1. Phase and Amplitude in Fourier Transforms, Meaning of frequencies

2. Shift Invariant Linear Systems
Superposition
Scaling
Shift Invariance

3. These can be arbitrary orthogonal or unitary transforms, not only Fourier

4. Remember – the idea is to use the same basis functions both ways – like in Walsh
With unitary transforms you do not need matrix inversion

5. Fourier Transform
What the base elements look like for 2D images?

6. What the base elements look like for 2D images?
Constant perpendicular to the direction
Sinusoid along the direction
To get some sense of what basis elements look like, we plot a basis element, or rather its real part – as a function of x, y for some fixed u,v
We get a function that is constant when (ux+vy) is constant
The magnitude of the vector (u,v) gives a frequency, and its direction gives an orientation.
The function is sinusoid with this frequency along the direction, and constant perpendicular to the direction.

7. How u and v look like Here u and v are larger than the previous slide
Here u and v are larger than the upper example
Higher frequency

8. Phase and magnitude of Fourier Transforms
Interesting property of NATURAL images

9. Cheetah Image
Fourier Magnitude (above)
Fourier Phase (below)

10. Zebra Image
Fourier Magnitude (above)
Fourier Phase (below)

11. Reconstruction with
Zebra phase,
Cheetah Magnitude
We see Zebra from phase, we lost cheetah from magnitude

12. Reconstruction with
Cheetah phase,
Zebra Magnitude
We see Cheetah from phase, we lost Zebra from magnitude

13. Suggested Reading
Chapter 7, David A. Forsyth and Jean Ponce, "Computer Vision: A Modern Approach"