1. Discrete Fourier Transform in 2D – Chapter 14

2. Discrete Fourier Transform – 1D Forward Inverse M is the length (number of discrete samples)

3. Discrete Fourier Transform – 2D After a bit of algebraic manipulation we find that the 2D Fourier Transform is nothing more than two 1D transforms Do a 1D DFT over the rows of the image Then do a 1D DFT over the columns of the row-wise DFT This is for an MxN (columns by rows) 1D DFT over row g(*,v)

4. What’s it all mean? Whereas for the 1D DFT we were adding together 1D sinusoidal waves… –For the 2D DFT we are adding together 2D sinusoidal surfaces Whereas for the 1D DFT we considered parameters of amplitude, frequency, and phase –For the 2D DFT we consider parameters of amplitude, frequency, phase, and orientation (angle)

5. Visualization A pixel in DFT space represents an orientation and frequency of the sinusoidal surface The corners each represent low frequency components which is inconvenient

6. Quadrant swapping Quadrant swapping brings all low frequency data to the center A D B C C B D A

7. A pixel in DFT space represents an orientation and frequency of the sinusoidal surface Visualization

8. The image is really a depiction of the frequency power spectrum and as such should be thought of as a surface Low frequencies are at the center, high frequencies are at the boundaries

9. Visualization Image coordinates represent the effective frequency… …and the orientation is the sampling interval

10. Something interesting If the DFT space is square then rotation in the spatial domain is rotation in the frequency domain

11. Artifacts Since spatial signal is assumed to be periodic, drastic differences (large gradients) at the opposing edges cause a strong vertical line in the DFT

12. No border differences