## Frequency Domain Techniques

Linear filters used for enhancement can also be implemented in the frequency domain by modifying the Fourier transform of the original image and taking the inverse Fourier transform. When an image g(m, n) is obtained by convolving an original image /(m, n) with a kernel w(m, n), g(m, n) = w(m, n) */(m, n), the convolution theorem states that G(m, v), the Fourier transform of g(m, n), is given by

G(M, V) = W(M, V)F(M, V), where W(m, v) and F(m, v) are the Fourier transforms of the kernel and the image, respectively. Therefore, enhancement can be achieved directly in the frequency domain by multiplying F(m, v), pixel-by-pixel, by an appropriate W(m, v) and forming the enhanced image with the inverse Fourier transform of the product. Noise suppression or image smoothing can be obtained by eliminating the high-frequency components of F(m, v), while edge enhancement can be achieved by eliminating its low-frequency components. Since the spectral filtering process depends on a selection of frequency parameters as high or low, each pair (m, v) is quantified with a measure of distance from the origin of the frequency plane,

where c is a coefficient that adjusts the position of the transition and n determines its steepness. If c = 1, these two functions take the value 0.5 when D(m, v) = DT. Another common choice for c is \[2 — 1, which yields 0.707 ( — 3 dB) at the cutoff Dt. The most common choice of n is 1; higher values yield steeper transitions.

The threshold DT is generally set by considering the power of the image that will be contained in the preserved frequencies. The set S of frequency parameters (m, v) that belong to the preserved region, i.e., D(m, v) < DT for low-pass and D(m, v) > Dt for high-pass, determines the amount of retained image power. The percentage of total power that the retained power constitutes is given by

and is used generally to guide the selection of the cutoff threshold. In Fig. 9a, circles with radii rp that correspond to five different B values are shown on the Fourier transform of an original MRI image in Fig. 9e. The m = v = 0 point of the transform is in the center of the image in Fig. 9a. The Butterworth low-pass filter obtained by setting Dt equal to rpfor p = 90% , with c = 1 and n = 1, is shown in Fig. 9b where bright points indicate high values of the function. The corresponding filtered image in Fig. 9f shows the effects of smoothing. A high-pass Butterworth filter with Dt set at the 95% level is shown in Fig. 9d, and its output in Fig. 9h highlights the highest frequency components that form 5% of the image power. Figure 9c shows a band-pass filter formed by the conjunction of a low-pass filter at 95% and a high-pass filter at 75%, while the output image of this band-pass filter is in Fig. 9g.

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