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Digtial Image Processing, Spring 2006
1
ECES 682 Digital Image ProcessingWeek 5
Oleh TretiakECE DepartmentDrexel 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
QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.f (x) =A0 + Ak cos(2πkx/ L)+ Bk sin(2πkx/ L)
k=1
∞
∑k=1
∞
∑
0 ≤x≤L
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
xg = xii=1
n
∏⎛⎝⎜⎞⎠⎟
1/n
xh =n
1 / xii=1
n
∑
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)
)f (x, y) =af(x,y) + (1−a) fl (x,y)
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:
g(x, y) = f (x,y) + n(x,y)
)f (x, y) =h(x,y)∗g(x,y)
minh (g(x, y) − f(x,y))2
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)
E(u,v) =1−H(u,v) 2 Rf (u,v) + H(u,v) 2 N
Ho(u,v) =Rf (u,v)
Rf (u,v) + N
E0 (u,v) =Rf (u,v)N
Rf (u,v) + N=
11 / Rf (u,v) +1 / N
Digtial Image Processing, Spring 2006 11
Inverse Filtering
• Model:
• Restoration
• Error spectrum
• Two kinds of error: distortion and noise amplification.
g(x, y) =h(x,y)∗f (x,y) + n(x,y)
)f (x, y) =hr (x,y)∗f(x,y)
E(u,v) =1−H(u,v)Hr (u,v)2 Rf (u,v) + Hr (u,v)
2 N
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
H ro(u,v) =H(u,v)Rf (u,v)
H(u,v)H(u,v)Rf (u,v) + N
