Classical signal and image processing uses linear processing techniques. These are methods based in the familiar Fourier, Z, and Laplace transforms. These methods assume that signal and image data may be processed by mapping them onto lower-dimensional orthogonal spaces resulting in solutions designed by decomposing the input into sinusoidal components and processing them individually. While mathematically elegant, this imposition of linearity results in a very limited set of processing operations compared to the total set of solutions possible, i.e., both linear and nonlinear. For example, techniques based on rank ordering of values, logical and geometric processing approaches can give excellent results, particularly for image processing applications. This approach should not be viewed as an alternative to the classical methods, but as a superset of techniques containing many new novel techniques as well as the linear techniques listed above.
The model chosen to convey these concepts is that of digital logic. This is because it can quite literally capture any processing operation, linear or nonlinear, that may be required. Many engineers and computer scientists are comfortable with its notation and concepts. Minimization techniques and software tools are available to reduce complex solutions into their simplest form, and the solutions translate readily into electronic hardware or software implementations.
Every digital signal or image processing operation can be viewed at its most basic level as the manipulation of a series of finite-length binary strings. Whether the operation is implemented on a processor through software or in dedicated hardware, the data and the algorithms are invariably mapped through electronic logic components, which are inherently binary in nature.
Therefore, every digital signal and image processing task can be cast in terms of a logical representation. It does not matter if the data is binary, grayscale, color, or multiband, nor whether the operation is linear or nonlinear. If it can be programmed, then it can be placed in the context of a logical representation.
In nonlinear image and signal processing, the design of operators is carried out by seeking the optimum mapping from one set of binary strings to another.