Circulant matrices and the stability of ring CNNs

The real parts of the eigenvalues of the feedback matrix associated with any CNN dynamical system provide useful information on the behaviour of its trajectories. In this paper we show how circulant and block circulant matrices arise as feedback matrices of ring CNNs. Such matrices possess many pleasant properties and we are able to give formulae for their eigenvalues and thus completely describe the spectrum of any feedback matrix associated with a ring CNN. Armed with this knowledge we are able to present a stability theorem for ring CNNs with a (specific) two-dimensional cloning template. This theorem provides a parameter range for which convergence of the CNN dynamical system is assured. Interestingly, this condition differs from the well-known diagonal dominance condition; moreover, it is of practical import since CNNs with feedback matrices which are diagonally dominant will not `transform´ bipolar images (such images lie in the basin of attraction of stable equilibria) and this must be considered a practical drawback to such a condition. The condition on the parameters presented here results in CNNs which will process bipolar images