Image Filtering

Filtering of an image or any signal, in general, is the process of emphasizing or attenuating some of its frequency components. Subsequently, the resulting image or signal retains features associated with those frequency components. Image filtering is often required in various applications such as reducing noise, extracting edges and boundaries of objects, interpolating functional values, etc. The term filtering itself is self-explanatory if we consider the representation of a function f(x) in the frequency domain by its Fourier transform,

say f̂(jω). Given a filtering function ĥ(jω), where the frequency component

at a frequency ω0 is modified by a factor of ĥ(jω0), the filtered output in the

frequency domain becomes ĝ(jω) = ĥ(jω)f̂(jω). Hence, the corresponding processed function in its original domain (say, temporal or spatial domain) becomes the inverse Fourier transform of ĝ(jω), which is represented in the

text as g(x). The filtering function in the frequency space ( ĥ(jω) in this example) is more commonly referred to as the frequency response or transfer function of the filter. The equivalent operation of filtering in the spatial or temporal domain (original domain of the function) is the convolution of

the function with h(x), which is the inverse Fourier transform of ĥ(jω). This has been already discussed in the previous chapter, where the convolutionmultiplication properties of Fourier transform as well as of discrete Fourier transform are discussed (see Sections 2.1.3.1 and 2.2.1.3). However, in this chapter, we briefly review this concept with respect to the analysis of a linear system, in particular a discrete linear system. This would be followed by adaptation of this computation in the block DCT domain, which is different from the frequency domain of the Fourier transform. Convolution-multiplication properties (CMP) with certain restrictions are also applicable to trigonometric transforms, as discussed before (see Chapter 2, Section 2.2.3). We also discuss their suitable adaptation for filtering images in this chapter. Here, initially, we limit our discussion to 1-D only. Later the concepts are extended to 2-D.