Image Restoration

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Image Restoration
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CONTENT

• Overview
• Noise Models
• Gaussian
• Salt-and-pepper
• Uniform
• Rayleigh
• Noise Removal using Spatial Filters
• Order filters
• Mean filters
• Geometric Transforms

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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

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• 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)

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• 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

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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

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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

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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

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• 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

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• 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

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Gaussian distribution
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• A bell-shapped
• 70 of all values fall within the range from one
  standard deviation (s) below the mean (m) to one
  above
• About 95 fall within two standard deviations

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Uniform distribution

• The gray level values of the noise are evenly
  distributed across a specific range

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Salt-and-pepper

• 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

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Rayleigh
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Original image without noise, and its histogram
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image with added Gaussian noise with mean 0 and
variance 600, and its histogram
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image with added uniform noise with mean 0 and
variance 600, and its histogram
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image with added salt-and-pepper noise with the
probability of each 0.08, and its histogram
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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

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• 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

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• 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

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• 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

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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

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• (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

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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)
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c) after median filtering with a 5x5 window, all
the noise is removed, but the image is blurry
acquiring the painted effect
c)
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• Two order filters are midpoint and alpha-trimmed
  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
• Most useful for Gaussian uniform noise

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• 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
• 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

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• Alpha-trimmed mean filter ranges from a mean to
  median filter, depending on the value selected
  for the T parameter

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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 salt-and-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 alpha-trimmed
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)
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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
• where N2 the number of pixels in the NxN
  window, W
• Smooths out local variations work best with
  Gaussian, gamma and uniform noise

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• Contra-harmonic mean filter works well for images
  containing salt OR pepper type noise, depending
  on the filter order, R
• where W is the NxN window under consideration
• Negative values of R, eliminates salt-type noise
• Positive values, eliminates pepper-type noise

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• 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

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• Harmonic mean filter also fails with pepper noise
  but works well for salt noise
• Retaining detail information better than the
  arithmetic mean filter

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• Yp mean filter is defined as follows

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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

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• 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

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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

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Geometric Distortion

• The original, undistorted image, I(r,c), and
  distorted (or degraded) image is
• Primary idea is to find a mathematical model for
  the geometric distortion process
• and apply the inverse process to find restored
  image

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• 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

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• Method to restore a geometrically distorted image
  consists of 3 steps
• Define quadriterals (4 sided polygons) with
  known, or best-guessed tiepoints for the entire
  image
• Find the equations
  for each set of tiepoints,
• Remap all the pixels within each quadrilateral
  subimage using the equations corresponding to
  those tiepoints

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1. 2 images are divided into subimages, defined by
   tiepoints (fig. 9.6.3 ab)
2. Using bilinear model for the mapping equations,
   these 4 points to generate the equations
3. Involves application of the mapping equations,
   , to all the (r,c) pairs

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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

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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

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Gray Level Interpolation

• Better results- uses bilinear interpolation with
  the equation
• where the gray level interpolating
  equation
• Example 9.6.3 9.6.4

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