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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2D Filtering Concepts Brought to you by Team Phantom Cruiser and the Power of Steam

Top: Original image. Bottom: Spectrum of original image.

Above we have both an image and it's spectral representation. But before we can work with filtering this image, we must first examine what this frequency content indicates. In a time-based signal, a low frequency signal is one which changes slowly, whereas a high frequency signal has a more rapid change. To extend this concept to a spatial signal, it is easy to see that low-frequency data occurs where intensity values change slowly, i.e. a smooth gradient, and high frequencies equate to a rapid change in intensity, i.e. a sharp edge. Armed with these concepts, we can now anticipate the results of filtering an image.

Top: Original image. Bottom: Image filtered with ideal lowpass filter on Y axis, normalized cutoff frequency .15. X axis is an allpass.

When we try to use an rectangular lowpass filter in the Y direction two things are illustrated. First, an ideal rectangular filter cannot be used because it creates "ringing" artifacts, the same as in a one-dimensional transform. The second and more important realization is that a filter varying only in the Y frequency direction, and equal across all X, has its effects only in the Y direction of the image. We expect this from the rotation property, and from this we can infer, properly it turns out, that a filter is just as seperable as the transform, and therefore the direction of a filter will be the direction of its effect. Notice the way the shadows ripple up and down from horizonal lines in the original image, whereas vertical lines such as the edge of the car door are unaffected.