Upload
polly-thomas
View
221
Download
2
Tags:
Embed Size (px)
Citation preview
October 29, 2013 Computer 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.
October 29, 2013 Computer Vision Lecture 13: Fourier Transform II
2
The Fourier Transform
Jean Baptiste Joseph Fourier (and his Fourier-transformed image)
October 29, 2013 Computer 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 numbersIt transforms this function into a weighted sum of sinusoidal terms.
October 29, 2013 Computer Vision Lecture 13: Fourier Transform II
4
=
3 sin(x) A
+ 1 sin(3x) B A+B
+ 0.8 sin(5x) CA+B+C
+ 0.4 sin(7x) DA+B+C+D
sin(x) A
Adding Sine Waves
October 29, 2013 Computer 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.
October 29, 2013 Computer Vision Lecture 13: Fourier Transform II
6
The Fourier TransformSo 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).
October 29, 2013 Computer Vision Lecture 13: Fourier Transform II
7
The Two-Dimensional Fourier Transform
)(2e ),(f),F( dxdyyxvu vyuxi
)(2e ),F(),( dudvvuyxf vyuxi
amplitude basis functionsand 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:
October 29, 2013 Computer Vision Lecture 13: Fourier Transform II
8
Refresher: Complex NumbersAs 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 exponentialfunction represents a rotation in the 2D spacespanned by the real andimaginary axes:
ei = cos + isin
cos
1
0
Im
Re0
sin
October 29, 2013 Computer Vision Lecture 13: Fourier Transform II
9
2D Discrete Fourier Transform
1
0
1
0
)(2),(
11),( N
x
M
y
M
vy
N
uxi
eyxfMN
vuF
(u = 0,..., N-1; v = 0,…,M-1)
1
0
1
0
)(2),(),( N
u
M
v
M
vy
N
uxi
evuFyxf
(x = 0,..., N-1; y = 0,…,M-1)
October 29, 2013 Computer Vision Lecture 13: Fourier Transform II
10
2D Discrete Fourier TransformWhat 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.
October 29, 2013 Computer Vision Lecture 13: Fourier Transform II
11
The 2D Basis Functions )(2 vyuxie
u=0, v=0 u=1, v=0 u=2, v=0u=-2, v=0 u=-1, v=0
u=0, v=1 u=1, v=1 u=2, v=1u=-2, v=1 u=-1, v=1
u=0, v=2 u=1, v=2 u=2, v=2u=-2, v=2 u=-1, v=2
u=0, v=-1 u=1, v=-1 u=2, v=-1u=-2, v=-1 u=-1, v=-1
u=0, v=-2 u=1, v=-2 u=2, v=-2u=-2, v=-2 u=-1, v=-2
U
V
October 29, 2013 Computer 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
October 29, 2013 Computer Vision Lecture 13: Fourier Transform II
13
Fourier Transform Example 1
cross section isometric display
October 29, 2013 Computer Vision Lecture 13: Fourier Transform II
14
Fourier Transform Example 2:
Cheetah Zebra
October 29, 2013 Computer Vision Lecture 13: Fourier Transform II
15
Fourier Transform of Cheetah Image
magnitude of cheetah phase of cheetah
October 29, 2013 Computer Vision Lecture 13: Fourier Transform II
16
Fourier Transform of Zebra Image
magnitude of zebra phase of zebra
October 29, 2013 Computer Vision Lecture 13: Fourier Transform II
17
Reconstruction with Zebra Phaseand Cheetah Magnitude
October 29, 2013 Computer Vision Lecture 13: Fourier Transform II
18
Reconstruction with Cheetah Phaseand Zebra Magnitude