14.1.1 Concept of Multidimensional Signals

The purpose of this section is to outline a generalised version of the theory that we have used so far in this book for the field of signals dependent on more than just a single variable. We shall define a multidimensional continuous signal as a scalar function of a continuous vector argument

f( x),

where the physical meaning of the function value and of the components of x is arbitrary. Probably the most common example of a multidimensional signal is a static greyscale plane image, which is described by the brightness (or reflectivity) function of two space coordinates, f( x, y). Examples of three-dimensional signals are a time-variable image f( x, y, t) or tomographic space image data f( x, y, z). In advanced tomography, four-dimensional time-dependent space image data f( x, y, z, t) is already dealt with. Contemporary computing technology enables the practical solution of many problems relating to multi-dimensional signals in spite of the enormous computational and memory requirements which follow from such tasks. We shall illustrate the discussion by some simple examples of this kind. It may be expected that future developments will make it possible to process and analyse multidimensional signals by highly sophisticated methods that would be capable of utilising complex inner relationships among elements and components of such signals, although their practical applications are...