The other commonly used method for local orientation estimation involves the use of two-dimensional quadrature filters. A quadrature filter is a complex filter whose real part is related to its imaginary part via a Hilbert transform along a particular axis through the origin. One example of a quadrature filter is the following function,
which is a 2D Gabor function oriented in the vertical direction (see Figure 2).
Figure 2: Intensity plot of the real (left) and imaginary (right) parts
of a vertically oriented 2D Gabor function.
The real part of a Gabor function, which is even, is sensitive to oriented lines while the imaginary part, which is odd, is sensitive to oriented edges. Another valid choice for the even and odd components of a quadrature filter may be obtained from the directional derivatives of 2D gaussians and their Hilbert transforms, as described by Freeman and Adelson in [FA91]. See Knutsson and Granlund [KG83], [GK95] for another example.
Figure 3.1.3 illustrates the result of filtering a simple test image with a horisontally oriented quadrature filter. (The filter used in this example is composed of the second directional derivative of a Gaussian and its Hilbert transform.)
Figure 3: Demonstrating the convolution of a test image with a
horizontally oriented quadrature filter.
The magnitude of the complex filtered image conveniently combines the sensitivity of both parts of the filter into a phase-independent measure of orientation strength. The phase of the complex filtered image, meanwhile, can be inspected to determine whether a given orientation arose from an edge, a light line on a dark background, a dark line on a white background, etc. In other words, the information about the type of structure that gives rise to a given orientation resides in the phase of the filtered image. This is one advantage of quadrature-filter based methods for orientation estimation over gradient-based methods. Another advantage is that quadrature-filter based methods can be readily extended to handle instances of multiple simultaneous orientations, as occur at the intersection of lines and corners [And92], [HF94], [MS94], [Per92].
Quadrature filter set methods for computing orientation maps generally consist of two main steps: (1) filtering the image with a set of directionally tuned quadrature filters which can sense both oriented lines and edges, and (2) vectorially combining the filtered images to obtain the orientation map. The details of such methods are fairly involved and will not be elaborated upon in this paper. The interested reader is referred to [GK95].