As we have seen in Chapter 8, the transformation that characterizes an invariant linear system is represented by the impulse response written hd(x, y), which satisfies the following formula:
where s(x,y) and w(x,y) respectively designate the two-dimensional input and output signals.
Depending on application requirements, we synthesize an impulse response that helps us to obtain a desired frequency response following typical shapes such as low-pass, high pass, cut-off band or passband filter. The correspondence between frequency domain representation and impulse response is assured by the 2-D Fourier transform.
Let us consider the truncated impulse response of the digital 2-D FIR filter defined by:
The quantities m and n represent the size of the impulse response in horizontal and vertical directions. To establish the link between discrete and continuous forms, equation (9.6) can be derived in a continuous spatial domain. We then have:
The quantities t1 and t2 represent the spatial boundaries associated with the non-null values of the impulse response h(x,y) and are directly proportional to m and ...