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Slides for a course on signal and image processing.
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Fourier
Processing
Gabriel Peyréhttp://www.ceremade.dauphine.fr/~peyre/numerical-tour/
Overview
•Continuous Fourier Basis
•Discrete Fourier Basis
•Sampling
•2D Fourier Basis
•Fourier Approximation
�m(x) = em(x) = e2i�mx
Continuous Fourier Bases
Continuous Fourier basis:
�m(x) = em(x) = e2i�mx
Continuous Fourier Bases
Continuous Fourier basis:
Fourier and Convolution
Fourier and Convolution
x! x+x 12
12
f
f
"1[! 12 ,
12 ]
Fourier and Convolution
x! x+x 12
12
f
f
"1[! 12 ,
12 ]
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Fourier and Convolution
x! x+x 12
12
f
f
"1[! 12 ,
12 ]
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Overview
•Continuous Fourier Basis
•Discrete Fourier Basis
•Sampling
•2D Fourier Basis
•Fourier Approximation
Discrete Fourier Transform
Discrete Fourier Transform
Discrete Fourier Transform
g[m] = f [m] · h[m]
Discrete Fourier Transform
g[m] = f [m] · h[m]
Overview
•Continuous Fourier Basis
•Discrete Fourier Basis
•Sampling
•2D Fourier Basis
•Fourier Approximation
Infinite continuous domains:
Periodic continuous domains:
Infinite discrete domains:
Periodic discrete domains:
f0(t), t � R
f0(t), t ⇥ [0, 1] � R/Z
The Four Settings
f [m] =N�1�
n=0
f [n]e�2i�N mn
f0(�) =� +⇥
�⇥f0(t)e�i�tdt
f0[m] =� 1
0f0(t)e�2i�mtdt
f(�) =�
n�Zf [n]ei�n
Note: for Fourier, bounded � periodic.
. . . . . .
. . .. . .f [n], n � Z
f [n], n ⇤ {0, . . . , N � 1} ⇥ Z/NZ
f [m] =N�1�
n=0
f [n]e�2i�N mn
Fourier Transforms
Discrete
Infinite Periodic
f [n], n � Z f [n], 0 � n < N
Periodization
Continuousf0(t), t � R f0(t), t � [0, 1]
f0(t) ⇥��
n f0(t + n)
Sam
plin
g
f0(�) ⇥� {f0(k)}k
Discrete
Infinite
Periodic
Continuous
Sampling
f [k], 0 � k < N
f0(�),� � R f0[k], k � Z Four
ier
tran
sfor
mIs
omet
ryf⇥�
f
f0(�) =� +⇥
�⇥f0(t)e�i�tdt f0[m] =
� 1
0f0(t)e�2i�mtdt
f(�) =�
n�Zf [n]ei�n
f(⇥),⇥ � [0, 2�]
Peri
odiz
atio
nf(⇥
)=
� k
f 0(N
(⇥+
2k�))
f[n
]=f 0
(n/N
)
Sampling and Periodization
(a)
(c)
(d)
(b)
1
0
Sampling and Periodization: Aliasing
(b)
(c)
(d)
(a)
0
1
Uniform Sampling and Smoothness
Uniform Sampling and Smoothness
Uniform Sampling and Smoothness
Uniform Sampling and Smoothness
Overview
•Continuous Fourier Basis
•Discrete Fourier Basis
•Sampling
•2D Fourier Basis
•Fourier Approximation
2D Fourier Basis
em[n] =1�N
e2i�N0
m1n1+ 2i�N0
m2n2 = em1 [n1]em2 [n2]
2D Fourier Basis
em[n] =1�N
e2i�N0
m1n1+ 2i�N0
m2n2 = em1 [n1]em2 [n2]
Overview
•Continuous Fourier Basis
•Discrete Fourier Basis
•Sampling
•2D Fourier Basis
•Fourier Approximation
1D Fourier Approximation
1D Fourier Approximation
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2D Fourier Approximation
