TABLE OF CONTENTS Chapter written by Philippe BOLON.
This chapter presents several digital filtering techniques applied to two-dimensional data. The most common applications are concerned with the processing of images. Other kinds of data can be processed using similar techniques, such as time-frequency representations and time-scale representations of mono-dimensional signals.
The fundamental principles of this kind of filtering are based on the 2-D sampling theorem and on the Fourier transform.
This chapter includes a brief reminder of continuous models and stationary 2-D linear filtering, since most of the later explanations make use of these. Then, we will introduce two-dimensional sampling techniques. Filtering operations will then be discussed in both spatial and frequency domains.
In a natural way and as with temporal signals, the usual model for representing two-dimensional signals is the functional model, which can possibly extend to distributions. Since we are most often dealing with images here, temporal coordinates are replaced by spatial coordinates, written as x and y.
| (8.1) |  |
Under normal conditions that is, for finite energy functions signals can be described in the Fourier domain by means of spatial frequencies u and v, using the bidimensional Fourier transform (FT):
| (8.2) |  |
It should be noticed that the 2-D transform is separable. The 2-D calculation is obtained by linking the two calculations of the one dimensional (1-D) transform by successively integrating them in relation to each of the two variables:
| (8.3) |  |
A linear filtering transforms the 2-D signal s(x,...
