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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Properties A few interesting properties of the 2D DFT.

The properties described here can be best seen with some simple examples. The exampled are laid out by giving the spatial domain representation followed by the magnitude of the frequency domain representation and (optionally) the phase of the frequency information. The phase for the larger squares is mostly an incomprehensible mess (one of the properties should make it clear why this is true) and so it is not included. The base image signal we will use for this illustration is the 2D analog of the square pulse. It is a square centered in the middle of the image as shown below:
Note that for a two dimensional square pulse, the magnitude of the frequency is a two dimensional sinc, which is analogous to its one dimensional counterpart.
Rotation Property See the property
As the math predicts, rotating the pulse by forty-five degrees also rotates its frequency content by the same amount.
Scale Property See the property
Analagous to time dilation in one dimensional signals, shrinking in the spatial domain will cause an expansion of the frequency content. In this extreme case, a two pixel square pulse expands the sinc pulse until only the central lobe is significant.
Shift Property See the property
Shifting the scaled pulse to the upper left-hand corner shifts the phase along the line y = -x while leaving the magnitude untouched.