1. October 29, 2013Computer Vision Lecture 13: Fourier Transform II 1 The Fourier Transform In the previous lecture, we discussed the Hough transform. There is another, much more common and fundamental type of transform: the Fourier transform. The Fourier transform maps a function onto its representation in the frequency domain.

2. October 29, 2013Computer Vision Lecture 13: Fourier Transform II 2 The Fourier Transform Jean Baptiste Joseph Fourier (and his Fourier-transformed image)

3. October 29, 2013Computer Vision Lecture 13: Fourier Transform II 3 The Fourier Transform Let us first look at one-dimensional functions. Later, we will move on to two-dimensional functions, such as images I(x, y). 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.

4. October 29, 2013Computer Vision Lecture 13: Fourier Transform II 4 = 3 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 sin(x) A Adding Sine Waves

5. October 29, 2013Computer Vision Lecture 13: Fourier Transform II 5 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.

6. October 29, 2013Computer Vision Lecture 13: Fourier Transform II 6 The Fourier Transform So for any given function f, we could define a function F s 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 F c 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), F s and F c describe the same function in the frequency domain (i.e., the extent to which different frequencies are included in the function).

7. October 29, 2013Computer Vision Lecture 13: Fourier Transform II 7 The Two-Dimensional Fourier Transform  amplitude basis functions and phase of required basis functions Two-Dimensional Fourier Transform: Note: F(u,v) is complex, while f(x, y) is real. Two-Dimensional Inverse Fourier Transform:

8. October 29, 2013Computer Vision Lecture 13: Fourier Transform II 8 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: e i  = cos  + i  sin  cos   1 0 Im Re 0 sin 

9. October 29, 2013Computer Vision Lecture 13: Fourier Transform II 9 2D Discrete Fourier Transform (u = 0,..., N-1; v = 0,…,M-1) (x = 0,..., N-1; y = 0,…,M-1)

10. October 29, 2013Computer Vision Lecture 13: Fourier Transform II 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.

11. October 29, 2013Computer Vision Lecture 13: Fourier Transform II 11 The 2D Basis Functions u=0, v=0 u=1, v=0u=2, v=0 u=-2, v=0u=-1, v=0 u=0, v=1u=1, v=1u=2, v=1 u=-2, v=1u=-1, v=1 u=0, v=2u=1, v=2u=2, v=2 u=-2, v=2u=-1, v=2 u=0, v=-1u=1, v=-1u=2, v=-1 u=-2, v=-1u=-1, v=-1 u=0, v=-2u=1, v=-2u=2, v=-2 u=-2, v=-2u=-1, v=-2 U V

12. October 29, 2013Computer Vision Lecture 13: Fourier Transform II 12 Fourier Transform Example 1 Two-dimensional rectangle function as an image d) Magnitude of Fourier spectrum of the 2-D rectangle

13. October 29, 2013Computer Vision Lecture 13: Fourier Transform II 13 Fourier Transform Example 1 cross sectionisometric display

14. October 29, 2013Computer Vision Lecture 13: Fourier Transform II 14 Fourier Transform Example 2: CheetahZebra

15. October 29, 2013Computer Vision Lecture 13: Fourier Transform II 15 Fourier Transform of Cheetah Image magnitude of cheetah phase of cheetah

16. October 29, 2013Computer Vision Lecture 13: Fourier Transform II 16 Fourier Transform of Zebra Image magnitude of zebraphase of zebra

17. October 29, 2013Computer Vision Lecture 13: Fourier Transform II 17 Reconstruction with Zebra Phase and Cheetah Magnitude

18. October 29, 2013Computer Vision Lecture 13: Fourier Transform II 18 Reconstruction with Cheetah Phase and Zebra Magnitude