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ECES 682 Digital Image Processing Week 5

This lecture from Drexel University's ECE Department explores advanced topics in digital image processing, focusing on analysis and restoration of images affected by noise and distortion. Key concepts include noise models, Fourier analysis, adaptive filtering, and Wiener filtering. The lecture covers various types of noise such as Gaussian, Poisson, and Laplacian, as well as methods for estimating and restoring images. Strategies for combining low-pass filtering and noise masking principles are also discussed, emphasizing practical applications in image restoration.

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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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