CS计算机代考程序代写 matlab Hive PowerPoint Presentation

PowerPoint Presentation

Fourier Series &
Fourier Transforms
Another way of thinking about frequency

Fourier Series

Animation by cmglee

Wikipedia – Fourier transform

Square wave (approx.)

Mehmet E. Yavuz

http://www.mathworks.com/matlabcentral/fileexchange/49533-fourier-series-animation-using-harmonic-circles
4

Sawtooth wave (approx.)

Mehmet E. Yavuz

One series in each of x and y

Generative Art, じゃがりきん, Video, 2018

Example: Music
We think of music in terms of frequencies at different magnitudes

Slide: Hoiem

Fourier analysis in images

Spatial domain images
Fourier decomposition frequency amplitude images
http://sharp.bu.edu/~slehar/fourier/fourier.html#filtering More: http://www.cs.unm.edu/~brayer/vision/fourier.html

8

Signals can be composed

+
=
http://sharp.bu.edu/~slehar/fourier/fourier.html#filtering More: http://www.cs.unm.edu/~brayer/vision/fourier.html
Spatial domain images
Fourier decomposition frequency amplitude images

Brian Pauw demo
Live Fourier decomposition images
Using FFT2 function

I hacked it a bit for MATLAB

New Live Fourier Transform code

Maru the cat: https://www.youtube.com/watch?v=2XID_W4neJo
10

Amplitude / Phase

Amplitude tells you “how much”
Phase tells you “where”

Translate the image?
Amplitude unchanged
Adds a constant to the phase.

Morse

Phase congruency edge detection: http://homepages.inf.ed.ac.uk/rbf/CVonline/LOCAL_COPIES/OWENS/LECT7/node2.html

Magnitude Amplitude of combined cosine and sine
Phase Relative proportions of sine and cosine

Bryan Morse, BYU Computer Science
http://www.astro.umd.edu/~lgm/ASTR410/ft_ref2.pdf

11

Fourier Transform
Stores the amplitude and phase at each frequency:
For mathematical convenience, this is often notated in terms of real and complex numbers
Related by Euler’s formula

Hays

Fourier Transform
Stores the amplitude and phase at each frequency:
For mathematical convenience, this is often notated in terms of real and complex numbers
Related by Euler’s formula

Hays

Hays
Phase encodes spatial information (indirectly):
Amplitude encodes how much signal there is at a particular frequency:

Fourier Transform
Stores the amplitude and phase at each frequency:
For mathematical convenience, this is often notated in terms of real and complex numbers
Related by Euler’s formula

What about phase?

Amplitude
Phase
Efros

What about phase?

Efros

What about phase?

Amplitude
Phase
Efros

Cheebra

Zebra phase, cheetah amplitude
Cheetah phase, zebra amplitude
Efros

Mathematical definition

is hiding our old friend:

So it’s just our signal f(x) times sine at frequency w

phase can be encoded
by sin/cos pair

Fourier Transform pairs

The Convolution Theorem

The Fourier transform of the convolution of two functions is the product of their Fourier transforms

The inverse Fourier transform of the product of two Fourier transforms is the convolution of the two inverse Fourier transforms

Convolution in spatial domain is equivalent to multiplication in frequency domain!
And vice versa!

21

The frequency amplitude of natural images are quite similar
Heavy in low frequencies, falling off in high frequencies
Will any image be like that, or is it a property of the world we live in?
Most information in the image is carried in the phase, not the amplitude
Not quite clear why
Efros

Smoothing with box filter
James Hays
1
1
1
1
1
1
1
1
1

23

Smoothing with Gaussian filter
James Hays

24

Comparison
James Hays
25

When sampling a signal at discrete intervals, the sampling frequency must be  2  fmax
fmax = max frequency of the input signal
This will allows to reconstruct the original perfectly from the sampled version

good
bad

Nyquist-Shannon Sampling Theorem

2D convolution theorem example

*
f(x,y)
h(x,y)
g(x,y)
|F(sx,sy)|

|H(sx,sy)|
|G(sx,sy)|

27

Why does the Gaussian give a nice smooth image, but the square filter give edgy artifacts?

Gaussian
Box filter
Filtering

Things we can’t understand without thinking in frequency
28

Gaussian

Box Filter

Filtering in spatial domain

-1
0
1
-2
0
2
-1
0
1

*
=
Hays

Filtering in frequency domain

FFT
FFT
Inverse FFT
=
Slide: Hoiem

Low Pass filter
A low-pass filter attenuates high frequencies and retains low frequencies unchanged. The result in the spatial domain is equivalent to that of a smoothing filter; as the blocked high frequencies correspond to sharp intensity changes, i.e. to the fine-scale details and noise in the spatial domain image.

33

High Pass filter
A highpass filter, on the other hand, yields edge enhancement or edge detection in the spatial domain, because edges contain many high frequencies. Areas of rather constant gray level consist of mainly low frequencies and are therefore suppressed.

34

A Low Pass Filter: Gaussian Filter
1D
2D
N=50 samples

35

Gaussian filter

g=fspecial(‘gaussian’,50,2.5);
surf(g);

36

Fourier Transforms
Gaussian filter

g=fspecial(‘gaussian’,50,2.5); h=fft2(g);
colormap(‘gray’);imagesc(g); colormap(‘gray’); imagesc(abs(h));

37

Fourier transforms
In Matlab lower frequencies are displayed at the corners
‘fftshift’ brings the origin to the center of the image

g=fspecial(‘gaussian’,50,2.5); h=fftshift(fft2(g));
colormap(‘gray’);imagesc(g); colormap(‘gray’); imagesc(abs(h));

38

Fourier transforms
Gaussian filter

g=fspecial(‘gaussian’,50,1.5); h=fftshift(fft2(g);
colormap(‘gray’);imagesc(g); colormap(‘gray’); imagesc(abs(h));

39

High Pass Filter

A high pass filter is the frequency complement of a low pass filter

40

A few questions
Why is frequency decomposition centered in middle, and duplicated and rotated?

From Euler:

Coefficients for negative frequencies
(i.e., ‘backwards traveling’ waves)

FFT of a real signal is conjugate symmetric
i.e., f(-x) = f*(x)

https://en.wikipedia.org/wiki/Euler%27s_formula

Negative Frequency and Its Physical Meaning


41

Fourier Transform
of important functions

spatial domain
frequency domain

42

Lavf57.25.100

2
2
)
Im(
)
Re(
j
j
+
±
=
A

)
Re(
)
Im(
tan
1
j
j
f

=

)
sin(
)
cos(
x
i
x
e
x
i
w
w
w
+
=

÷
÷
ø
ö
ç
ç
è
æ

=

+
±
=
)
+
=
+

Q
P
Q
P
Α
x
A
x
Q
x
P
1
2
2
tan
sin(
)
sin(
)
cos(
f
f

)
+
f
w
x
sin(

]
F[
]
F[
]
F[
h
g
h
g
=
*

]
[
F
]
[
F
]
[
F
1
1
1
h
g
gh



*
=

]
,
[
×
×
f

/docProps/thumbnail.jpeg