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

Higher frequencies due to sharp image variations (e.g., edges, noise, etc.)

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

Properties of the Fourier Transform – Developed on the board… (e.g., separability of the 2D transform, linearity, scaling/shrinking, derivative, rotation, shift  phase-change, periodicity of the discrete transform, etc.) We also developed the Fourier Transform of various commonly used functions, and discussed applications which are not contained in the slides (motion, etc.)

2D Image2D Image - Rotated Fourier Spectrum

Image Domain Frequency Domain Fourier Transform -- Examples

Image Domain Frequency Domain Fourier Transform -- Examples

Image Domain Frequency Domain Fourier Transform -- Examples

Image Domain Frequency Domain Fourier Transform -- Examples

Image Fourier spectrum Fourier Transform -- Examples

Image Fourier spectrum Fourier Transform -- Examples

Image Fourier spectrum Fourier Transform -- Examples

Image Fourier spectrum Fourier Transform -- Examples

Image Fourier spectrum Fourier Transform -- Examples