This document discusses Fourier processing and covers topics such as continuous and discrete Fourier bases, sampling, 2D Fourier bases, and Fourier approximation. The continuous Fourier basis uses complex exponentials while the discrete Fourier basis uses a finite number of complex exponentials. Sampling and periodization allow transforming between continuous and discrete settings. The 2D Fourier basis is a product of 1D bases. Fourier approximation represents functions as a sum of complex exponentials.

1. Fourier
Processing
Gabriel Peyré
http://www.ceremade.dauphine.fr/~peyre/numerical-tour/

2. Overview
• Continuous Fourier Basis
• Discrete Fourier Basis
• Sampling
• 2D Fourier Basis
• Fourier Approximation

3. Continuous Fourier Bases
Continuous Fourier basis:
m (x) = em (x) = e2i mx

4. Continuous Fourier Bases
Continuous Fourier basis:
m (x) = em (x) = e2i mx

5. Fourier and Convolution

6. Fourier and Convolution
f ∗1[ − 1 , 1 ]
2 2
f
x− 1
2
x x+ 1
2

7. Fourier and Convolution
f ∗1[ − 1 , 1 ]
2 2
f
x− 1
2
x x+ 1
2
1
0.8
0.6
0.4
0.2
0
1
0.8
0.6
0.4
0.2
0
1
0.8
0.6
0.4
0.2
0

8. Fourier and Convolution
f ∗1[ − 1 , 1 ]
2 2
f
x− 1
2
x x+ 1
2
1
0.8
0.6
0.4
0.2
0
1
0.8
0.6
0.4
0.2
0
1
0.8
0.6
0.4
0.2
0

9. Overview
• Continuous Fourier Basis
• Discrete Fourier Basis
• Sampling
• 2D Fourier Basis
• Fourier Approximation

10. Discrete Fourier Transform

11. Discrete Fourier Transform

12. Discrete Fourier Transform
ˆ ˆ
g [m] = f [m] · h[m]
ˆ

13. Discrete Fourier Transform
ˆ ˆ
g [m] = f [m] · h[m]
ˆ

14. Overview
• Continuous Fourier Basis
• Discrete Fourier Basis
• Sampling
• 2D Fourier Basis
• Fourier Approximation

15. The Four Settings
Note: for Fourier, bounded periodic.
Infinite continuous domains: f0 (t), t R ... ...
+⇥
ˆ
f0 ( ) = f0 (t)e i t
dt
⇥
Periodic continuous domains: f0 (t), t ⇥ [0, 1] R/Z
1
ˆ
f0 [m] = f0 (t)e 2i mt
dt
0
Infinite discrete domains: f [n], n Z ... ...
ˆ
f( ) = f [n]ei n
n Z
Periodic discrete domains: f [n], n ⇤ {0, . . . , N 1} ⇥ Z/N Z
N 1
ˆ
f [m] = f [n]e
2i
N mn
n=0

16. Fourier Transforms
Periodization f0 (t) ⇥ n f0 (t + n)
f [n] = f0 (n/N )
Sampling
f0 (t), t R f0 (t), t [0, 1] Continuous
f [n], n Z f [n], 0 n<N Discrete
Fourier transform
Isometry f ⇥ f ˆ
Infinite Periodic
f0 (N (⇥ + 2k ))
ˆ
Sampling f0 ( ) ⇥ ˆ
{f0 (k)}k
ˆ
f0 ( ), ˆ
f0 [k], k Z Infinite
Periodization
R
ˆ ˆ
ˆ
f (⇥), ⇥ [0, 2 ] f [k], 0 k<N Periodic
k
f (⇥) =
Continuous Discrete
ˆ
+⇥ 1
ˆ
f0 ( ) = f0 (t)e i t
dt ˆ
f0 [m] = f0 (t)e 2i mt
dt
⇥ 0
N 1
ˆ
f( ) = f [n]e i n ˆ
f [m] = f [n]e
2i
N mn
n Z n=0

17. Sampling and Periodization
(a)
(b)
1
(c) 0
(d)

18. Sampling and Periodization: Aliasing
(a)
(b)
1
(c) 0
(d)

19. Uniform Sampling and Smoothness

20. Uniform Sampling and Smoothness

21. Uniform Sampling and Smoothness

22. Uniform Sampling and Smoothness

23. Overview
• Continuous Fourier Basis
• Discrete Fourier Basis
• Sampling
• 2D Fourier Basis
• Fourier Approximation

24. 2D Fourier Basis
1 2i m1 n1 + 2i m2 n2
em [n] = e N0 N0
= em1 [n1 ]em2 [n2 ]
N

25. 2D Fourier Basis
1 2i m1 n1 + 2i m2 n2
em [n] = e N0 N0
= em1 [n1 ]em2 [n2 ]
N

26. Overview
• Continuous Fourier Basis
• Discrete Fourier Basis
• Sampling
• 2D Fourier Basis
• Fourier Approximation

27. 1D Fourier Approximation

28. 1D Fourier Approximation
1 1
0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2
0 0
1 1
0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2
0 0

29. 2D Fourier Approximation