Outline Computer Vision &

Outline
• Periodic noise
• Estimation of noise parameters
• Restoration in the presence of noise only –spatial
filtering
Computer Vision &
Digital Image Processing
Image Restoration and
Reconstruction II
Electrical & Computer Engineering
Dr. D. J. Jackson Lecture 12-1
Periodic noise
Electrical & Computer Engineering
Dr. D. J. Jackson Lecture 12-2
Sample periodic images and their spectra
• Periodic noise
typically arises
from interference
during image
acquisition
• Spatially
dependent noise
type
• Can be
effectively
reduced via
frequency
domain filtering
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Sample periodic images and their spectra
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Electrical & Computer Engineering
Dr. D. J. Jackson Lecture 12-4
Estimation of noise parameters
• Noise parameters can often be estimated by
observing the Fourier spectrum of the image
– Periodic noise tends to produce frequency spikes
• Parameters of noise PDFs may be known (partially)
from sensor specification
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Dr. D. J. Jackson Lecture 12-3
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– Can still estimate them for a particular imaging setup
– One method
• Capture a set of “flat” images from a known setup (i.e. a uniform
gray surface under uniform illumination)
• Study characteristics of resulting image(s) to develop an indicator
of system noise
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Dr. D. J. Jackson Lecture 12-5
Electrical & Computer Engineering
Dr. D. J. Jackson Lecture 12-6
Estimation of noise parameters (continued)
Estimation of noise parameters (continued)
• If only a set of images already generated by a sensor are
available, estimate the PDF function of the noise from small
strips of reasonably constant background intensity
• Consider a subimage (S) and let
ps(zi), i=0,1,2,…L-1
• denote the probability estimates of the intensities of the
pixels in S.
• L is the number of possible intensities in the image
• The mean and the variance of the pixels in S are given by:
• The shape of the noise histogram identifies the
closest PDF match
L −1
L −1
i =0
i =0
z = ∑ zi ps ( zi ) and σ 2 = ∑ ( zi − z ) 2 ps ( zi )
Electrical & Computer Engineering
Dr. D. J. Jackson Lecture 12-7
Histograms from noisy strips of an area of an
image
– If the shape is Gaussian, then the mean and variance are
all that is needed to construct a model for the noise (i.e.
the mean and the variance completely define the
Gaussian PDF)
– If the shape is Rayleigh, then the Rayleigh shape
parameters (a and b) can be calculated using the mean
and variance
– If the noise is impulse, then a constant (with the exception
of the noise) area of the image is needed to calculate Pa
and Pb probabilities for the impulse PDF
Electrical & Computer Engineering
Dr. D. J. Jackson Lecture 12-8
Restoration in the presence of noise only –spatial
filtering
• When only additive random noise is present, spatial
filtering is commonly used to restore images
• Common types
– Mean filters
– Order-Statistic filters
– Adaptive filters
Electrical & Computer Engineering
Dr. D. J. Jackson Lecture 12-9
Electrical & Computer Engineering
Dr. D. J. Jackson Lecture 12-10
Mean filters (arithmetic)
Mean filters (geometric)
• Arithmetic mean filter
• Geometric mean filter
– Computes the average value of a corrupted image g(x,y)
in the area defined by a window (neighborhood)
1
fˆ ( x, y ) =
∑ g ( s, t )
mn ( s ,t )∈S xy
1
⎡
⎤ mn
fˆ ( x, y ) = ⎢ ∏ g ( s, t )⎥
⎣⎢( s ,t )∈S xy
⎦⎥
– The operation is generally implemented using a spatial
filter of size m*n in which all coefficients have value 1/mn
– A mean filter smoothes local variations in an image
– Noise is reduced as a result of blurring
Electrical & Computer Engineering
– A restored pixel is given by the product of the pixels in an
area defined by a window (neighborhood), raised to the
power 1/mn
Dr. D. J. Jackson Lecture 12-11
– Achieves smoothing comparable to the arithmetic mean
filter, but tends to loose less detail in the process
Electrical & Computer Engineering
Dr. D. J. Jackson Lecture 12-12
Arithmetic and geometric mean filter examples
Mean filters (harmonic)
• Harmonic mean filter
– A restored pixel is given by the expression
fˆ ( x, y ) =
mn
∑
( s ,t )∈S xy
1
g ( s, t )
– Works well for salt noise (fails for pepper noise)
– Works well for Gaussian noise also
Electrical & Computer Engineering
Dr. D. J. Jackson Lecture 12-13
Mean filters (contraharmonic)
Electrical & Computer Engineering
Dr. D. J. Jackson Lecture 12-14
Contraharmonic mean filter examples
• Contraharmonic mean filter
– A restored pixel is given by the expression
fˆ ( x, y ) =
∑ g ( s, t )
Q +1
( s ,t )∈S xy
∑ g ( s, t )
Q
( s ,t )∈S xy
–
–
–
–
–
Q is the order of the filter
Works well for salt and pepper noise (cannot do both simultaneously)
+Q eliminates pepper noise, -Q eliminates salt noise
Q=0 → arithmetic mean filter
Q=-1 → harmonic mean filter
Electrical & Computer Engineering
Dr. D. J. Jackson Lecture 12-15
Contraharmonic mean filter examples
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Dr. D. J. Jackson Lecture 12-17
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Dr. D. J. Jackson Lecture 12-16