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ECES 682 Digital Image Processing Week 5

ECES 682 Digital Image Processing Week 5. Oleh Tretiak ECE Department Drexel University. Mr. Joseph Fourier. To analyze a heat transient problem, Fourier proposed to express an arbitrary function by the formula. Image Distortion Model. Restoration depends on distortion

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ECES 682 Digital Image Processing Week 5

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  1. ECES 682 Digital Image ProcessingWeek 5 Oleh Tretiak ECE Department Drexel University Digtial Image Processing, Spring 2006

  2. Mr. Joseph Fourier • To analyze a heat transient problem, Fourier proposed to express an arbitrary function by the formula Digtial Image Processing, Spring 2006

  3. Image Distortion Model • Restoration depends on distortion • Common model: convolve plus noise • Special case: noise alone (no convolution) Digtial Image Processing, Spring 2006

  4. Noise Models • Another noise: Poisson Digtial Image Processing, Spring 2006

  5. Noise Reduction • Model: s(i) = a + n(i) i = 1 ... n • n(i) Gaussian, independent • Best estimate of a: arithmetic average • When is the arithmetic average not good? • Long tailed distribution • If n(i) is Cauchy, average has no effect • If n(i) is Laplacian, median is the best estimate Digtial Image Processing, Spring 2006

  6. Other Averages • Geometric mean • Harmonic mean • These are generalization of the arithmetic average Digtial Image Processing, Spring 2006

  7. Adaptive Filters • Filter changes parameters • Simple model: • fl(x, y) low pass filtered version of f • a - adaptation parameter • a = 1: no noise filtering • 0 = 1: full noise filtering (low pass image) Digtial Image Processing, Spring 2006

  8. Ideas for Adaptation • Noise masking as an adaptation principle: • f(x, y) = constant (low frequency) —> a = 0 (noise visible) • f(x, y) highly variable —> a = 1 (image detail is masking the noise) • Fancier versions • Diffusion filtering • different low pass filtering in different directions • Wavelet filtering • estimate frequency content, treat each wavelet coefficient independently Digtial Image Processing, Spring 2006

  9. “Wiener” Filtering • Signal model: • f(x,y) zero mean stationary random process with autocorrelation function Rf(x,y), power spectrum Sf(u, v), n(x, y) uncorrelated zero mean stationary noise, variance N, Sn(u, v) = N. • Restoration model: • Error criterion: Digtial Image Processing, Spring 2006

  10. Analysis Result • Error spectrum • Best filter • Optimal noise spectrum • Principle: • R(u, v) > N, H = 1, E = N. • R(u, v) < N, H = 0, E = R(u, v) Digtial Image Processing, Spring 2006

  11. Inverse Filtering • Model: • Restoration • Error spectrum • Two kinds of error: distortion and noise amplification. Digtial Image Processing, Spring 2006

  12. “Wiener” Inverse Filter • Optimal filter • Adaptation principle • |H(u,v)|2R(u,v)>N, Hr(u, v) = (H(u, v))-1 • |H(u,v)|2R(u,v)<N, Hr(u, v)<N, Hr(u,v) = 0 Digtial Image Processing, Spring 2006

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