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Fourier Transform J.B. Fourier 1768-1830. Image Enhancement in the Frequency Domain 1-D. A. A. sin(x). 3 sin(x). B. + 1 sin(3x). A+B. + 0.8 sin(5x). C. A+B+C. + 0.4 sin(7x). D. A+B+C+D. A sum of sines and cosines. =. Fourier spectrum of step function.

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## Fourier Transform J.B. Fourier 1768-1830

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**Image Enhancement in the**Frequency Domain 1-D**A**A sin(x) 3 sin(x) B + 1 sin(3x) A+B + 0.8 sin(5x) C A+B+C + 0.4 sin(7x) D A+B+C+D A sum of sines and cosines =**The Continuous Fourier Transform**The InverseFourier Transform The Fourier Transform 2D Continuous Fourier Transform: 1D Continuous Fourier Transform: The Inverse Transform The Transform**Discrete Functions**f(x) f(n) = f(x0 + nDx) f(x0+2Dx) f(x0+3Dx) f(x0+Dx) f(x0) 0 1 2 3 ... N-1 x0+2Dx x0+3Dx x0 x0+Dx The discrete function f: { f(0), f(1), f(2), … , f(N-1) }**The Discrete Fourier Transform**2D Discrete Fourier Transform: (u = 0,..., N-1; v = 0,…,M-1) (x = 0,..., N-1; y = 0,…,M-1) 1D Discrete Fourier Transform: (u = 0,..., N-1) (x = 0,..., N-1)**The 2D Basis Functions**V u=-2, v=2 u=-1, v=2 u=0, v=2 u=1, v=2 u=2, v=2 u=-2, v=1 u=-1, v=1 u=0, v=1 u=1, v=1 u=2, v=1 U u=0, v=0 u=-2, v=0 u=-1, v=0 u=1, v=0 u=2, v=0 u=-2, v=-1 u=-1, v=-1 u=0, v=-1 u=1, v=-1 u=2, v=-1 u=-2, v=-2 u=-1, v=-2 u=0, v=-2 u=1, v=-2 u=2, v=-2 The wavelength is . The direction is u/v .**The Fourier Transform**Jean Baptiste Joseph Fourier**Original: Real, imaginary, amplidute**• F.T.; Real, imaginary, amplitude • Reconstructed**Fourier spectrum |F(u,v)|**The Fourier Image Fourier spectrum log(1 + |F(u,v)|) Image f**Frequency Bands**Image Fourier Spectrum Percentage of image power enclosed in circles (small to large) : 90%, 95%, 98%, 99%, 99.5%, 99.9%**Low pass Filtering**90% 95% 98% 99% 99.5% 99.9%**Noise-cleaned image**Fourier Spectrum Noise Removal Noisy image**Higher frequencies dueto sharp image variations**(e.g., edges, noise, etc.)**High Pass Filtering**Original High Pass Filtered**High Frequency Emphasis**+ Original High Pass Filtered**High Frequency Emphasis**Original High Frequency Emphasis High Frequency Emphasis Original**High pass Filter**High Frequency Emphasis High Frequency Emphasis + Histogram Equalization High Frequency Emphasis Original**2D Image - Rotated**Fourier Spectrum Fourier Spectrum Rotation 2D Image**Fourier Transform -- Examples**Image Domain Frequency Domain**Fourier Transform -- Examples**Image Fourier spectrum**Chapter 4**Image Enhancement in the Frequency Domain**Chapter 4**Image Enhancement in the Frequency Domain**Chapter 4**Image Enhancement in the Frequency Domain**Chapter 4**Image Enhancement in the Frequency Domain**Chapter 4**Image Enhancement in the Frequency Domain**Chapter 4**Image Enhancement in the Frequency Domain**Chapter 4**Image Enhancement in the Frequency Domain**Chapter 4**Image Enhancement in the Frequency Domain**Chapter 4**Image Enhancement in the Frequency Domain

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