where F *(«, v) is the complex conjugate of f(m, v). This power spectrum is expressed in polar coordinates as a new function Q(r, 6) where each pixel now is indicated by a distance r = Vm2 + v2 from the origin and an angle 6 = tan-1(v/w). The distance r is the frequency of the pixel, and the angle 6 provides its orientation in the spatial domain. For a texture with a given periodicity and direction, the spectrum exhibits a peak at the corresponding frequency r and orientation 6.
The presence of texture with a given periodicity in any direction can be quantified by forming the sum based on the corresponding spatial frequency rt
The limits of this summation may need to be restricted if texture in a selective range of orientations is relevant. Texture of any size in a desired orientation 6t can be measured with
r = ri where r; and rm are the lower and maximal frequencies of interest which should be selected to represent the largest and smallest texture grain sizes respectively. The two functions T(rt) and T(0t) obtained by varying the selected rt or 0t convey comprehensive information of the texture in the analyzed region. Statistical analysis of these two functions allow further data reduction and parsimonious quantification of image texture. For example the highest value in each of these functions indicate the dominant periodicity and orientation, wheras their means can provide more global representation when periodicity or orientation are distributed. The variance and other statistical moments of these functions can also be used as descriptors.
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