Fourier Processing Gabriel Peyré http://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 1 2 1 2 f f ∗1[− 1 2 , 1 2 ]

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

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

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

Discrete Fourier Transform

Discrete Fourier Transform

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

Discrete Fourier Transform ˆ g[m] = ˆ f[m] · ˆ 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 ˆ f[m] = N 1 n=0 f[n]e 2i N mn ˆ f0 ( ) = +⇥ ⇥ f0 (t)e i tdt ˆ f0 [m] = 1 0 f0 (t)e 2i mtdt ˆ f( ) = n Z f[n]ei n Note: for Fourier, bounded periodic. .. . .. . .. . .. . f[n], n Z f[n], n ⇤ {0, . . . , N 1} ⇥ Z/NZ

ˆ 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 Continuous f0 (t), t R f0 (t), t [0, 1] f0 (t) ⇥ n f0 (t + n) Sampling ˆ f0 ( ) ⇥ { ˆ f0 (k)}k Discrete Infinite Periodic Continuous Sampling ˆ f[k], 0 k < N ˆ f0 ( ), R ˆ f0 [k], k Z Fourier transform Isometry f ⇥ ˆ f ˆ f0 ( ) = +⇥ ⇥ f0 (t)e i tdt ˆ f0 [m] = 1 0 f0 (t)e 2i mtdt ˆ f( ) = n Z f[n]ei n ˆ f(⇥), ⇥ [0, 2 ] Periodization ˆ f(⇥) = k ˆ f0 (N(⇥ + 2k )) f[n] = f0 (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 e 2i N0 m1n1+ 2i N0 m2n2 = em1 [n1 ]em2 [n2 ]

2D Fourier Basis em [n] = 1 N e 2i 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 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

2D Fourier Approximation