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Image Restoration
CONTENT
• Overview
• Noise Models
–
–
–
–
Gaussian
Salt-and-pepper
Uniform
Rayleigh
• Noise Removal using Spatial Filters
– Order filters
– Mean filters
• Geometric Transforms
Overview
• Used to improve the appearance of an image by
application of a restoration process that uses a
mathematical model for image degradation
• Types of degradation :– Blurring caused by motion or atmospheric
disturbance
– Geometric distortion caused by imperfect lenses
– Superimposed interference patterns caused by
mechanical systems
– Noise from electronic sources
• We see that sample degraded images and
knowledge of the image acquisition process are
inputs to development of a degradation model
• After the model has been developed, the next step
is the formulation of the inverse process
• This inverse degradation process is then applied to
the degraded image, d(r,c), which results in the
output image, Î(r,c),
• The output image Î(r,c), is the restored image which
represents an estimate of original image, I(r,c)
d̂
• Once the estimated image has been created, any
knowledge gained by observation & analysis of this
image is used as additional input for further
development of degradation model
• This process continues until satisfactory results are
achieved
• With this perspective, can define image restoration
as the process of finding an approximation to the
degradation process & finding the appropriate
inverse process to estimate the original image
System Model
• Degradation process model consists of 2 parts,
the degradation function & the noise function
• General mode in spatial domain :
d(r,c) = h(r,c) * I(r,c) + n(r,c)
where
d(r,c) = degraded image
h(r,c) = degradation function
I(r,c) = original image
n(r,c) = additive noise function
System Model
• Frequency domain :
D(u,v) = H(u,v) * I(u,v) + N(u,v)
where
D(u,v) = Fourier transform of the degraded image
H(u,v) = Fourier transform of the degradation
function
I(u,v) = Fourier transform of the original image
N(u,v) = Fourier transform of the additive noise
function
Noise Models
• Any undesired information that contaminates an
image
• noise models is a random variable with a
probability density function (PDF) that describes
its shape and distribution
• The actual distribution of noise in a specific image
is the histogram of the noise
• Noise can be modeled with Gaussian (“normal”),
uniform, salt-and-pepper (“impulse”), or Rayleigh
distribution
• Gaussian model – occur from electronic noise in
image acquisition system
– Most problematic with poor lighting conditions or
vary high temperatures
– Also valid for film grain noise
• Salt-and-pepper noise (also called impulse noise,
shot noise or spike noise) typically caused by
malfunctioning pixel element in camera sensors,
faulty memory locations, or timing errors in
digitization process
• Uniform noise is useful - it can be used to
generate any other type of noise distribution, and
is often used to degrade images for the
evaluation of image restoration algo since
provides the most unbiased or neutral noise
model
Gaussian distribution
1
Hg 
2
2
e
 ( g  m ) 2 / 2 2
g  gray level
m  mean (average)
  standard deviation
  var iance
2
• A bell-shapped
• 70% of all values fall within the range from
one standard deviation (σ) below the mean
(m) to one above
• About 95% fall within two standard deviations
Uniform distribution
• The gray level values of the noise are evenly
distributed across a specific range
H Uniform
mean
 1

 b  a

0

a b
2
(b - a) 2
variance 
12
, for a  g  b
elsewhere
Salt-and-pepper
A
H

salt& paper  B
for g  a ("pepper")
for g  b ("salt")
• There are only 2 possible values, a and b, and the
probability of each is typically less than 0.2 – with
numbers greater than this the noise will swamp
out the image
Rayleigh
H Rayleigh 
2g

e
g 2 / 
where :
mean 

varia nce 
4
 (4 -  )
4
Original image without noise, and
its histogram
image with added Gaussian noise with
mean = 0 and variance = 600, and its
histogram
image with added uniform noise with
mean = 0 and variance = 600, and its
histogram
image with added salt-and-pepper noise
with the probability of each 0.08, and its
histogram
Noise Removal Using Spatial Filters
• Spatial filters can be effectively used to remove
various types of noise
• Operate on small neighborhoods, 3x3 to 11x11
• Will use the degradation model with the
assumption that h(r,c) causes no degradation
where the only corruption to the image is
caused by additive noise
d(r,c) = I(r,c) + n(r,c)
where
d(r,c) = degraded image
I(r,c) = original image
n(r,c) = additive noise function
• Two primary categories; order filters and mean
filters
• Order filters – implemented by arranging the
neighborhood pixels in order from smallest to
largest gray level value, and using this ordering to
select the “correct” value
• Mean filters determine, in one sense or another,
an average value
• Mean filters work best with Gaussian or
uniform noise
• Order filters work best with salt-and-pepper,
negative exponential, or Rayleigh noise
• Mean filters have disad of blurring the image
edges, or details
• Order filters such as mean can be used to
smooth images
Order Filters
• Operate on small subimages, windows, and
replace the center pixel value (similar to
convolution process)
• Given an N x N wondow, W, the pixel values can
be ordered as follows
I1 ,  I 2  I 3  ......  I N
2
where
I , I , I ,......., I  are the Intensity
2
1
2
3
N
(gray level) values
110 110 114
100 104 104


95 88 85 
• (85, 88, 95, 100, 104, 104, 110, 110, 114)
• Min = 85, Med = 104, max = 114 (will be
replaced at the center value)
• Median filter is most useful
• Max & min filters can eliminate salt or pepper
noise
a) Image with added salt-and-pepper noise, the probability
for salt = probability for pepper = 0.10, b) after median
filtering with a 3x3 window, all the noise is not removed
a)
b)
c) after median filtering with a 5x5 window, all the noise is
removed, but the image is blurry acquiring the “painted”
effect
c)
• Two order filters are midpoint and alphatrimmed mean filters – both order and mean
filters since they rely on ordering the pixels
values, but are then calculated by an averaging
process
• Midpoint filter – the average of max & min within
the window;
Ordered set  I1 ,  I 2  I 3  ......  I N
Midpoint 
2
I1  I N 2
2
• Most useful for Gaussian & uniform noise
• Alpha-trimmed mean is the average of pixel values
within the window, but with some of the endpoint
ranked excluded
• Useful for images containing multiple types of
noise, Gaussian and salt-and-pepper noise
Ordered set  I1 ,  I 2  I 3  ......  I N
1
Alpha - trimmed mean  2
N  2T
2
N 2 T
I
i T 1
i
where T is the number of pixel values excluded at each
end of the ordered set, and can range from 0 to (N2 – 1)/2
• Alpha-trimmed mean filter ranges from a
mean to median filter, depending on the value
selected for the T parameter
Figure 9.3-5 Alpha-Trimmed Mean. This filter can vary between a mean filter and a median
filter. a) Image with added noise: zero-mean Gaussian noise with a variance of 200, and saltand-pepper noise with probability of each = 0.03, b) result of alpha-trimmed mean filter, mask
size = 3x3, T = 1, c) result of alpha-trimmed mean filter, mask size = 3x3, T = 2, d) result of alphatrimmed mean filter, mask size = 3x3, T = 4. As the T parameter increases the filter becomes
more like a median filter, so becomes more effective at removing the salt-and pepper noise.
a)
b)
c)
d)
Mean Filters
• Function by finding some form of an average
within the NxN window, using sliding window
concept to process entire image
• The most basic – arithmetic mean filter which
finds the arithmetic average of pixel values ;
1
Arithmetic mean  2
N
2
 d (r , c)
r ,c )W
where N = the number of pixels in (the
NxN window, W
• Smooths out local variations & work best with
Gaussian, gamma and uniform noise
• Contra-harmonic mean filter works well for
images containing salt OR pepper type noise,
depending on the filter order, R:
 d(r, c)
 d(r, c)
R 1
Contra - harmonic mean 
( r ,c )W
R
( r ,c )W
where W is the NxN window under consideration
• Negative values of R, eliminates salt-type noise
• Positive values, eliminates pepper-type noise
• Geometric mean filter works best with Gaussian
noise, & retains detail information better than an
arithmetic mean filter
• Defined as the product of pixel values within
window, raised to the 1/N2 power:
Geometric mean 
 I(r, c)
( r ,c )W
1
N2
• Harmonic mean filter also fails with pepper
noise but works well for salt noise;
Purata H 
N2

( r , c ) W
1
d(r, c)
• Retaining detail information better than the
arithmetic mean filter
• Yp mean filter is defined as follows:
1/ p
P

d ( r , c) 
Yp mean   

2
( r ,c )W N

Geometric Transforms
• Images that have been spatially, or geometrically,
distorted
• Used to modify the location of pixel values within
an image, typically to correct images that have
been spatially warped
• Often referred as rubber-sheet transforms image is modeled as a sheet of rubber and
stretched and shrunk
• Because of defective optics in image acquisition system,
distortion in image display devices, or 2D imaging of 3D
surfaces
• This methods are used in map making, image
registration, image morphing, and other applications
requiring spatial modification
• Simplest – translate, rotate, zoom & shrink
• More sophisticated – 1) spatial transform & 2) gray level
interpolation
Input Image
Spatial
Transform
Gray Level
Interpolation
Output Image
Spatial Transforms
• Used to map the input image location to a
location in the output image; it defines how the
pixel values in output image are to be arranged
I(r,c)
rˆ  Rˆ ( r , c )
d ( rˆ, cˆ)
cˆ  Cˆ ( r , c )
• The original, undistorted image, I(r,c), and distorted
(or degraded) image is
d ( rˆ, cˆ)
rˆ  Rˆ (r , c), defines the row coordinate for the distorted image
cˆ  Cˆ (r , c), defines the column coordinate for the distorted image
• Primary idea is to find a mathematical model for the
geometric distortion process – Rˆ (r , c) and Cˆ (r , c)
and apply the inverse process to find restored image
• Different equations for different portions of
the image
• To determine the necessary equations, need
to identify a set of points in the original image
that match points in the distorted image
• These sets of points are called tiepoints, used
to define the equations Rˆ (r , c) and Cˆ (r , c)
• Method to restore a geometrically distorted
image consists of 3 steps:
1. Define quadriterals (4 sided polygons) with known,
or best-guessed tiepoints for the entire image
2. Find the equations Rˆ (r , c) and Cˆ (r , c) for each set of
tiepoints,
3. Remap all the pixels within each quadrilateral
subimage using the equations corresponding to
those tiepoints
1. 2 images are divided into subimages, defined
by tiepoints (fig. 9.6.3 a&b)
2. Using bilinear model for the mapping
equations, these 4 points to generate the
ˆ ( r , c )  k r  k c  k rc  k  r
ˆ
R
equations : Cˆ (r , c)  k r  k c  k rc  k  cˆ
3. Involves application of the mapping
equations, Rˆ (r , c) and Cˆ (r , c) , to all the (r,c) pairs
1
5
2
6
3
7
4
8
Exercise
• Example 9.6.1 & 9.6.2
• The difficulty in above example arises when
we try to determine the value of d(41.4, 20.6)
– Since digital images are defined only at integer
values for Î(r,c) as an estimate to the original
image I(r,c) to represent the restored image
Gray Level Interpolation
• The simplest – nearest neighbor method, where
the pixel is assigned the value of the closest pixel
in the distorted image
– Î(2,3) is set to the value of d(41,21), the row and
column values determined by rounding
– Easy to implement and computationally fast
• More advance is to interpolate the value
– More computationally extensive but more visually
pleasing results
– Easiest - neighborhood average. Provide smoother
object edges but slightly blurry
Gray Level Interpolation
• Better results- uses bilinear interpolation with the
equation: g (rˆ, cˆ)  k1rˆ  k2cˆ  k3rˆcˆ  k4
where g (rˆ, cˆ) = the gray level interpolating
equation
• Example 9.6.3 & 9.6.4