The Fourier Transform Jean Baptiste Joseph Fourier

= 3 sin(x) A + 1 sin(3x) B A+B sin(5x) C A+B+C sin(7x)D A+B+C+D A sum of sines and cosines sin(x) A

The Continuous Fourier Transform

Complex Numbers Real Imaginary Z=(a,b) a b |Z| 

x – The wavelength is 1/u. – The frequency is u. 1 The 1D Basis Functions 1/u

The Fourier Transform 1D Continuous Fourier Transform: The Inverse Fourier Transform The Continuous Fourier Transform 2D Continuous Fourier Transform: The Inverse Transform The Transform

The wavelength is. The direction is u/v. The 2D Basis Functions u=0, v=0 u=1, v=0u=2, v=0 u=-2, v=0u=-1, v=0 u=0, v=1u=1, v=1u=2, v=1 u=-2, v=1u=-1, v=1 u=0, v=2u=1, v=2u=2, v=2 u=-2, v=2u=-1, v=2 u=0, v=-1u=1, v=-1u=2, v=-1 u=-2, v=-1u=-1, v=-1 u=0, v=-2u=1, v=-2u=2, v=-2 u=-2, v=-2u=-1, v=-2 U V

Discrete Functions N-1 f(x) f(x 0 ) f(x 0 +  x) f(x 0 +2  x) f(x 0 +3  x) f(n) = f(x 0 + n  x) x0x0 x0+xx0+x x 0 +2  xx 0 +3  x The discrete function f: { f(0), f(1), f(2), …, f(N-1) }

(u = 0,..., N-1) (x = 0,..., N-1) 1D Discrete Fourier Transform: The Discrete Fourier Transform 2D Discrete Fourier Transform: (x = 0,..., N-1; y = 0,…,M-1) (u = 0,..., N-1; v = 0,…,M-1)

Fourier spectrum log(1 + |F(u,v)|) Image f The Fourier Image Fourier spectrum |F(u,v)|

Frequency Bands Percentage of image power enclosed in circles (small to large) : 90%, 95%, 98%, 99%, 99.5%, 99.9% ImageFourier Spectrum

Low pass Filtering 90% 95% 98% 99% 99.5% 99.9%

Noise Removal Noisy image Fourier Spectrum Noise-cleaned image

Noise Removal Noisy imageFourier SpectrumNoise-cleaned image

High Pass Filtering OriginalHigh Pass Filtered

High Frequency Emphasis + OriginalHigh Pass Filtered

High Frequency Emphasis OriginalHigh Frequency Emphasis Original High Frequency Emphasis

OriginalHigh pass Filter High Frequency Emphasis High Frequency Emphasis + Histogram Equalization High Frequency Emphasis

2D Image2D Image - Rotated Fourier Spectrum Rotation

Image Domain Frequency Domain Fourier Transform -- Examples

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