Abstract
Since its introduction to the research community in 1988, the Cellular Neural Network (CNN) (Chua & Yang, 1988) paradigm has become a fruitful soil for engineers and physicists, producing over 1,000 published scientific papers and books in less than 20 years (Chua & Roska, 2002), mostly related to Digital Image Processing (DIP). This Artificial Neural Network (ANN) offers a remarkable ability of integrating complex computing processes into compact, real-time programmable analogic VLSI circuits as the ACE16k (Rodríguez et al., 2004) and, more recently, into FPGA devices (Perko et al., 2000). CNN is the core of the revolutionary Analogic Cellular Computer (Roska et al., 1999), a programmable system based on the so-called CNN Universal Machine (CNN-UM) (Roska & Chua, 1993). Analogic CNN computers mimic the anatomy and physiology of many sensory and processing biological organs (Chua & Roska, 2002). This article continues the review started in this Encyclopaedia under the title Basic Cellular Neural Network Image Processing.
TopBackground
The standard CNN architecture consists of an M × N rectangular array of cells C(i,j) with Cartesian coordinates (i,j), i = 1, 2, …, M, j = 1, 2, …, N. Each cell or neuron C(i,j) is bounded to a sphere of influence Sr(i,j) of positive integer radius r, defined by:
(1)This set is referred as a (2r +1) × (2r +1) neighbourhood. The parameter r controls the connectivity of a cell. When r > N /2 and M = N, a fully connected CNN is obtained, a case that corresponds to the classic Hopfield ANN model.
The state equation of any cell C(i,j) in the M × N array structure of the standard CNN may be described by:
(2)where C and R are values that control the transient response of the neuron circuit (just like an RC filter), I is generally a constant value that biases the state matrix Z = {zij}, and Sr is the local neighbourhood defined in (1), which controls the influence of the input data X = {xij} and the network output Y = {yij} for time t.
This means that both input and output planes interact with the state of a cell through the definition of a set of real-valued weights, A(i, j; k, l) and B(i, j; k, l), whose size is determined by r. The cloning templates A and B are called the feedback and feed-forward operators, respectively.
An isotropic CNN is typically defined with constant values for r, I, A and B, implying that for an input image X, a neuron C(i, j) is provided for each pixel (i, j), with constant weighted circuits defined by the feedback and feed-forward templates A and B. The neuron state value zij is adjusted with the bias parameter I, and passed as input to an output function of the form:
(3) Key Terms in this Chapter
Dynamic Range: A term used to describe the ratio between the smallest and largest possible values of a variable quantity.
VLSI: Acronym that stands for Very Large Scale Integration. It is the process of creating integrated circuits by combining thousands (nowadays hundreds of millions) of transistor-based circuits into a single chip. A typical VLSI device is the microprocessor.
Chebyshev Polynomial: An important type of polynomials used in data interpolation, providing the best approximation of a continuous function under the maximum norm.
Spatial Convolution: A term used to identify the linear combination of a series of discrete 2D data (a digital image) with a few coefficients or weights. In the Fourier theory, a convolution in space is equivalent to (spatial) frequency filtering.
FPGA: Acronym that stands for Field-Programmable Gate Array, a semiconductor device invented in 1984 by R. Freeman that contains programmable interfaces and logic components called “logic blocks” used to perform the function of basic logic gates (e.g. XOR) or more complex combination functions such as decoders.
Bionics: The application of methods and systems found in nature to the study and design of engineering systems. The word seems to have been formed from “biology” and “electronics” and was first used by J. E. Steele in 1958.
Piecewise Linear Function: A function f(x) that can be split into a number of linear segments, each of which is defined for a non-overlapping interval of x.
Template: Also known as kernel, or convolution kernel, is the set of coefficients used to perform a spatial filter operation over a digital image via the spatial convolution operator.