Curvature can be recognised in an orientation map wherever the angles of the orientation vectors change as a function of position. If we assume that there is only one type of curvature (at least locally) in a given image, then we can approximate the angle of the orientation vectors to be of the form
where 
represents a doubled-angle
orientation map. An example of such an orientation field is shown in
Figure 4.
Figure 4: Examples of doubled-angle orientation fields with linear
curvature.
The constants a and b represent the rate of angular change as a function of position and c represents a global orientation offset. Using this approximation, we can apply the shifting theorem for the Fourier transform to observe that the 2D Fourier transform of f(x,y) will simply be a translated version of the Fourier transform of r(x,y). Since r(x,y) is strictly real, its transform is even, which means that its center of gravity will be located at the origin. As such, the task of orientation estimation on the orientation map consists of estimating the center of gravity of the spectrum of f(x,y). The further away from the origin the center of gravity is, the larger the magnitude of curvature. The angle formed by the center of gravity, the origin and the horizontal frequency axis, meanwhile, represents the direction of curvature.
One method for achieving this kind of curvature estimation is to design a set of bandpass filters which collect the energy of the Fourier transform of f(x,y) in specified regions around a ring in the spatial frequency domain. A specific variety of filters which accomplish this task is covered in [BGK90].