Example of two functions in quadrature. Two functions are said to be in quadrature when the are each others Hilbert transform. The black line is a sine modulated with a Gaussian window, the blue one is a cosine modulated with a gaussian window.
We will investigate why in signal processing, one often uses combined filter sets in quadrature. We will investigate a onedimensional example and a twodimensional example.
Basically, there are two types of filters: even filters and odd filters.
The blue function in the figure above is an even filter and will have maxima at even signal structures, like delta peaks. The twodimensional variant of this filter is also sometimes called a line detector, because it will have maximum response at line-like features.
The black function in the figure above is an odd filter and will have maxima at odd signal structures, like step edges. The twodimensional variant of this filter is also sometimes called an edge detector, because it will have maximum response at step edge-like features.
Next, we show a synthetic test signal (black) that contains a delta peak, a step edge and a combined step edge + delta peak. We also show the responses of even and odd filters for this signal.
Image (black) + output of an even filter (blue). The delta peak response is a minimum, the step edge response is a zero crossing. We find something strange in between for the combined edge.
Image (black) + output of an odd filter (blue). The delta peak response is a zero crossing, the step edge response is a maximum. Again, we find something strange in between for the combined edge.
Image filtered with a set of quadrature filters. All the responses of features of interest are maxima, for even features, odd features and features in between.
We have a classical test image:
Using the outputs of a steerable pyramid, we can calculate the local dominant orientation (see also Freeman & Adelson: "The design and use of steerable filters"). We assigned a color code to the orientation map of the image: red represents vertical edges (90o), green represents edges under 30o and blue edges under 150o. For edges in between, the colors were interpolated. When we only use a steerable pyramid containing odd filters, we obtain the following picture:
where we can clearly see that the response depends both on the local phase of the signal as well on the local orientation of the signal.
However, we combine the output of two steerable pyramids in quadrature, we obtain the following result:
where we have a response only dependent on the local orientation in the image.