Digital Filters Design for Signal and Image Processing

9.2. Introduction to 2-D FIR filters

The most basic digital forms of the impulse responses of 2-D FIR filters are given by squared matrices of small size [3×3] or [5×5]. The coefficients of these matrices represent the spatial form of the digital filter that is applied according to an implementation mode shown in equation (8.17). Many currently available forms of software dedicated to image retouching make use of these basic forms. For example, let us look at the following low-pass filters:

As shown in equation (8.19) in Chapter 8, because of their separability, these masks decompose into two 1-D filters that are respectively applied according to horizontal and vertical directions. Thus, we have:

Convolution masks M₁ and M₂ thus have coefficients that are characterized by 1-D amplitude profiles that are respectively rectangular and triangular. As shown in Figure 9.1, these low-pass filters do not have tunable cut-off frequencies. The only adjustable feature, from the frequency domain point of view, is the position of the first passage through zero of the response, which can be modified ...