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

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

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

Digtial Image Processing, Spring Noise Models Another noise: Poisson

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

Digtial Image Processing, Spring 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)

Digtial Image Processing, Spring 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 “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:

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

Digtial Image Processing, Spring “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