckl|Nr (i,j)

Bf.*/

(i, j ; k, l )ukl#

&

¨Ckl|Nr (i,j)

Bf.!9

(i, j ; k, l )ukl

where Af.*/

(i, j; k, l), Af.!9

(i, j; k, l), Bf.*/

(i, j; k, l ) and Bf.!9

(i, j; k, l), are entries of fuzzy feedback MINtemplate, fuzzy feedback MAX template, fuzzy feed-forward MIN template, and fuzzy feedforward MAXtemplate, respectively. A(i, j ; k, l ) and B (i, j ; k, l ) are entries of feedback template and feedforward template,respectively, I is the bias, §3 and 3̈ denote fuzzy AND and fuzzy OR, respectively. In this paper, we let§3 "min and 3̈ "max.

The parameters of the multiplicative FCNN for implementing erosion with a flat structuring element aregiven by

A"0, B"0, I"!h, Af.*/

"0, Af.!9

"0, Bf.!9

"0, Bf.*/

"S (22)

where h is the height of the flat structuring element. Since S is the domain set of the grey-scale structuringelement, it should be noted that if s(i, j)"h then S (i, j)¢1 and if s(i, j ) is undefined then S (i, j ) is alsoundefined and then the corresponding B

f.*/synaptic weight is non-existent.

The parameters of the multiplicative FCNN for implementing dilation with a flat structuring element aregiven by

A"0, B"0, I"h, Af.*/

"0, Af.!9

"0, Bf.!9

"S*, Bf.*/

"0 (23)

where S*"M!x Dx3SN is the reflected set of S with respect to the origin of S. Also, if s* (i, j )"h thenS*(i, j )¢1 and if s* (i, j ) is undefined then S*(i, j ) is also undefined and then the corresponding B

f.!9synaptic

weight is non-existent.

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22 T. YANG E¹ A¸.


Fig. 3. The output results of FCNN-based mathematical morphological operations proposed in References 4 and 5: (a) the output of theerosion FCNN in equation (27) at instant t"3 s: (b) the output of the dilation FCNN in equation (28) at instant t"3 s

The following additive FCNN can be used to implement erosion and dilation with any kind of structuringelement:

xRij"!x

ij# +

Ckl|Nr (i, j)

A(i, j ; k, l)ykl# +

Ckl|Nr (i,j)

B(i, j ; k, l)ukl#I (24)

#

&

§Ckl|Nr (i,j)

(Bf.*/

(i, j ; k, l )#ukl)#

&

¨Ckl|Nr (i,j)

(Bf.!9

(i, j ; k, l)#ukl)

The parameters of an additive FCNN for implementing erosion are given by

A"0, B"0, I"0, Bf.*/

"!s (25)

where s denotes the grey value set which is defined in the domain, S, of the structuring element. In thisstructure we do not need the B

f.!9template.

The parameters of an additive FCNN for implementing dilation are given by

A"0, B"0, I"0, Bf.!9

"s* (26)

where s* denotes the grey value set defined in set S*. Similarly, in this structure we do not need the Bf.*/

template.In this paper, we only discuss the performances of the additive FCNN-based mathematical morphology

operations because the additive FCNN is more unified in implementing mathematical morphology opera-tions than the multiplicative FCNN. When the structuring element is of size 3]3, we can use the following

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additive FCNN to implement erosion:

xRij"!x

ij# min

Ckl|N1(i,j)(u

kl!S

ij(kl)) (27)

where Sij(kl) is defined in equation (5).

The output equation is the same as that in equation (6), and the following additive FCNN is used toimplement dilation:

xRij"!x

ij# max

Ckl|N1 (i, j)(u

kl!S*

ij(kl)) (28)

where S*ij(kl) is defined in equation (9).

The output equation is the same as that in equation (6). The above two FCNNs are globally andasymptotically stable. On the other hand, a local max/min operation is much easier to be implemented thannine programmable thresholding non-linear synaptic laws used in conventional CNN-based dilation anderosion. The output results of the FCNN in equations (27) and (28) are error-free.

Figure 3(a) shows the output of the FCNN in equation (27) at instant t"3 s. The structuring element is thesame as that in equation (10). Figure 3(b) shows the output of the FCNN in equation (28) at instant t"3 s. Inboth simulations, the input is the image shown in Figure 1(a). The initial conditions are arbitrary. We findthat the FCNN-based erosion and dilation give error-free results.

4. CONCLUSIONS

In this paper, the performances of the conventional CNN and FCNN implementations of mathematicalmorphological operations are studied. We find that the conventional CNN-based mathematical morphologi-cal operations proposed in Reference 12 provide results with time-varying errors, which are input image and

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24 T. YANG E¹ A¸.


stop-moment-dependent, and the CNN structures in Reference 12 is structurally unstable. The error-free andglobally asymptotically stable implementations of morphological operations are based on the new concept ofFCNN as the authors had been proposed in References 5 and 6.

Although we do not argue that the FCNN-based mathematical morphological operations are the onlychoices for the future CNN-based mathematical morphological engine, if one want to get the error-free andstable results, the FCNN is the best choice.

REFERENCES

1. H. J. A. M. Heijmans, Morphological Image Operators, Academic Press, New York 1994.2. J. Serra, Image Analysis and Mathematical Morphology, Academic press, New York, 1982.3. L. O. Chua and L. Yang, ‘Cellular neural networks: theory’, IEEE ¹rans. Circuits Systems, 35, 1257—1272 (1988).4. H. Harrer and J. A. Nossek, ‘Discrete-time cellular neural networks’, Int. J. Circuit ¹heory Appl., 20, 453—467 (1992).5. T. Yang, and L. B. Yang, ‘Applications of fuzzy cellular neural network to Euclidean distance transformations’, IEEE ¹rans.

Circuits Systems 44, 242—246 (1997).6. T. Yang and L. B. Yang, ‘Applications of fuzzy cellular neural network to morphological grey-scale reconstruction’, Int. J. Circuit

¹heory Appl., 25, 153—165 (1997).7. T. Yang and L. B. Yang, ‘Fuzzy cellular neural network: A new paradigm for image processing’, Int. J. Circuit ¹heory Appl., in press.8. T. Yang and L. B. Yang, ‘The global stability of fuzzy cellular neural network’, IEEE ¹rans. Circuits Systems 43, 880—883 (1996).9. T. Yang, L. B. Yang, C. W. Wu and L. O. Chua, ‘Fuzzy cellular neural networks: theory’, Proc. 4th IEEE Int. ¼orkshop on Cellular

Neural Networks and their Applications (CNNA’96), Sevilla, Spain, 24—26 June 1996, pp. 181—186.10. T. Yang, L. B. Yang, C. W. Wu and L. O. Chua, ‘Fuzzy cellular neural networks: applications’, Proc. 4th IEEE Int. ¼orkshop on

Cellular Neural Networks and their Applications (CNNA’96), Sevilla, Spain, 24—26 June 1996, pp. 225—230.11. L.-B. Yang, T. Yang, K. R. Crounse and L. O. Chua, ‘Implementation of binary mathematical morphology using discrete-time

cellular neural networks’, Proc. 4th IEEE Int. ¼orkshop on Cellular Neural Networks and their Applications (CNNA’96), Sevilla,Spain, 24—26 June 1996, pp. 7—12.

12. A. Zarandy, A. Stoffels, T. Roska and L. O. Chua, ‘Implementation of binary and grey-scale mathematical morphology on the CNNuniversal machine’, Proc. 4th IEEE Int. ¼orkshop on Cellular Neural Networks and their Applications (CNNA’96), Sevilla, Spain,24—26 June 1996, pp. 151—156.

13. T. Yang, ‘Blind signal separation using cellular neural networks’, Int. J. Circuit ¹heor. Appl., 22, 399—408 (1994).

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