Jinxiang Chai Composite Transformations and Forward Kinematics 0.
Computational Photography: Fourier Transform Jinxiang Chai.
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Transcript of Computational Photography: Fourier Transform Jinxiang Chai.
Image Scaling
This image is too big tofit on the screen. Howcan we reduce it?
How to generate a half-sized version?
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
Really Bad in Video
Click here
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 and vision? 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
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?
Periodic Functions
• A periodic function is a function defined in an interval that repeats itself outside the interval
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
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(
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)()(
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
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
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
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
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
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
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)?
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
?
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)(
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)
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
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)?
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)?
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
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
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
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
Properties
Linearity:
Time-shift:
)()()()( ubGuaFxbgxaf
dxexxf uxi 20 )(
dxexgbdxexfadxexbgxaf uxiuxiuxi 222 )()())()((
Properties
Linearity:
Time-shift:
)()()()( ubGuaFxbgxaf
dxexxf uxi 20 )(
dxexgbdxexfadxexbgxaf uxiuxiuxi 222 )()())()((
Properties
Linearity:
Time-shift:
)()()()( ubGuaFxbgxaf
dxexxf uxi 20 )(
defedef iuxixui 22)(2 )()( 00
)()( 020 uFexxf uxi
dxexgbdxexfadxexbgxaf uxiuxiuxi 222 )()())()((
Properties
Linearity:
Time shift:
Derivative:
Integration:
Convolution:
)()()()( ubGuaFxbgxaf
)()( 020 uFexxf uxi
)()(
uuFx
xdf
u
uFdxxf
)()(
)()()()( uGuFxgxf
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
Filtering in Frequency Domain
=
=
=
Low-pass
High-pass
Band-pass
F G H
So what happens in the spatial domain? Convolution
Convolution
Compute the integral of the product between f and a reversed and translated version of g
dtgftgtftC )()()()()(
Convolution
Compute the integral of the product between f and a reversed and translated version of g
dtgftgtftC )()()()()(
Convolution
Compute the integral of the product between f and a reversed and translated version of g
dtgftgtftC )()()()()(
Reversed and translated function g
Convolution
Compute the integral of the product between f and a reversed and translated version of g
dtgftgtftC )()()()()(
Reversed and translated function g
Convolution
Compute the integral of the product between f and a reversed and translated version of g
dtgftgtftC )()()()()(
t=2.5
Convolution
Compute the integral of the product between f and a reversed and translated version of g
dtgftgtftC )()()()()(
t=4.6
Convolution
Compute the integral of the product between f and a reversed and translated version of g
dtgftgtftC )()()()()(
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
Convolution Theory
dxedxgf uxi 2))()((
dedgf ui )(2))()((
ddegef uiui 22 )()(
degdef uiui 22 )()(
)(uF )(uG
x
Qualitative Property
The spectrum of a functions tells us the relative amounts of high and low frequencies– Sharp edges give high frequencies– Smooth variations give low frequencies
A function is band-limited if its spectrum has no frequencies above a maximum limit– sine, cosine are bandlimited– Box, Gaussian, etc are not
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.