Enhancement vs.
Restoration
Lecture 5: Image Restoration
o Same goal:
improve image in some predefined sense
o Image enhancement
• Subjective process
• Heuristic procedures
BE 244: Biomedical Image Analysis • Example: contrast stretching
o Image restoration
• Objective process
• Criterion for image goodness
• Example: removal of image blur
Original slides by Tracy McKnight, modified by Piotr Habas, UCSF, 2009 Original slides by Tracy McKnight, modified by Piotr Habas, UCSF, 2009 2
Image Degradation Image Restoration
o g(x,y) = H[f(x,y)] + (x,y) o Given g(x,y) and some a priori information about H and
(x,y), obtain an estimate f’(x,y) of the original image
• f(x,y): original input image
• H(): degradation function o We want the estimate f’(x,y) to be as close as possible
to the original input image f(x,y)
• (x,y): additive noise
• g(x,y): degraded output image o The more we know about H and the closer f’(x,y) will
(x,y) be to f(x,y)
(x,y)
f(x,y) H + g(x,y)
f(x,y) H + g(x,y) ? f’(x,y)
Original slides by Tracy McKnight, modified by Piotr Habas, UCSF, 2009 3 Original slides by Tracy McKnight, modified by Piotr Habas, UCSF, 2009 4
Noise Probability Density Functions Noise PDFs
o Gaussian Noise (z )2
1 2 2
p( z ) e
o Noise is introduced into 2
images during the • “Normal” noise distribution
acquisition and/or • Electronic or sensor noise
transmission processes • Common noise model – often
abused
o Noise can be correlated
or uncorrelated with o Rayleigh Noise
2 ( z a )2 b
spatial coordinates z ae for z a
p( z ) b a b4
0 for z a
2 b4
• Useful for histogram analysis of 4
images with significant
background component (ie
skewed)
Original slides by Tracy McKnight, modified by Piotr Habas, UCSF, 2009 5 Original slides by Tracy McKnight, modified by Piotr Habas, UCSF, 2009 6
1
Noise PDFs Noise PDFs
oUniform Noise
1
for a z b
p( z ) b a
0 otherwise
o Impulse Noise
Pa for z a
p( z ) Pb for z b a b
0 otherwise 2
2
• “Bipolar” , “Salt-and-Pepper”, 2 b a
“Shot”, or “Spike” noise 12
• Impulses can be negative or
positive and are typically at
saturation levels
Original slides by Tracy McKnight, modified by Piotr Habas, UCSF, 2009 7 Original slides by Tracy McKnight, modified by Piotr Habas, UCSF, 2009 8
Noise PDFs Noise Estimation
o Noise characteristics may be estimated from the Fourier spectrum
of an image
• Periodic (spatially correlated) noise will appear as frequency spikes
o Characteristics may be empirically derived for a given acquistion
system by imaging a flat (typically all black) environment
o Regions of interest in existing images may also be used to
characterize noise
Original slides by Tracy McKnight, modified by Piotr Habas, UCSF, 2009 9 Original slides by Tracy McKnight, modified by Piotr Habas, UCSF, 2009 10
Noise reduction Image Degradation
o Mean filters
• Arithmetic mean filter
o H is linear
• Geometric mean filter H[k1f1(x,y) + k2f2(x,y)] = k1H[f1(x,y)] + k2H[f2(x,y)]
• Harmonic mean filter (salt noise, Gaussian noise)
and position-invariant
o Order-statistics filters
• Median filter (salt and pepper noise) H[f(x - ,y - )] = g(x - ,y - )
• Min filter (salt noise) o Spatial domain
• Max filter (pepper noise)
• Midpoint filter (uniform noise, Gaussian noise)
g(x,y) = h(x,y) * f(x,y) + (x,y)
o Frequency domain
G(u,v) = H(u,v) F(u,v) + N(u,v)
Original slides by Tracy McKnight, modified by Piotr Habas, UCSF, 2009 11 Original slides by Tracy McKnight, modified by Piotr Habas, UCSF, 2009 12
2
Image Degradation, example Image Restoration
o Estimation of degradation function H
• Image observation, Hs(u,v) = Gs(u,v) / Fs’(u,v)
• Experimentation
= * • Mathematical modeling
g(x,y) = h(x,y)
* f(x,y) o Direct inverse filtering
F’(u,v) = G(u,v) / H(u,v)
f’(x,y) = F-1 [F’(u,v)] = F--1 [G(u,v) / H(u,v)]
Original slides by Tracy McKnight, modified by Piotr Habas, UCSF, 2009 13 Original slides by Tracy McKnight, modified by Piotr Habas, UCSF, 2009 14
Image Restoration, example Inverse Filtering, problems
o F’(u,v) = G(u,v) / H(u,v)
50 F’(u,v) = F(u,v) + N(u,v) / H(u,v)
100
F(u,v) = F’(u,v) - N(u,v) / H(u,v)
150
N(u,v) = ?
o H(u,v) small -> N(u,v) / H(u,v) large
f’(x,y)
200
f(x,y) g(x,y)
f(x,y) blurred with a 7x7 mean f(x,y) restored with the inverse
may dominate the estimate F’(u,v)
250
50 100 150 200 250 filter filter
we need to limit the analysis to frequencies near the
origin H(0,0)
Original slides by Tracy McKnight, modified by Piotr Habas, UCSF, 2009 15 Original slides by Tracy McKnight, modified by Piotr Habas, UCSF, 2009 16
Wiener Filter
Wiener Filter
o For g = Hf + n
1 | H (u, v) |2
F ' (u, v) G (u, v)
H (u, v) H (u, v) 2 S n (u, v) / S f (u, v)
S n (u, v) | N (u, v) |2 noise power spectrum
S f (u, v) | F (u, v) |2 original image power spectrum
f g = Hf gn = Hf + n
o Also called “least squares filter” because it minimizes
2 = E{[f(u,v) - f’(u,v)]}
o For S (u,v) = 0 => inverse filter
o For “white noise” Sn(u,v) => constant
inverse(g) inverse(gn) wiener(gn)
Original slides by Tracy McKnight, modified by Piotr Habas, UCSF, 2009 17 Original slides by Tracy McKnight, modified by Piotr Habas, UCSF, 2009 18
