Discrete Fourier Transform

1. From continuous to discrete
    when proper samples are obtained, one can usually treat these samples as an ordered discrete signal without considering sampling intervals in time or spatial domain, making these true digital signals.
2. 1D Discrete Fourier Transform

3. Fast Fourier Transform
    1D DFT can be implemented by some fast algorithm that is called Fast Fourier Transform.
4. From 1D DFT to 2D DFT

5. 2D DFT Properties
    The 2D DFT can be implemented by two 2 separable 1D DFT because the exponential base is separable

    The purple-colored summation inside the bracket is a 1D DFT can be applied to mth image line in a line by line fashion.
    The second 1D DFT can be applied to the results of the first 1D DFT in a column- by- column fashion.
6. DFT Periodicity and DFT Spectrum

7. DFT Pairs of Image Patterns
http://www.cs.unm.edu/~brayer/vision/fourier.html