2D Frequency Domain Filtering and the 2D DFT
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The 2D DFT:
o The Transforms
o Frequency Content Location
o Properties of 2D DFT
o Examples of Properties

Frequency Domain Image Filters:
o 2D Filtering Concepts
o Smoothing
o Edge Detection
o Sharpening
o Filter Design



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Designing Filters in 2D Brought to you by Team Phantom Cruiser and the Power of Steam

Designing filters in 2D simply involves taking basic filter design techniques and basing them around the two dimensional frequency, which is the Cartesian distance from the origin, equal to the square root of the sum of the squares.

In a circularly oriented filter, i.e. one which is equally potent in all directions, the cutoff frequency remains a fixed number. However, when designing elliptically oriented filters, i.e. filters which have varying effectiveness at different angles, the cutoff frequency becomes a function of the angle.

One of the easiest class of filters to work with in two dimensions are Butterworth filters. Beginning with the equation for an ellipse, we now rotate our coordinate axes from (x,y) to (M,m) where M is the direction of the major axis and m the direction of the minor axis.

From here, we substitute the elliptical radius at this point into the equation for the magnitude of a Butterworth filter. Notice that we are designing these filters as magnitude only, no phase, due to the fact that phase spreading will strongly degrade the image.

And now we continue to combine terms and substitute in two cutoff frequencies instead of one: one for the major axis, and one for the minor.

This gives us our completed formula for design of a lowpass Butterworth filter of elliptical shape in 2D. By using an elliptical filter instead of the circular ones we used in previous examples, we can create strongly directional filters.

Top: Our car, filtered with a lowpass Butterworth filter with cutoff frequency .03 along the minor axis, .75 along the major, at a rotation of 60 degrees. Bottom: The same image with the same filter, only at a rotation of 120 degrees.

Once we have the basic lowpass filter, a highpass filter can easily be created, as the frequency response of a highpass filter = 1 - lowpass filter. Likewise, bandpass filters can be created just as easily by multiplying a highpass filter by a lowpass filter.