Filters in the Frequency Domain

 Image Smoothing Using Frequency Domain Filters: ◦ Ideal Lowpass Filters ◦ Butterworth Lowpass Filters ◦ Gaussian Lowpass Filters.  Image sharpening Using Frequency Domain Filters: ◦ Ideal Highpass Filters ◦ Butterworth Highpass Filters ◦ Gaussian Highpass Filters. ◦ The Laplacian in the Frequency Domain

 We’ll begin with lowpass filters.  Edges and other sharp intensity transitions (such as noise) in an image contribute significantly to the high-frequency content of its fourier transform. Hence, smoothing (blurring) is achieved in the frequency domain by high frequency attenuation; that is, by lowpass filtering.  In this section we consider three tuype of lowpass filters: ideal, Butterworth, and Gaussian.  These three categories cover from very sharp (ideal) to very smooth (Gaussian) filtering.  The Butterworth has a parameter called the filter order filter. For high order values it approaches the ideal filter. Foe lower order values, it is more like the Gaussian filter.

 A 2-D lowpass filter that passes without attenuation all frequencies within a circle of radius D 0 from the origin and “cuts off” all frequencies outside this circle is called an ideal lowpass filter (ILPF)  It is specified by the function: Where D(u,v) is the Distance from point (u,v) from the origin of the frequency rectangle.

 The name ideal indicates that all frequencies on or inside a circle of radius D 0 are passed without attenuation, where as all frequencies outside the circle are completely attenuated (filtered out).  ILPFs have blurring and ringing properties as shown in figure 4.42

 The transfer function of a Butterworth lowpass filter (BLPF) of order n, and with cutoff frequency at a distance D 0 from the origin is defined as:

Unlike the ILPF, the BLPF transfer function does not have a sharp discontinuity that gives a clear cut off between passed and filtered frequencies.

 Gaussian lowpass filters (GLPFs) if 2-D is given by:

The GLPF achieved slightly less smoothing than the BLPF of order 2 for the same value of cut off frequency. GLPF ensure that there is no ringing at all. In cases where tight control of the transition between low and high frequencies about the cut off frequency are needed, then the BLPF presents a more suitable choice.

In figure 4.51 the objective here is to blur out as much details as possible while leaving large features recognizable.

 In this section edges and other abrupt changes in intensities are associated with high- frequency components.  image sharpening can be achieved in the frequency domain by highpass filtering, which attenuates the low-frequency components without disturbing high-frequency information in the Fourier transform.  In this section, we will consider ideal, Butterworth, and Gaussian highpass filters. As before we’ll see that Butterworth filters represent a transition between the sharpness of ideal the ideal filter and the broad smoothness of the Gaussian filter.

 A 2-D ideal highpass filter (IHPF) is defined as: Where D 0 is the cutoff frequency. The IHPF is the opposite of the ILPF. IHPF has the same ringing properties as the ILPF.

 A 2-D Butterworth highpass (BHPF) of order n and cutoff frequency D 0 is defined as:

 The transfer function of the (GHPF) is given by: The results obtained in figure 4.56 are more gradual than with the previous two filters. Even the filtering of the smaller objects and thin bars is cleaner with the Gaussian filter.

 The Laplacian is used in spatial filtering for image enhancement and it yields equivalent results using frequency domain teqniques.