Discrete Fourier Transform in 2D – Chapter 14

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 C B B C D A

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

8. Visualization • 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