1. Computer Vision Lecture 7: The Fourier Transform
Last week we talked about convolution as a technique for a variety of image processing applications (and we will see many more).
Convolution works in the spatial domain, i.e., it uses spatial filters that are shifted across an input image.
Sometimes it is useful to perform filtering in the frequency domain instead.
To understand what this means and to perform such filtering, we need to take a look at the Fourier transform.
September 30, 2014
Computer Vision Lecture 7: The Fourier Transform

2. The Fourier Transform
Jean Baptiste Joseph Fourier (and his Fourier-transformed image)
September 30, 2014
Computer Vision Lecture 7: The Fourier Transform

3. Computer Vision Lecture 7: The Fourier Transform
The Fourier transform receives a function as its input and outputs another function.
The Fourier transform assumes that the input function is periodic, i.e., repetitive, and defined for all real numbers.
It transforms this function into a weighted sum of sinusoidal terms.
Let us first look at one-dimensional functions.
Later, we will move on to two-dimensional functions such as images.
September 30, 2014
Computer Vision Lecture 7: The Fourier Transform

4. Computer Vision Lecture 7: The Fourier Transform
Adding Sine Waves =
3 sin(x) A
sin(x) A
+ 1 sin(3x) B
A+B
+ 0.8 sin(5x) C
A+B+C
+ 0.4 sin(7x) D
A+B+C+D
September 30, 2014
Computer Vision Lecture 7: The Fourier Transform

5. Computer Vision Lecture 7: The Fourier Transform
By adding a number of sine waves of different frequencies and amplitudes, we can approximate any given periodic function.
However, if we limit ourselves to only sine waves with no offset, i.e., difference in phase at x = 0, we cannot perfectly reconstruct every function. For instance, the output of our function at x = 0 must always be 0.
We could add a phase to each sine wave, but as Monsieur Fourier found out, it is sufficient to use both sine and cosine waves (which describe the same wave, but with a 90 phase difference) and keep all phases at 0.
September 30, 2014
Computer Vision Lecture 7: The Fourier Transform

6. Computer Vision Lecture 7: The Fourier Transform
So for any given function f, we could define a function Fs that assigns to any frequency an amplitude (i.e., weight) that the sine wave of that frequency will receive in our sum of sine waves.
Similarly, we could define a function Fc that does the same for the cosine waves.
In other words, these functions describe the contribution of different frequencies to the overall signal, our function f.
While f describes our function in the spatial domain (i.e., what value does it have in what position x), Fs and Fc describe the same function in the frequency domain (i.e., the extent to which different frequencies are included in the function).
September 30, 2014
Computer Vision Lecture 7: The Fourier Transform

7. Refresher: Complex Numbers
As you certainly remember, complex numbers consist of a real and an imaginary part, which we will use to represent the contribution of cosine and sine waves, respectively, to a given function. For example, 5 + 3i means Re = 5 and Im = 3.
Importantly, the exponential function represents a rotation in the 2D space spanned by the real and imaginary axes:
ei = cos  + isin  Im Re
sin  1 
cos 
September 30, 2014
Computer Vision Lecture 7: The Fourier Transform

8. The Two-Dimensional Fourier Transform
Note: F(u,v) is complex, while f(x, y) is real.
Two-Dimensional Inverse Fourier Transform:
 
amplitude basis functions
and phase
of required basis functions
September 30, 2014
Computer Vision Lecture 7: The Fourier Transform

9. 2D Discrete Fourier Transform
(u = 0,..., N-1; v = 0,…,M-1)
(x = 0,..., N-1; y = 0,…,M-1)
September 30, 2014
Computer Vision Lecture 7: The Fourier Transform

10. 2D Discrete Fourier Transform
What do these equations mean?
Well, if you look at the inverse transform, you see that each F(u, v) is multiplied by an exponential term for each pixel, and in sum they give us the original image.
In other words, each exponential function represents one particular wave, and the sum of these waves, weighted by F, is our original image f.
In the exponential term, you see an expression such as (ux + vy). This means that the variables u and v determine the “speed” with which the term increases when x and y, respectively, grow.
Since this term determines the current angle of a wave, u and v determine the frequency of the waves in horizontal and vertical direction, respectively.
September 30, 2014
Computer Vision Lecture 7: The Fourier Transform

11. Computer Vision Lecture 7: The Fourier Transform
September 30, 2014
Computer Vision Lecture 7: The Fourier Transform

12. Magnitude And Phase Spectra
Remember that we have both Fs and Fc to describe the sine and cosine contributions, respectively, to the spatial function.
They can be used to compute the magnitude m:
… and the phase :
September 30, 2014
Computer Vision Lecture 7: The Fourier Transform

13. Fourier Transform: Rectangle
September 30, 2014
Computer Vision Lecture 7: The Fourier Transform

14. Fourier Transform: Spatial Shifting
September 30, 2014
Computer Vision Lecture 7: The Fourier Transform

15. Fourier Transform: Rotation
September 30, 2014
Computer Vision Lecture 7: The Fourier Transform

16. Fourier Transform: Multiplicity
September 30, 2014
Computer Vision Lecture 7: The Fourier Transform

17. Fourier Transform: Real-World Images
September 30, 2014
Computer Vision Lecture 7: The Fourier Transform

18. Fourier Transform: Real-World Images
September 30, 2014
Computer Vision Lecture 7: The Fourier Transform

19. Fourier Transform: Real-World Images
September 30, 2014
Computer Vision Lecture 7: The Fourier Transform

20. Fourier Transform: Low-Pass Filtering
September 30, 2014
Computer Vision Lecture 7: The Fourier Transform

21. Fourier Transform: High-Pass Filtering
September 30, 2014
Computer Vision Lecture 7: The Fourier Transform

22. Fourier Transform: Noise Removal
September 30, 2014
Computer Vision Lecture 7: The Fourier Transform