904 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 49, NO.7, JULY 2002 A Piecewise-Linear Simplicial Coupling Cell for CNN Gray-Level Image Processing Pedro Julián, Member, IEEE, Radu Dogaru, Member, IEEE, and Leon O. Chua, Fellow, IEEE Abstract—In this paper, we propose a universal piecewise-linear (PWL) CNN coupling cell, the simplicial cell, which is intended to work with binary as well as gray-level inputs. The construction of the cell is based on the theory of canonical simplicial PWL rep- resentations. As a consequence, the coupling function is endowed with important numerical features, namely: the representation of the characteristic cell function is sparse; the familiy of coupling functions constitutes a Hilbert space; powerful solution algorithms have been developed for the approximation of nonlinear functions, which is particularly useful when the CNN parameters need to be tuned from examples; the parameters can be extracted from a truth table when the CNN is specified analytically. Index Terms—Cellular neural networks, image processing, piecewise-linear approximation. I. INTRODUCTION C ELLULAR nonlinear networks (CNN) were originally in- troduced by Chua and Yang [1] as an array of dynamical systems, called cells (see also [2]). In a two-dimensional (2-D) situation, each cell can be described by the CNN equation [3] (1) where and are, respectively, the set of outputs and inputs of the cells within the sphere of influence of cell . Without loss of generality, we will consider the case of a 3 3 sphere of influence, i.e., , so that and Manuscript received May 14, 2001; revised February 21, 2002. This paper was recommended by Associate Editor P. Szolgay. P. Julián is with the Department of Electrical Engineering and Computer Sci- ences, University of California, Berkeley, CA 94720 USA, and also with CON- ICET (Consejo Nacional de Investigaciones Cientificas y Técnicas), Cap. Fed. 1033, Argentina, on leave from the Departamento de Ingenieria Eléctrica and Computadoras, Universidad Nacional del Sur, Bahia Blanca 8000, Argentina (e-mail: pjulian@ieee.org). R. Dogaru is with the Department of Applied Electronics and Information Engineering, Polytechnic University of Bucharest, Bucharest 75547, Romania. L. O. Chua is with the University of California, Department of Electrical En- gineering and Computer Sciences, Berkeley, CA 94720 USA. Publisher Item Identifier 10.1109/TCSI.2002.800464. The output of each cell is given by the expression , and the functions , (which in the standard CNN are linear functions) are called the feedback and feedforward coupling functions, respectively, since they de- fine the coupling with neighbor cells. Throughout the paper, we will focus only on feedforward CNN, i.e., the class of CNN where the feedback term in (1) is null. The coupling functions play a major role on the CNN signal- processing capabilities, since they are the core of the cell con- stitutive differential equation, as it can be deduced from (1). The starting point of this paper is the following question: What are the factors to be considered in the design or selection of a cell? In [3], five basic features of a coupling cell were identified, namely: 1) It should be able to implement any of the possible Boolean functions in Boolean interval logic, i.e., any function , where the input set is 1 and the output set is (2) 2) The algorithm to generate the parameters should be tractable from a computational point of view and should have guaranteed convergence. 3) It should be possible to train the cell from examples, even when a truth table is not available. 4) It should be optimal with respect to the required informa- tion, in the sense that the average information needed to specify the parameters should be equal to the information needed to specify any Boolean function . 5) It should have low computational complexity. Depending on the type of signals to be processed, extra constraints can be added. For instance, in the present case, we will require that the cell be able to process gray-level signals, i.e., when the input and output sets are expanded to (3) and respectively. 1 For the purposes of this paper and in accordance with the CNN literature, the set of possible values of a binary signal will be assumed to be . In other words, this implies that, in Boolean algebra, we are considering 1 as the FALSE digital value and 1 as the TRUE digital value. 1057-7122/02$17.00 © 2002 IEEE