trasformata di fourier - unisa · example of lsis defocused image ( g ) is a processed version of...
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Trasformata di Fourier
Michele Nappi, Ph.D
Università degli Studi di Salerno
biplab.unisa.it
089-963334
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IndexLinear Systems
• Definitions & Properties
• Shift Invariant Linear Systems
• Linear Systems and Convolutions
• Linear Systems and sinusoids
• Complex Numbers and Complex Exponentials
• Linear Systems - Frequency Response
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Linear System
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Linear System Properties
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Linear Shift Invariant Systems (LSIS)
Linearity:
1f 1g 2f 2g
21 ff 21 gg
Shift invariance:
axf axg
a a
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Example of LSIS
Defocused image ( g ) is a
processed version of the
focused image ( f )
g f
Ideal lens is a LSIS xf xgLSIS
Linearity: Brightness variation
Shift invariance: Scene movement
(not valid for lenses with non-linear distortions)
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Shift Invariant Linear System (cont.)
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Convolution in Continous Case
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1 2-1-2
xc
-1 1
1
xb
-1 1
1 xa
bac
1
Convolution - Example
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Convolution Kernel – Impulse Response
f gh hfg
• What h will give us g = f ?
Dirac Delta Function (Unit Impulse)
x
21
0
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Impulse Sequence
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Shift-Invariant Linear
System and Convolution
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Complex Number
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Complex Number (cont.)
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Complex Number (cont.)
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The (Co-)Sinusoid
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The (Co-)Sinusoid (cont.)
• The wavelength of
sin(2πωx) is 1/ω .
• The frequency is ω .
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The (Co-)Sinusoid (cont.)
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The (Co-)Sinusoid (cont.)
• If we add a Sine wave
to a Cosine wave with
the same frequency
we get a scaled and
shifted (Co-)Sine
wave with the same
frequency
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The (Co-)Sinusoid (cont.)
Phase tg
Amplitude R
sin cossinsincos
:obtain we(1) from Thus
sin and cos
:such that exists There
1
Since
(1) cossinbcosasin
:Proof
1-
22
2222
2222
2
22
2
22
2222
22
a
b
ba
xbaxxba
ba
b
ba
a
ba
b
ba
a
xba
bx
ba
abaxx
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The response of Shift-Invariant
Linear System to a Sine wave
• The response of a
shift-invariant linear
system to a sine wave
is a shifted and
scaled sine wave with
the same frequency.
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The response of Shift-Invariant
Linear System to a Sine wave (cont.)
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Jean Baptiste Joseph 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
– Possibly the greatest tool
used in Engineering
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The Fourier Transform (cont.)
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Time and Frequency
• example : g(t) = sin(2pf t) + (1/3)sin(2p(3f) t)
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Time and Frequency
• example : g(t) = sin(2pf t) + (1/3)sin(2p(3f) t)
= +
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Frequency Spectra
• example : g(t) = sin(2pf t) + (1/3)sin(2p(3f) t)
= +
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Frequency Spectra
• Usually, frequency is more interesting than the phase
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= +
=
Frequency Spectra
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= +
=
Frequency Spectra
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= +
=
Frequency Spectra
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= +
=
Frequency Spectra
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= +
=
Frequency Spectra
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= 1
1sin(2 )
k
A ktk
Frequency Spectra
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Frequency Spectra
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Frequency Analysis
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The Fourier Transform
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The Fourier Transform (cont.)
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The Fourier Transform (cont.)
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The Fourier Transform (cont.)
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Frequency Domain
Filtering
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2D Fourier Transform
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Discrete Fourier Transform
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Discrete Fourier Transform (cont.)
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Discrete Fourier Transform (cont.)
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Discrete Fourier Transform (cont.)
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Discrete Fourier Transform (cont.)
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Discrete Fourier Transform (cont.)
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Discrete Fourier Transform (cont.)
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Discrete Fourier Transform (cont.)
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Spatial vs Frequency Domain
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Spatial vs Frequency Domain (cont.)
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Spatial vs Frequency Domain (cont.)
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Low Pass Filter
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Sharpening (High Pass) Filter
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Band Pass Filter
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Filtering Example (cont.)
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Filtering Example
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30/03/2016 Michele Nappi 59
Filtering Example (cont.)
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30/03/2016 Michele Nappi 60
Filtering Example (cont.)
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30/03/2016 Michele Nappi 61
Frequency Bands
• Percentage of image
power enclosed in
circles (small to
large):
90, 95, 98, 99, 99.5, 99.9
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30/03/2016 Michele Nappi 62
Fourier Transform
• Computing Time
– O(n2)
• Fast Fourier Transform (FFT)
– O(nlogn)