124.

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Page 209

(5.34)

For a discrete N×N digital image, the discrete Fourier transform (Brigham, 1974) is used. This is defined by:

(5.35)

Both F(u, v) and the magnitude function |F(u, v)| are related by the expression

(5.36)

The magnitude function |F(u, v)| is called the Fourier spectrum of f(x, y), and (u, v) is the phase spectrum. The square of the amplitude spectrum is generally called the Fourier power spectrum, denoted by P(u, v):

(5.37)

It is well known that the angular distribution of values in P(u, v) is sensitive to the directionality of structure in f(x, y). If f(x, y) contains many lines or edges with some direction represented by angle θ, P(u, v) will contain high values concentrated around the direction perpendicular to θ, i.e. θ+π/2. If f(x, y) shows a non-directional structure, P(u, v) will be non-directional as well. Besides being sensitive to texture directions, the radial distribution of values in P(u, v) is also known to be sensitive to the roughness of the structure in f(x, y). A fine structure in f(x, y) will result in high values distant from the origin of P(u, v) while for a rough structure the high values in P(u, v) are concentrated near to the origin (Figure 5.14). Note that the co-ordinate origin of the amplitude spectrum is the centre of the image, not the upper or lower left corner.

Based on these properties of P(u, v), it is possible to design filters to extract radial and directional information concerning image structure from P(u, v). Two filters that are often used are the wedge (W) and ring (R) filters. The W and R filters in the case of a discrete image surface can be expressed as:

(5.38)

(5.39)

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