ABSTRACT
Two-dimensional (2-D) digital filters find applications in many fields such as image processing and seismic signal processing. Design of 2-D digital filters is more complicated than that of 1-D digital filters because of the increase in the number of coefficients with the increase in the dimension and also the difficulty of testing their stability. Fortunately, 2-D frequency responses possess many types of symmetries and the presence of these symmetries can be used to reduce the complexity of the design as well as the implementation of these filters. Symmetry in the frequency response of a filter induces certain constraints on the coefficients of the filter, which in turn reduces the filter design complexity [1-28]. Therefore, a study of the symmetries of the filter frequency responses and the resulting constraints on the filter coefficients is undertaken in this chapter. As there is a close relationship between digital and analog filter functions, symmetry properties are discussed in this chapter for both analog and digital domain functions. In addition to filter design, symmetry can also be applied in the computation of Fourier transform of 2-D signals. To facilitate this, the symmetry properties in the Fourier transform pairs will be presented. It will also be shown that the presence of symmetry will reduce the complexity in the implementation of the fast Fourier transform (FFT) of 2-D signals [29]. The following is the layout of the sections. First, the symmetries are defined. Then, the
Fourier transform pairs with symmetry and the use of symmetry in the implementation of FFT are presented. Next, symmetry in the magnitude functions of filters is discussed. This is followed by the symmetry constraints on polynomials and a procedure to design 2-D filters employing the constraints. Finally, several examples are given to illustrate the application of the symmetry-based filter design procedure.
