【作者】 周冬明；

【导师】 张立明；

【作者基本信息】 复旦大学 ， 电路与系统， 2004， 博士

【摘要】 细胞神经网络（CNNs）的理论和应用研究已经成为了新的研究热点。CNNs是由许多细胞单元通过局部连接组成的网络，其中每个细胞可由线性和非线性电路构成，可用VLSI实现，进行大规模的并行计算，因此它可应用于解决图像处理问题，尤其是可应用于解决传统方法还不能很好地解决的问题，比如：计算复杂性比较大的动态轮廓线目标分割和视频光流场的运动估计问题。然而，进行视频信号处理必须使用时滞细胞神经网络（DCNNs）。 细胞神经网络的主要功能是把一个输入图像转换成一个相应的输出图像。它为了完成这个功能，它必须是一个完全稳定的网络，即所有输出轨迹必须收敛到一个稳定的平衡点。因此稳定性是细胞神经网络可靠工作的前提。 本文主要研究了细胞神经网络无时滞和有时滞的几类模型的稳定性和基于离散细胞神经网络在图像处理方面的应用。稳定性研究主要是提出了在各种情况下（无时滞，常时滞，变时滞和不同激活函数等）神经网络的Lyapunov泛函，利用Lyapunov泛函方法证明了网络能稳定收敛到唯一的平衡点，给出了网络稳定的系数充分条件，并举例说明了所提出的条件优于其它文献的条件。图像处理应用研究主要是利用离散细胞神经网络来实现GVF场和光流场，实现图像分割的目的。 具体地，本论文的主要创新点如下： 利用Lyapunov函数方法、非奇异M-矩阵和不等式a²+b²≥2ab对无时滞的细胞神经网络状态方程进行了全局渐近稳定性和全局指数稳定性分析，给出了模板设计的稳定性判据，所得稳定性判据不要求模板是对称模板，并且所得出的全局指数稳定性判据对神经元的激活函数没有严格限于分段线性函数，而可以使用其它形式的非线性函数，只需要激活函数满足Lipschitz条件。 构造了新的Lyapunov泛函，使用了矩阵不等式2X^TY≤X^TPX+Y^T P^-1Y，X，Y∈Rⁿ为任意向量，矩阵P∈R^n×n为正定，研究了具有常时滞的细胞神经网络的平衡点的唯一性和全局渐近稳定性。得到了具有常时滞细胞神经网络的平衡点的唯一性和全局渐近稳定性的新的充分判据。所得稳定性的充分条件提供了一些参数来适当地弥补了反馈矩阵和时滞反馈矩阵之间所需要的平衡关系。该稳定性判据可以容易被用来设计和检验全局稳定的网络。而且，所得的稳定性条件不会受到时滞参数的影响。进行了计算机仿真实验，并与已有文献的结果作了比较，复旦大学博士学位论文结果表明:所得出的稳定性的充分条件改善和扩展了相关文献的结果，更有利于细胞神经网络的模板设计。 研究了具有变时滞的细胞神经网络的稳定性。构造了新的切apunov函数，使用不等式3ab。毛扩十驴+护（a，b，c>0）和矩阵不等式Zx丁Y‘x丁Px+Y了尸一，Y，x，YoRn为任意向量，矩阵尸任Rnx”为正定，并利用推广的Halanay不等式，对具有变时滞细胞神经网络的平衡点存在性和全局指数稳定性进行了分析。通过对细胞神经网络的激活函数进行三种不同的假设，分别得出了3个定理，在第三个假设中，不要求神经元的激活函数是可微的、有界的和单调递增的，对这些新的稳定性充分判据与已有的相关文献结果进行了比较，并进行了计算机仿真，结果表明我们的稳定性条件扩展和改善了已有的相关文献的结果。 讨论了具有一般激活函数和连续分布时滞的回归神经网络的稳定性问题。所讨论的模型是更一般的模型，得出了全局渐近稳定性的充分判据。这样使得在神经网络的设计中可以更灵活地使用激活函数;所得结果扩展和改善了已有文献的结果。 针对在图像处理应用中使用离散细胞神经网络，讨论了离散细胞神经网络的模板设计的收敛性问题。并研究了具有时滞的离散细胞神经的稳定性，得出了一个稳定性的充分条件，即I一（}川+】川）a是一个非奇异的林矩阵，则该网络是一个全局指数稳定的网络，式中I为单位矩阵，}川，}川表示反馈矩阵A和时滞反馈矩阵B的绝对值，在这里细胞激活函数只需满足Lipschitz条件即可。 最后，利用本文提出的稳定性理论结果，应用到动态轮廓的图像分割和视频光流场的运动检测的CNNS模板设计上，利用多层细胞神经网络实现GVF场，并与扩展、细化的细胞神经网络相结合来实现动态轮廓的图像分割，解决传统串行算法复杂性大，不能实时处理的问题，又克服了梯度场作为CNNS的外力驱动方法的局部最小问题。在图像处理过程中，初始轮廓由外部图像的GVF信息引导，最后收敛到所期望的目标位置。实验结果表明，该方法在不同的输入图像条件下均获得了比Vilarin。等人提出的方法更好的分割结果。 在文献[93]提出的一种改进的光流场计算方法的基础上，设计了双层带时滞的离散细胞神经网络来进行光流场的估计方法。并用本文的稳定性条件对以上两个应用的细胞神经网络的稳定性作了验证。更多还原

【Abstract】 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 3abc<a3+b3+c3 (a, b, c>0) 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更多还原