Sj = Wzj * X i i = O...input j = l...hidden (1)

Where, input and hidden are the numbers of inputs and hidden layers in our system. These values are constant and equal to 10. The output of each hidden layer is computed using a signed func- tion as follows:

j ,

j = l...hidden (2) h . - (1 + e - s j ) - l 3 -

Inserting the value of S, from eqn. 1, the main outputs of the neu- ral network is as follows:

When h, is computed from eqn. 2 and the output is a constant value equal to 6. The output of the network is:

0, = (1 + e-‘J)--l j = l...output (4)

1,

3

0 L 00 800 1200

Fig. 1 Results of counting vehicles (heuristic approach) no. of vehicles 183111/

Results: The algorithm has been implemented in a user friendly package in an MS Windows environment. The results of counting vehicles by a heuristic method is shown in Fig. 1, while Fig. 2 rep- resents the results of vehicle counting by a neural network approach. As can be seen, this approach provides better results. To further improve the accuracy of the vehicle counting operation, the windows are again divided into two sections and a neural net- work with 20 inputs is used for analysis. For this training net- work, 150 training samples were selected and the network produced a designated response after lOOK training. With this approach the accuracy achieved was > 97%

L

z 3 2 2

1

a

L

8 3 e 2

1

0 Loo 800 izoo b no. of vehicles

Fig. 2 Results of counting vehicles by neural network approach and dividing window into two halves a Neural network approach b Dividing window into two halves

Conclusions: In this Letter, a novel neural network and image processing approach was described to analyse and measure road traffic parameters. To further increase the accuracy in the case of vehicle not moving on their lanes, half size windows were used. Using neural networks is more accurate than using the heuristic approach as considering all the states of the window pattern is dif- ficult in a real-world traffic scene because the vehicles move with different speeds and directions. In this case, an intelligent approach such as the neural network, which is trained for many traffic scenes and situations, can compute the parameters more accurately.

0 IEE 1997 Electronics Letters Online No: 19970646 M.Y. Siyal, M. Fathy and F. Dorry (School of EEE, Nanyang Technologicul University, Nanyang Avenue, Singapore 639798, Singapore)

2 April 1997

References

DICKINSON, K.W., and WATERFALL, R.C.: ‘Image processing applied to traffic: Practical experiences’, Traflic Eng. Control, 1984, pp. 60- 67 FATHY, M., and SIYAL, M Y : ‘An image detection technique based on morphological edge detection and background differencing for real-time traffic analysis’, Pattern Recognit. Lett., 1995, 16, pp. 1321-1330 HOOSE, N.: ‘Computer image processing in traffic engineering’ (Research Studies Press, Taunton, 1991) ALI, A.T., and DAGLES, E.L.: ‘Recent progress in real-time image analysis for real-world traffic analysis’. ICARVC’92, 2nd Int. Conf. Automation, Robotics and Computer Vision Proc., 1992,

FATHY, M., and SIYAL, M.Y.: ‘A real-time image processing approach to measure traffic queue parameters’, IEE Pvoc. Image, Vision, Signal Process., 1995, 142, (5), pp. 297-303 FATHY, M., and SIYAL, M.Y.: ‘A window-based edge detection technique for measuring road traffic parameters in real-time’ in ‘Real-Time Imaging 1’ (Academic Press, 1995), pp. 297-305

Vol. 1, pp. 10-15

Practical stability criteria for cellular neural networks

P.P. Civalleri and M. Gilli

Indexing terms: Cellular neural nets, Stability criteria

New sufficient conditions for the existence of stable equilibrium points in cellular neural networks (CNNs), described by space- invariant templates, are presented. Extensive simulations have shown that such conditions also guarantee the complete stability of the network.

Cellular neural networks (CNNs) are arrays of dynamical cells [l, 21 that are suitable for the formulation and solution of many com- plex computational problems. Most applications (e.g. image processing) require the network to be completely stable, i.e. that every trajectory converges to an equilibrium point. The study of the stability of such networks (that are large-scale dynamical sys- tems) is a cumbersome task and for this reason only a few results are available [3 - 81. Two considerations are needed: (i) the simu- lations show that the class of stable CNNs is much larger than the subclass, for which a rigorous proof of stability is already availa- ble; (ii) CNN design requires simple stability criteria that should be directly expressed in terms of the template elements.

In this Letter we present some sufficient conditions for the exist- ence of stable equilibrium points in both 1D and 2D template CNNs. Such conditions are different from those reported in [7, 81 and in several cases give rise to less strong constraints. Moreover, they can be checked by simply looking at the template elements and therefore are suitable for design.

We have verified, by extensive simulations, that our suficient conditions seem to also guarantee the stability of the network.

We consider an autonomous CNN described by a space-invari- ant template and assume that the cells are arranged on a rectangu- lar matrix, composed by N rows and M columns. We also assume that the input terms are null. (The results, however can be easily extended to constant inputs.)

The dynamics of the network is governed by the following set of normalised equations:

k l (z , r ) max(1- i, - T ) ka( i , r ) = min(N - i , ~ )

I l ( j , r ) = m a x ( l - j , - r ) Iz(j ,r) = m i n ( M - j , r )

970 ELECTRONICS LETTERS 22nd May 1997 Vol. 33 No. I 1

where xzl denotes the state of the cell located at the crossing between the ith row and the j th column of the network; A is the feedback cloning template, defined by a (2,r+l) x (2r+l) real matrix; the outputs yil are expressed through the well-known satu- ration function, introduced in 111:

(’4 1

Y d t ) = @ , 3 ( q + 11 - IZ i , j ( t ) - 11)

For simplicity, we will not distinguish between complete stability and almost complete stability; we simply say that a CNN is stable if each trajectory, with the exception of a set of zero measure, con- verges to an equilibrium point. It is known 181 that the stability properties of the CNNs are related to the existence of equilibrium points; in particular we formulate the following.

Conjecture: A CNN described by eqn. 1 that exhibits at least one stable equilibrium point, is stable.

The above conjecture has already been stated in [8] and has, so far, been verified by all the simulations available in the wide CNN literature. It represents a good tool for finding simple stability cri- teria suitable for design purposes. Since the conjecture does not admit a counterexample, we expect that the stability criteria, derived from it, are valid in most CNN applications.

We now report two theorems, which yield sufficient conditions for the existence of at least one stable equilibrium point in a CNN; they represent the fundamental contribution of this brief. The proofs of the theorems are strictly based on the behaviour of the cells located on the CNN boundary; for the sake of brevity they are not reported here. Theorem 1: A CNN described by the 1D template A,, 111 I r, such that

admits at least one stable equilibrium point lying in a saturation region (i.e. in a region where all the cell outputs are saturated at the values kl).

Theorem 2: A CNN described by the 2D template A!<,[, /kl 2 r, 111 5 r, such that

/ r ,.

T T -1 -T T T

k z - 1 I=-r l=-r 1=1 k=-T k=l r r - 1 -r T

(4)

admits at least one stable equilibrium point located in a saturation region.

According to the conjecture stated previously, we expect that the sufficient conditions given in theorems 1 and 2 are also useful criteria for establishing the CNNs stability.

To show the usefulness of theorems 1 and 2, we discuss two simple examples. As a first case, we consider the CNN described by the 1D template A = [-sI, p, s,], s, > 0, s, > 0, p > 0. Theorem 1 yields the following condition for the existence of at least one stable equilibrium point:

If s, = s, = s, such a criterion coincides with the well-known stabil- ity results reported in [6]. If si # s,, we have verified, by extensive simulations, that the inequality in eqn. 5 identifies exactly the range of parameters for which the network is stable. A compari- son with the conditions given in [8] for the existence of at least one stable equilibrium point, permits us to state the following: (i) For a 2-cell CNN with si = s, = s, both eqn. 5 and theorem 1 of [8] give rise to the same constraint p l > s;

p - 1 > min(s1,sZ) ( 5 )

(ii) For a 3 4 1 CNN with s, = s, = s, eqn. 5 still yields p l > s, whereas theorem 1 of 181 gives p l > d2s, which represents a con- straint stronger than eqn. 5; (iii) For a 2 4 1 CNN with sI # s,, the condition resulting from theorem 1 of 181 is p-1 > d(s,s,), that is still a constraint stronger than eqn. 5; (iv) For a CNN composed by N cells, theorem 1 of [8] requires that p l > 2d(s,s2)cos[d(N+1)]. Such a constraint, regardless of the number of cells, is always worse than eqn. 5.

As a second example, we consider the CNN described by the 2D template A , , with the neighbourhood r set to 1.

We have verified by extensive simulations that the constraint of eqn. 4 always guarantees the stability of the network. This leads to conjecture that the set of stable 2D templates can be considerably enlarged.

We refer in particular to the three classes of sign-symmetric templates reported in the conclusion of 131, for which no stability result is known. As expected, the simulations confirm that if such templates satisfy eqn. 4, the corresponding CNN is stable. Moreo- ver the stability criterion, derived from eqn. 4 can also be applied to the opposite-sign 2D templates, that are not considered in [3].

Finally we remark that the conditions of eqns. 3 and 4 can be checked by direct examination of the template, regardless of the number of cells. This seems to be an advantage with respect to [8], where computation of the principal minors of the feedback matrix is required.

0 IEE 1997 Electronics Letters Online No: 19970640 P.P. Civalleri and M. Gilli (Cattedra di Elettrotecnica, Dipartimento di Elettronica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 1-10129 Torino, Italy) E-mail: gilli@polito.it

20 March 1997

References

CHUA, L.o., and YANG, L.: ‘Cellular neural networks: theory’, ZEEE Trans. Circuits Syst., 1988, 35, pp. 1257-1272 CHUA, L.o., and ROSKA, T.: ‘The CNN paradigm’, IEEE Trans.

CHUA, L.o., and wu, c.w.: ‘The universe of stable CNN templates’, Int. J. Circuit Theory Appl., 1992, 20, pp. 497-517 GILLI, M.: ‘Stability of cellular neural networks and delayed cellular neural networks with nonpositive templates and non-monotonic output functions’, ZEEE Trans. Circuits Syst. I, 1994, 41, (8), pp. 5 18-528 SAVACI, F.A., and VANDEWALLE, J.: ‘On the stability analysis of cellular neural networks’, IEEE Trans. Circuits Syst. I, 1993, 40, pp. 213-215 JOY, M.P., and TAVSANOGLU, v.: ‘A new parameter range for the stability of opposite sign cellular neural networks’, ZEEE Trans. Circuits Syst. I, 1993, 40, pp. 204207 THIRAN, P.: ‘Influence of the boundary conditions on the behavior of cellular neural networks’, IEEE Trans. Circuits Syst. L 1993, 40,

ARIK, s., and TAVSANOGLU, v.: ‘Equilibrium analysis of non- symmetric CNNs’, Int. J . Circuit Theory Appl., 1996, 24, pp. 269- 214

Circuits Syst. I, 1993, 40, pp. 147-156

pp. 207-212

10Gbit/s optical single sideband system

M. Sieben, J. Conradi, D. Dodds, B. Davies and S. Walklin

Indexinn terms: Optical modulation, Optical communication

An approach for transmitting a digital optical single sideband signal in a self-homodyne detection system has been developed. Simulation results show that lOGbitis of data can be transmitted through conventional singlemode fibre over distances > lOOO!un with post-detection electrical dispersion compensation; initial experimental results show transmission of lOGbit/s over distances up to 200km.

ELECTRONICS LETTERS 22nd May 1997 Vol. 33 No. I I 97 1