| Theory and applications of the cellular neural networks (CNNs) have been a new focus recently. It is known that the CNNs is composed of many units called cells with local interconnections. Each cell in the CNNs is formed by linear and nonlinear circuit elements. The CNNs is well suited for VLSI implementation and parallel computing, thus it can be used to solve some image processing problems, especially, to solve some complex problems that traditional methods cannot do well such as object segmentation using active contour, optical flow estimation for video image etc. In video signal processing it is necessary to consider using delayed cellular neural networks (DCNNs).The primary function of CNNs is to transform an input image into a corresponding output image. To implement this function, it must be a complete stable network, namely, its all output tracks must converge at a stable equilibrium point. Therefore, the stability is precondition of credibility work for CNNs.In this paper, some theorems of stability for several kinds of CNNs are proved and applications in image processing are also obtained. In theory some new different Lyapunov functions are proposed for all kinds of conditions such as CNNs without delay, with constant delays, with variable delays and with different activation functions, that ensure the networks to converge at a unique equilibrium point by using proposed Lyapunov functions. Numerical examples show that our results are superior to other ones. For the image processing applications, the implementation of Gradient Vector Flow field and optical flow field by using multiplayer discrete-time CNNs are proposed for image segmentation.Concretely, major innovations of this dissertation are shown as follows:First, For CNNs without delay, its the global asymptotic stability and the global exponential stability are analyzed by using Lyapunov function method, non-singular M-matrix and inequality a2+b2>2ab, the stability criteria are obtained, and these stable conditions do not require the cloning templates to be symmetric and the activation function of cells does not strictly limit to a piecewise linear function. It can use other nonlinear function, but the function is Lipschitz continuous.Second, the uniqueness and the global asymptotic stability of the equilibrium point for CNNs with constant delays are proposed by constructing new Lyapunov functional and combining with the inequality of matrix 2XTY < X1PX + YTP-1Y , in which X, Y Rn is arbitrary vectors, the matrix P Rnxn is positive definite. A newsufficient condition ensuring the uniqueness and the global asymptotic stability of the equilibrium point for CNNs with constant delays is obtained, which provides some parameters to appropriately compensate relation between feedback matrix and delayed feedback matrix. This criterion can easily be used to design and verify globally stable networks. Furthermore, the condition presented here is independent of the delay parameter. Some computer simulations demonstrate that our results improve and generalize other ones, and are propitious to design the cloning templates of CNNs.Third, study the stability of CNNs with variable delays. By constructing new Lyapunov function, using the inequality 3abc0) and the inequality of matrix 2XTY < XTPX + YTP-1Y ,(X,Y Rn is arbitrary vectors, and the matrixP Rnxn is positive definite), and employing the extended Halanay's delay differential inequality, we analyze the existence and the global exponential stability of the equilibrium point for CNNs with variable delays. Three theorems are obtained in three different hypotheses of the activation functions of cells. In the third hypothesis, the activation functions are not necessary to satisfy the monotonic, bounded and differentiable conditions. Computer simulations show that our results improve the previous ones.Fourth, the stability problem is discussed for recurrent neural networks with a general class of activation functions and distributed delays. The neural networks model consider...
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