csce 641 computer graphics: fourier transform jinxiang chai
Post on 20-Jan-2016
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![Page 1: CSCE 641 Computer Graphics: Fourier Transform Jinxiang Chai](https://reader035.vdocuments.net/reader035/viewer/2022062222/56649d555503460f94a3355e/html5/thumbnails/1.jpg)
CSCE 641 Computer Graphics:Fourier Transform
Jinxiang Chai
![Page 2: CSCE 641 Computer Graphics: Fourier Transform Jinxiang Chai](https://reader035.vdocuments.net/reader035/viewer/2022062222/56649d555503460f94a3355e/html5/thumbnails/2.jpg)
Outline
• Aliasing
• Fourier transform
• Filtering
![Page 3: CSCE 641 Computer Graphics: Fourier Transform Jinxiang Chai](https://reader035.vdocuments.net/reader035/viewer/2022062222/56649d555503460f94a3355e/html5/thumbnails/3.jpg)
Image Scaling
This image is too big tofit on the screen. Howcan we reduce it?
How to generate a half-sized version?
![Page 4: CSCE 641 Computer Graphics: Fourier Transform Jinxiang Chai](https://reader035.vdocuments.net/reader035/viewer/2022062222/56649d555503460f94a3355e/html5/thumbnails/4.jpg)
Image Sub-sampling
Throw away every other row and
column to create a 1/2 size image- called image sub-sampling
1/4
1/8
![Page 5: CSCE 641 Computer Graphics: Fourier Transform Jinxiang Chai](https://reader035.vdocuments.net/reader035/viewer/2022062222/56649d555503460f94a3355e/html5/thumbnails/5.jpg)
Image Sub-sampling
1/4 (2x zoom) 1/8 (4x zoom)
Why does this look so crufty?
1/2
![Page 6: CSCE 641 Computer Graphics: Fourier Transform Jinxiang Chai](https://reader035.vdocuments.net/reader035/viewer/2022062222/56649d555503460f94a3355e/html5/thumbnails/6.jpg)
With/Without Aliasing
![Page 7: CSCE 641 Computer Graphics: Fourier Transform Jinxiang Chai](https://reader035.vdocuments.net/reader035/viewer/2022062222/56649d555503460f94a3355e/html5/thumbnails/7.jpg)
Even Worse for Synthetic Images
![Page 8: CSCE 641 Computer Graphics: Fourier Transform Jinxiang Chai](https://reader035.vdocuments.net/reader035/viewer/2022062222/56649d555503460f94a3355e/html5/thumbnails/8.jpg)
Difference between Lines
![Page 10: CSCE 641 Computer Graphics: Fourier Transform Jinxiang Chai](https://reader035.vdocuments.net/reader035/viewer/2022062222/56649d555503460f94a3355e/html5/thumbnails/10.jpg)
Aliasingoccurs when your sampling rate is not high enough to capture the
amount of detail in your image
Can give you the wrong signal/image—an alias
Where can it happen in computer graphics? During image synthesis:
– sampling continuous signal into discrete signal– e.g., ray tracing, line drawing, function plotting, etc.
During image processing: – resampling discrete signal at a different rate– e.g. Image warping, zooming in, zooming out, etc.
As well as other data such animation synthesisTo do sampling right, need to understand the structure of your
signal/image– signal processing
![Page 11: CSCE 641 Computer Graphics: Fourier Transform Jinxiang Chai](https://reader035.vdocuments.net/reader035/viewer/2022062222/56649d555503460f94a3355e/html5/thumbnails/11.jpg)
Signal Processing
Analysis, interpretation, and manipulation of signals
- images, videos, geometric and motion data
- sampling and reconstruction of the signals.
- minimal sampling rate for avoiding aliasing artifacts
- how to use filtering to remove the aliasing artifacts?
![Page 12: CSCE 641 Computer Graphics: Fourier Transform Jinxiang Chai](https://reader035.vdocuments.net/reader035/viewer/2022062222/56649d555503460f94a3355e/html5/thumbnails/12.jpg)
Outline
• Aliasing
• Fourier transform
• Filtering
![Page 13: CSCE 641 Computer Graphics: Fourier Transform Jinxiang Chai](https://reader035.vdocuments.net/reader035/viewer/2022062222/56649d555503460f94a3355e/html5/thumbnails/13.jpg)
Periodic Functions
• A periodic function is a function defined in an interval that repeats itself outside the interval
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Sine Waves
xAsin(
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Sine Waves
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Jean Baptiste Fourier (1768-1830)
had crazy idea (1807):
Any periodic function can be rewritten as a weighted sum of sines and cosines of different frequencies.
Don’t believe it? – Neither did Lagrange,
Laplace, Poisson and other big wigs
– Not translated into English until 1878!
But it’s true!– called Fourier Series
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A Sum of Sine WavesOur building block:
Add enough of them to get any signal f(x) you want!
xAsin(
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A Sum of Sine WavesOur building block:
Add enough of them to get any signal f(x) you want!
xAsin(
![Page 19: CSCE 641 Computer Graphics: Fourier Transform Jinxiang Chai](https://reader035.vdocuments.net/reader035/viewer/2022062222/56649d555503460f94a3355e/html5/thumbnails/19.jpg)
A Sum of Sine WavesOur building block:
Add enough of them to get any signal f(x) you want!
xAsin(
![Page 20: CSCE 641 Computer Graphics: Fourier Transform Jinxiang Chai](https://reader035.vdocuments.net/reader035/viewer/2022062222/56649d555503460f94a3355e/html5/thumbnails/20.jpg)
A Sum of Sine WavesOur building block:
Add enough of them to get any signal f(x) you want!
xAsin(
![Page 21: CSCE 641 Computer Graphics: Fourier Transform Jinxiang Chai](https://reader035.vdocuments.net/reader035/viewer/2022062222/56649d555503460f94a3355e/html5/thumbnails/21.jpg)
A Sum of Sine WavesOur building block:
Add enough of them to get any signal f(x) you want!
How many degrees of freedom?
What does each control?
Which one encodes the coarse vs. fine structure of the signal?
xAsin(
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How about Non-peoriodic Function?
• A non-periodic function can also be represented as a sum of sin’s and cos’s
• But we must use all frequencies, not just multiples of the period
• The sum is replaced by an integral.
dueuFxf uxi 2)()(
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Fourier Transform
A function f(x) can be represented as a sum of phase-shifted sine waves
dueuFxf uxi 2)()(
1
2sin2cos2
i
uxiuxe uxi
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Fourier Transform
A function f(x) can be represented as a sum of phase-shifted sine waves
How to compute F(u)?
dueuFxf uxi 2)()(
1
2sin2cos2
i
uxiuxe uxi
![Page 25: CSCE 641 Computer Graphics: Fourier Transform Jinxiang Chai](https://reader035.vdocuments.net/reader035/viewer/2022062222/56649d555503460f94a3355e/html5/thumbnails/25.jpg)
Fourier Transform
A function f(x) can be represented as a sum of phase-shifted sine waves
How to compute F(u)?
dxexfuF uxi 2)()(
dueuFxf uxi 2)()(
1
2sin2cos2
i
uxiuxe uxi
![Page 26: CSCE 641 Computer Graphics: Fourier Transform Jinxiang Chai](https://reader035.vdocuments.net/reader035/viewer/2022062222/56649d555503460f94a3355e/html5/thumbnails/26.jpg)
Fourier Transform
A function f(x) can be represented as a sum of phase-shifted sine waves
How to compute F(u)?
dxexfuF uxi 2)()(
dueuFxf uxi 2)()(
)()()( uiIuRuF
)()()( 22 uIuRuF
)(
)(tan)( 1
uR
uIu
Amplitude: Phase angle:
1
2sin2cos2
i
uxiuxe uxi
![Page 27: CSCE 641 Computer Graphics: Fourier Transform Jinxiang Chai](https://reader035.vdocuments.net/reader035/viewer/2022062222/56649d555503460f94a3355e/html5/thumbnails/27.jpg)
Fourier Transform
A function f(x) can be represented as a sum of phase-shifted sine waves
How to compute F(u)?
dxexfuF uxi 2)()(
dueuFxf uxi 2)()(
)()()( uiIuRuF
)()()( 22 uIuRuF
)(
)(tan)( 1
uR
uIu
Amplitude: Phase angle:
Inverse Fourier Transform
Fourier Transform
1
2sin2cos2
i
uxiuxe uxi
![Page 28: CSCE 641 Computer Graphics: Fourier Transform Jinxiang Chai](https://reader035.vdocuments.net/reader035/viewer/2022062222/56649d555503460f94a3355e/html5/thumbnails/28.jpg)
Fourier Transform
A function f(x) can be represented as a sum of phase-shifted sine waves
How to compute F(u)?
dxexfuF uxi 2)()(
dueuFxf uxi 2)()(
)()()( uiIuRuF
)()()( 22 uIuRuF
)(
)(tan)( 1
uR
uIu
Amplitude: Phase angle:
Inverse Fourier Transform
Fourier Transform
Dual property for Fourier transform and its inverse transform
1
2sin2cos2
i
uxiuxe uxi
![Page 29: CSCE 641 Computer Graphics: Fourier Transform Jinxiang Chai](https://reader035.vdocuments.net/reader035/viewer/2022062222/56649d555503460f94a3355e/html5/thumbnails/29.jpg)
Fourier Transform
Magnitude against frequency:
f(x) |F(u)|
How much of the sine wave with the frequency u appear in the original signal f(x)?
![Page 30: CSCE 641 Computer Graphics: Fourier Transform Jinxiang Chai](https://reader035.vdocuments.net/reader035/viewer/2022062222/56649d555503460f94a3355e/html5/thumbnails/30.jpg)
Fourier Transform
Magnitude against frequency:
f(x) |F(u)|
How much of the sine wave with the frequency u appear in the original signal f(x)?
5
?
![Page 31: CSCE 641 Computer Graphics: Fourier Transform Jinxiang Chai](https://reader035.vdocuments.net/reader035/viewer/2022062222/56649d555503460f94a3355e/html5/thumbnails/31.jpg)
Fourier Transform
Magnitude against frequency:
f(x) |F(u)|
How much of the sine wave with the frequency u appear in the original signal f(x)?
5
x
xi dxexf 10)(
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Fourier Transform
f(x)
f(x)
|F(u)|
|F(u)|
![Page 33: CSCE 641 Computer Graphics: Fourier Transform Jinxiang Chai](https://reader035.vdocuments.net/reader035/viewer/2022062222/56649d555503460f94a3355e/html5/thumbnails/33.jpg)
Fourier Transform
f(x)
f(x)
|F(u)|
|F(u)|
![Page 34: CSCE 641 Computer Graphics: Fourier Transform Jinxiang Chai](https://reader035.vdocuments.net/reader035/viewer/2022062222/56649d555503460f94a3355e/html5/thumbnails/34.jpg)
Fourier Transform
f(x)
f(x)
|F(u)|
|F(u)|
![Page 35: CSCE 641 Computer Graphics: Fourier Transform Jinxiang Chai](https://reader035.vdocuments.net/reader035/viewer/2022062222/56649d555503460f94a3355e/html5/thumbnails/35.jpg)
Fourier Transform
f(x)
f(x)
|F(u)|
|F(u)|
![Page 36: CSCE 641 Computer Graphics: Fourier Transform Jinxiang Chai](https://reader035.vdocuments.net/reader035/viewer/2022062222/56649d555503460f94a3355e/html5/thumbnails/36.jpg)
Box Function and Its Transform
21 02
1 1)(
x
xxf
x
f(x)
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Box Function and Its Transform
21 02
1 1)(
x
xxf
u
u
ui
ui
ui
ee
ui
e
ui
de
ui
uxidedxe
dxexf
xixiuxi
uxiuxiuxi
uxi
sin
2
sin2
22
22
2
)(
21
21
2
21
21
221
21
221
21
2
2
x
f(x)
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Box Function and Its Transform
21 02
1 1)(
x
xxf
)(sinsin
)( ucu
uuF
x
u
f(x)
|F(u)|
If f(x) is bounded, F(u) is unbounded
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Another Example
dxexf uxi 2)(
If the fourier transform of a function f(x) is F(u), what is the fourier transform of f(-x)?
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Another Example
)()(
)(
)(
*
2
2
uFuF
dzezf
dzezf
uzi
uzi
z
dxexf uxi 2)(
If the fourier transform of a function f(x) is F(u), what is the fourier transform of f(-x)?
![Page 41: CSCE 641 Computer Graphics: Fourier Transform Jinxiang Chai](https://reader035.vdocuments.net/reader035/viewer/2022062222/56649d555503460f94a3355e/html5/thumbnails/41.jpg)
Dirac Delta and Its Transform
00
0)(
x
xx
1)( dxx
x
f(x)
)0()()( fdxxxf
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Dirac Delta and Its Transform
00
0)(
x
xx
x
1)( uF1
u
f(x)
|F(u)|
Fourier transform and inverse Fourier transform arequalitatively the same, so knowing one direction gives you the other
1)( dxx
)0()()( fdxxxf
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Cosine and Its Transform
-1.5
-1
-0.5
0
0.5
1
1.5
1-1
If f(x) is even, so is F(u)
)2cos( x
![Page 44: CSCE 641 Computer Graphics: Fourier Transform Jinxiang Chai](https://reader035.vdocuments.net/reader035/viewer/2022062222/56649d555503460f94a3355e/html5/thumbnails/44.jpg)
Sine and Its Transform
-1.5
-1
-0.5
0
0.5
1
1.5
1
-1
-
If f(x) is odd, so is F(u)
)2sin( x
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Gaussian and Its Transform
2222 ue -0.02
0.03
0.08
0.13
0.18
-0.02
0.03
0.08
0.13
0.18
2
2
2
2
1
x
e
If f(x) is gaussian, F(u) is also guassian.
2222 ue
![Page 46: CSCE 641 Computer Graphics: Fourier Transform Jinxiang Chai](https://reader035.vdocuments.net/reader035/viewer/2022062222/56649d555503460f94a3355e/html5/thumbnails/46.jpg)
Gaussian and Its Transform
2222 ue -0.02
0.03
0.08
0.13
0.18
-0.02
0.03
0.08
0.13
0.18
2
2
2
2
1
x
e
If f(x) is gaussian, F(u) is also guassian.what’s the relationship of their variances?
2222 ue
![Page 47: CSCE 641 Computer Graphics: Fourier Transform Jinxiang Chai](https://reader035.vdocuments.net/reader035/viewer/2022062222/56649d555503460f94a3355e/html5/thumbnails/47.jpg)
Gaussian and Its Transform
2222 ue
constux
-0.02
0.03
0.08
0.13
0.18
-0.02
0.03
0.08
0.13
0.18
2
2
2
2
1
x
e
If f(x) is gaussian, F(u) is also guassian.what’s the relationship of their variances?
2222 ue
![Page 48: CSCE 641 Computer Graphics: Fourier Transform Jinxiang Chai](https://reader035.vdocuments.net/reader035/viewer/2022062222/56649d555503460f94a3355e/html5/thumbnails/48.jpg)
Properties
Linearity:
)()()()( ubGuaFxbgxaf
dxexgbdxexfadxexbgxaf uxiuxiuxi 222 )()())()((
![Page 49: CSCE 641 Computer Graphics: Fourier Transform Jinxiang Chai](https://reader035.vdocuments.net/reader035/viewer/2022062222/56649d555503460f94a3355e/html5/thumbnails/49.jpg)
Properties
Linearity:
Time-shift:
)()()()( ubGuaFxbgxaf
dxexxf uxi 20 )(
dxexgbdxexfadxexbgxaf uxiuxiuxi 222 )()())()((
![Page 50: CSCE 641 Computer Graphics: Fourier Transform Jinxiang Chai](https://reader035.vdocuments.net/reader035/viewer/2022062222/56649d555503460f94a3355e/html5/thumbnails/50.jpg)
Properties
Linearity:
Time-shift:
)()()()( ubGuaFxbgxaf
dxexxf uxi 20 )(
dxexgbdxexfadxexbgxaf uxiuxiuxi 222 )()())()((
![Page 51: CSCE 641 Computer Graphics: Fourier Transform Jinxiang Chai](https://reader035.vdocuments.net/reader035/viewer/2022062222/56649d555503460f94a3355e/html5/thumbnails/51.jpg)
Properties
Linearity:
Time-shift:
)()()()( ubGuaFxbgxaf
dxexxf uxi 20 )(
defedef iuxixui 22)(2 )()( 00
)()( 020 uFexxf uxi
dxexgbdxexfadxexbgxaf uxiuxiuxi 222 )()())()((
![Page 52: CSCE 641 Computer Graphics: Fourier Transform Jinxiang Chai](https://reader035.vdocuments.net/reader035/viewer/2022062222/56649d555503460f94a3355e/html5/thumbnails/52.jpg)
Properties
Linearity:
Time shift:
Derivative:
Integration:
Convolution:
)()()()( ubGuaFxbgxaf
)()( 020 uFexxf uxi
)()(
uuFx
xdf
u
uFdxxf
)()(
)()()()( uGuFxgxf
![Page 53: CSCE 641 Computer Graphics: Fourier Transform Jinxiang Chai](https://reader035.vdocuments.net/reader035/viewer/2022062222/56649d555503460f94a3355e/html5/thumbnails/53.jpg)
Outline
• Aliasing
• Fourier transform
• Filtering
![Page 54: CSCE 641 Computer Graphics: Fourier Transform Jinxiang Chai](https://reader035.vdocuments.net/reader035/viewer/2022062222/56649d555503460f94a3355e/html5/thumbnails/54.jpg)
1D Signal Filtering
![Page 55: CSCE 641 Computer Graphics: Fourier Transform Jinxiang Chai](https://reader035.vdocuments.net/reader035/viewer/2022062222/56649d555503460f94a3355e/html5/thumbnails/55.jpg)
2D Image Filtering
![Page 56: CSCE 641 Computer Graphics: Fourier Transform Jinxiang Chai](https://reader035.vdocuments.net/reader035/viewer/2022062222/56649d555503460f94a3355e/html5/thumbnails/56.jpg)
2D Image Filtering
![Page 57: CSCE 641 Computer Graphics: Fourier Transform Jinxiang Chai](https://reader035.vdocuments.net/reader035/viewer/2022062222/56649d555503460f94a3355e/html5/thumbnails/57.jpg)
Animation Filtering
• Cartoon animation filter
Click here
![Page 58: CSCE 641 Computer Graphics: Fourier Transform Jinxiang Chai](https://reader035.vdocuments.net/reader035/viewer/2022062222/56649d555503460f94a3355e/html5/thumbnails/58.jpg)
Signal Filtering
A filter is something that attenuates or enhances particular frequencies
![Page 59: CSCE 641 Computer Graphics: Fourier Transform Jinxiang Chai](https://reader035.vdocuments.net/reader035/viewer/2022062222/56649d555503460f94a3355e/html5/thumbnails/59.jpg)
Signal Filtering
A filter is something that attenuates or enhances particular frequencies
Easiest to visualize in the frequency domain, where filtering is defined as multiplication:
)()()( uGuFuH
Spectrum of the function
Spectrum of the filter
Spectrum of the filtered
function
![Page 60: CSCE 641 Computer Graphics: Fourier Transform Jinxiang Chai](https://reader035.vdocuments.net/reader035/viewer/2022062222/56649d555503460f94a3355e/html5/thumbnails/60.jpg)
Filtering
=
=
=
Low-pass
High-pass
Band-pass
F G H
![Page 61: CSCE 641 Computer Graphics: Fourier Transform Jinxiang Chai](https://reader035.vdocuments.net/reader035/viewer/2022062222/56649d555503460f94a3355e/html5/thumbnails/61.jpg)
Filtering
Identify filtered images from low-pass filter, high-pass filter, and band-pass filter?
![Page 62: CSCE 641 Computer Graphics: Fourier Transform Jinxiang Chai](https://reader035.vdocuments.net/reader035/viewer/2022062222/56649d555503460f94a3355e/html5/thumbnails/62.jpg)
Filtering in Frequency Domain
=
=
=
Low-pass
High-pass
Band-pass
F G H
So what happens in the spatial domain? Convolution
![Page 63: CSCE 641 Computer Graphics: Fourier Transform Jinxiang Chai](https://reader035.vdocuments.net/reader035/viewer/2022062222/56649d555503460f94a3355e/html5/thumbnails/63.jpg)
Convolution
Compute the integral of the product between f and a reversed and translated version of g
dtgftgtfth )()()()()(
![Page 64: CSCE 641 Computer Graphics: Fourier Transform Jinxiang Chai](https://reader035.vdocuments.net/reader035/viewer/2022062222/56649d555503460f94a3355e/html5/thumbnails/64.jpg)
Convolution
Compute the integral of the product between f and a reversed and translated version of g
dtgftgtfth )()()()()(
![Page 65: CSCE 641 Computer Graphics: Fourier Transform Jinxiang Chai](https://reader035.vdocuments.net/reader035/viewer/2022062222/56649d555503460f94a3355e/html5/thumbnails/65.jpg)
Convolution
Compute the integral of the product between f and a reversed and translated version of g
dtgftgtfth )()()()()(
Reversed and translated function g
![Page 66: CSCE 641 Computer Graphics: Fourier Transform Jinxiang Chai](https://reader035.vdocuments.net/reader035/viewer/2022062222/56649d555503460f94a3355e/html5/thumbnails/66.jpg)
Convolution
Compute the integral of the product between f and a reversed and translated version of g
dtgftgtfth )()()()()(
Reversed and translated function g
![Page 67: CSCE 641 Computer Graphics: Fourier Transform Jinxiang Chai](https://reader035.vdocuments.net/reader035/viewer/2022062222/56649d555503460f94a3355e/html5/thumbnails/67.jpg)
Convolution
Compute the integral of the product between f and a reversed and translated version of g
dtgftgtfth )()()()()(
t=2.5
![Page 68: CSCE 641 Computer Graphics: Fourier Transform Jinxiang Chai](https://reader035.vdocuments.net/reader035/viewer/2022062222/56649d555503460f94a3355e/html5/thumbnails/68.jpg)
Convolution
Compute the integral of the product between f and a reversed and translated version of g
dtgftgtfth )()()()()(
t=4.6
![Page 69: CSCE 641 Computer Graphics: Fourier Transform Jinxiang Chai](https://reader035.vdocuments.net/reader035/viewer/2022062222/56649d555503460f94a3355e/html5/thumbnails/69.jpg)
Convolution
Compute the integral of the product between f and a reversed and translated version of g
dtgftgtfth )()()()()(
t=4.6
For discrete signals such as images, this is just a weighted sum of all function values in the window
- weights and window sizes are filter dependent
![Page 70: CSCE 641 Computer Graphics: Fourier Transform Jinxiang Chai](https://reader035.vdocuments.net/reader035/viewer/2022062222/56649d555503460f94a3355e/html5/thumbnails/70.jpg)
Convolution
For discrete signals such as images, this is just a weighted sum of all function values in the window
- weights and window sizes are filter dependent
![Page 71: CSCE 641 Computer Graphics: Fourier Transform Jinxiang Chai](https://reader035.vdocuments.net/reader035/viewer/2022062222/56649d555503460f94a3355e/html5/thumbnails/71.jpg)
Convolution Theory
dtedtgf uti 2))()((
h(t)
)(uH
![Page 72: CSCE 641 Computer Graphics: Fourier Transform Jinxiang Chai](https://reader035.vdocuments.net/reader035/viewer/2022062222/56649d555503460f94a3355e/html5/thumbnails/72.jpg)
Convolution Theory
dtedtgf uti 2))()((
dedgf ui )(2))()((
t)(uH
![Page 73: CSCE 641 Computer Graphics: Fourier Transform Jinxiang Chai](https://reader035.vdocuments.net/reader035/viewer/2022062222/56649d555503460f94a3355e/html5/thumbnails/73.jpg)
Convolution Theory
dtedtgf uti 2))()((
dedgf ui )(2))()((
ddegef uiui 22 )()(
t)(uH
![Page 74: CSCE 641 Computer Graphics: Fourier Transform Jinxiang Chai](https://reader035.vdocuments.net/reader035/viewer/2022062222/56649d555503460f94a3355e/html5/thumbnails/74.jpg)
Convolution Theory
dtedtgf uti 2))()((
dedgf ui )(2))()((
ddegef uiui 22 )()(
degdef uiui 22 )()(
t)(uH
![Page 75: CSCE 641 Computer Graphics: Fourier Transform Jinxiang Chai](https://reader035.vdocuments.net/reader035/viewer/2022062222/56649d555503460f94a3355e/html5/thumbnails/75.jpg)
Convolution Theory
dtedtgf uti 2))()((
dedgf ui )(2))()((
ddegef uiui 22 )()(
degdef uiui 22 )()(
)(uF )(uG
t)(uH
![Page 76: CSCE 641 Computer Graphics: Fourier Transform Jinxiang Chai](https://reader035.vdocuments.net/reader035/viewer/2022062222/56649d555503460f94a3355e/html5/thumbnails/76.jpg)
Qualitative Property
The spectrum of a function tells us the relative amounts of high and low frequencies– Sharp edges imply high frequencies– Smooth variations give low frequencies– The zero frequency term is the average value of function
A function is band-limited if its spectrum has no frequencies above a maximum limit– sine, cosine are bandlimited– Box, Gaussian, etc. are not
![Page 77: CSCE 641 Computer Graphics: Fourier Transform Jinxiang Chai](https://reader035.vdocuments.net/reader035/viewer/2022062222/56649d555503460f94a3355e/html5/thumbnails/77.jpg)
Qualitative Property
The spectrum of a function tells us the relative amounts of high and low frequencies– Sharp edges imply high frequencies– Smooth variations give low frequencies– The zero frequency term is the average value of function
A function is band-limited if its spectrum has no frequencies above a maximum limit– sine, cosine are bandlimited– Box, Gaussian, etc are not
![Page 78: CSCE 641 Computer Graphics: Fourier Transform Jinxiang Chai](https://reader035.vdocuments.net/reader035/viewer/2022062222/56649d555503460f94a3355e/html5/thumbnails/78.jpg)
Summary
• Convert a function from the spatial domain to the frequency domain and vice versa.
• Interpret a function in either domain, e.g., filtering and correlation.
• Build your intuition about functions and their spectra.
![Page 79: CSCE 641 Computer Graphics: Fourier Transform Jinxiang Chai](https://reader035.vdocuments.net/reader035/viewer/2022062222/56649d555503460f94a3355e/html5/thumbnails/79.jpg)
Next lecture
• Fourier transform for 2D signals (images)
• Sampling and reconstruction
• How to avoid aliasing
Fourier transform