Digital Filters Design for Signal and Image Processing

8.3. Discrete models

8.3.1. 2-D sampling

For digital images, data are stored in the form of 2-D tables. The usual representation is that of real functions defined on Z².

where k represents the index of columns and l the index of rows. The link between the continuous model and the discrete model is established by the sampling operation, represented mathematically by the product of the continuous image by a bidimensional Dirac comb. Here we will assume that the multiplication conditions of the distributions are satisfied; that is, that the function representing the analog image is sufficiently regular.

For the sake of simplicity and without loss of information, we will assume that the sampling steps in horizontal and vertical directions are identical and equal to the unity. The bidimensional Dirac comb is written as:

where δ(x,y) is the bidimensional Dirac distribution centered on the origin:

where ψ (x,y) is a test function.

The sampled signal associated with the analog signal s(x,y) is then given by the following formula:

As with temporal signals, it is possible to characterize 2-D ...