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

2. Digtial Image Processing, Spring 20062 Mr. Joseph Fourier To analyze a heat transient problem, Fourier proposed to express an arbitrary function by the formula

3. Digtial Image Processing, Spring 20063 Image Distortion Model Restoration depends on distortion  Common model: convolve plus noise  Special case: noise alone (no convolution)

4. Digtial Image Processing, Spring 20064 Noise Models Another noise: Poisson

5. Digtial Image Processing, Spring 20065 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

6. Digtial Image Processing, Spring 20066 Other Averages Geometric mean Harmonic mean These are generalization of the arithmetic average

7. Digtial Image Processing, Spring 20067 Adaptive Filters Filter changes parameters Simple model: f l (x, y) low pass filtered version of f a - adaptation parameter  a = 1: no noise filtering  0 = 1: full noise filtering (low pass image)

8. Digtial Image Processing, Spring 20068 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

9. Digtial Image Processing, Spring 20069 “Wiener” Filtering Signal model: f(x,y) zero mean stationary random process with autocorrelation function R f (x,y), power spectrum S f (u, v), n(x, y) uncorrelated zero mean stationary noise, variance N, S n (u, v) = N. Restoration model: Error criterion:

10. Digtial Image Processing, Spring 200610 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)

11. Digtial Image Processing, Spring 200611 Inverse Filtering Model: Restoration Error spectrum Two kinds of error: distortion and noise amplification.

12. Digtial Image Processing, Spring 200612 “Wiener” Inverse Filter Optimal filter Adaptation principle  |H(u,v)| 2 R(u,v)>N, H r (u, v) = (H(u, v)) -1  |H(u,v)| 2 R(u,v)<N, H r (u, v)<N, H r (u,v) = 0