Digital Filters Design for Signal and Image Processing

9.3. Synthesizing with the two-dimensional windowing method

9.3.1. Principles of method

As we have seen in Chapter 8, the transformation that characterizes an invariant linear system is represented by the impulse response written h_d(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 ...