TABLE OF CONTENTS In order to develop the capability of processing signals using optics, especially two-dimensional signals (i.e., images), a fundamental mathematical understanding of two-dimensional (2-D) linear systems theory is required. Two-dimensional linear systems, which are described most succinctly and generally by Equation (1.1), obey essentially the same rules and properties that one-dimensional systems do. The important properties of linear systems that we should recall are linearity, shift invariance, stability, and causality, which will hold with restrictions noted below.
A system can be defined as linear if its output is determined purely by addition as described in Equation (1.2). Linearity is usually assumed to hold except when nonlinear electro-optical interfaces are used. Nonlinear effects on the signal output may result from raising the input signal level into the saturation region of an otherwise linear spatial light modulator, or a component may be deliberately nonlinear as in the case of square-law photodetectors or logarithmic amplifiers (designed to accommodate large changes in input signal power). These topics will be covered in Chapters 2, 7, and 8. Shift invariance will hold over limited regions in the ( x, y) plane (known as isoplanatic patches in optics). This is an important consideration in real optical systems that are designed with large apertures or wide fields of view. In image processing the term describing nonisoplanatic systems is "space variant." Stability will be required for the same reasons as in one-dimensional systems: A bounded input must yield a bounded output. Instances where this may...
