Chapter 3: Image Restoration

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Chapter 3 Image Restoration

• Noise Removal Using Spatial Filters

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Overview

• Spatial filters can be used to remove various
  types of noise in digital images.
• These spatial filters typically operate on small
  neighborhood, between 3x3 to 11x11.
• We will use the degradation model defined before,
  but we assume that h(r,c) causes no degradation.

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Overview

• Therefore, corruption on the image is only caused
  by additive noise, n(r,c).
• d(r,c) I(r,c) n(r,c)
• There are two primary categories of spatial
  filters for noise removal.
• Order filters arrange the pixels from smallest
  to largest and select the correct value.
• Mean filters calculate the average value.

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Overview

• The mean filters work best with gaussian or
  uniform noise.
• The order filters work best with salt-and-pepper,
  negative exponential, or Rayleigh noise.
• The mean filters are essentially low pass
  filters
• They tend to blur the edges or details.

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Overview

• The order filters are nonlinear filters
• The results are sometimes unpredictable.
• In general, there is a tradeoff between
  preservation of image detail and noise
  elimination.
• In practical applications, a good approach is to
  use an adaptive filter (a filter that can adapt
  itself to the underlying pixel values).

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Order Filters

• Order filters are based on a specific type of
  image statistics called order statistics.
• Order statistics is a technique that arranges all
  the pixels in sequential order, based on
  gray-level value.
• The placement of the value within this ordered
  set is referred to as the rank.

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Order Filters

• Given an NxN window, the pixel values can be
  ordered from smallest to largest as follows
• I1? I2 ? I3?.....? IN2
• Where I1,I2,I3,.....,IN2 are the gray-level
  values of the subset of pixels in the image, that
  are in the NxN window.
• Different types of order filters select different
  values from the ordered pixel list.

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Order Filters

• Median filter
• Select the middle pixel value from the ordered
  set.
• Used to remove salt-and-pepper noise.
• Maximum filter
• Select the highest pixel value from the ordered
  set.
• Remove pepper-type noise.

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Order Filters

• Minimum filter
• Select the lowest pixel value from the ordered
  set.
• Remove salt-type noise.
• As the size of the window gets bigger, the more
  information loss occurs.
• With windows larger than about 5x5, the image
  acquires an artificial, painted, effect.

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Order Filters
Minimum Filter
Image with salt noise Probability .04
Result of minimum filtering Mask 3 x 3
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Order Filters
Minimum filtering Mask 5 x 5
Minimum filtering Mask 9 x 9
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Order Filters
Maximum Filter
Maximum filtering Mask 3 x 3
Image with pepper noise Probability .04
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Order Filters
Maximum filtering Mask 9 x 9
Maximum filtering Mask 5 x 5
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Order Filters

• Order filters can also be defined to select a
  specific pixel rank within the ordered set.
• For example, we may find the second highest value
  is the better choice than the maximum value for
  certain pepper noise.
• This type of ordered selection is application
  specific.
• Minimum filter tend to darken the image and
  maximum filter tend to brighten the image.

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Order Filters

• Midpoint filter
• Average of the maximum and minimum within the
  window.
• Useful for removing gaussian and uniform noise.

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Order Filters
Image with gaussian noise. Variance 300, mean
0
Result of midpoint filter Mask size 3
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Order Filters
Result of midpoint filter Mask size 3
Image with uniform noise. Variance 300, mean
0
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Order Filters

• Alpha-trimmed mean filter
• The average of the pixel values within the
  window, but with some endpoint-ranked values
  excluded.
• T is the number of pixels excluded at each end of
  the ordered set

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Order Filters

• The alpha-trimmed mean filter ranges from a mean
  to median filter, depending on the value selected
  for the T parameter.
• If T 0, ? mean filter.
• If T (N2 1) / 2, ? median filter.
• The alpha-trimmed mean filter is useful for
  images containing multiple types of noise.
• Example Gaussian salt-and-pepper.

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Order Filters
Image with gaussian noise Variance 200, mean
0. Salt-and-pepper noise probability 0.02
Result of alpha-trimmed mean filter Mask size
3 Trim size 0
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Order Filters
Result of alpha-trimmed mean filter Mask size
3 Trim size 1
Result of alpha-trimmed mean filter Mask size
3 Trim size 4
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Mean Filters

• The mean filters function by finding some form of
  an average within the NxN window.
• The most basic of these filters is the arithmetic
  mean filter.
• This filter mitigates the noise effect, but at
  the same time tend to blur the image.
• The blurring effect is not desirable, and
  therefore other mean filters are designed to
  minimize this loss of detail information.

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Mean Filters

• Arithmetic mean filter
• Find the arithmetic average of the pixel values
  in the window.
• Smooth out local variations in an image.
• Tend to blur the image.
• Works best with gaussian and uniform noise.

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Mean Filters
Image with gaussian noise Variance300, mean 0
Result of arithmetic mean filter Mask size 3
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Mean Filters
Result of arithmetic mean filter Mask size 5
Result of arithmetic mean filter Mask size 9
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Mean Filters
Image with gamma noise Variance300, mean 0
Result of arithmetic mean filter Mask size 3
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Mean Filters
Result of arithmetic mean filter Mask size 5
Result of arithmetic mean filter Mask size 9
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Mean Filters

• Contra-harmonic mean filter
• Works for salt OR pepper noise, depending on the
  filter order R.
• Negative R ? Eliminate salt-type noise.
• Positive R ? Eliminate pepper-type noise.

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Mean Filters
Image with salt noise Probability .04
Result of contra-harmonic filter Mask size 3
order 0
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Mean Filters
Result of contra-harmonic filter Mask size 3
order -1
Result of contra-harmonic filter Mask size 3
order -5
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Mean Filters
Image with pepper noise Probability .04
Result of contra-harmonic filter Mask size 3
order 0
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Mean Filters
Result of contra harmonic filter Mask size 3
order 5
Result of contra harmonic filter Mask size 3
order 1
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Mean Filters

• Geometric mean filter
• Works best with gaussian noise.
• Retains detail better than arithmetic mean
  filter.
• Ineffective in the presence of pepper noise (if
  very low values present in the window, the
  equation will return a very small number).

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Mean Filters
Image with gaussian noise Variance 300, mean
0
Result of geometric filter Mask size 3
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Mean Filters
Image with pepper noise Probability .04
Result of geometric filter Mask size 3
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Mean Filters
Image with salt noise Probability.04
Result of geometric filter Mask size 3
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Mean Filters

• Harmonic mean filter
• Works with gaussian noise.
• Retains detail better than arithmetic mean
  filter.
• Works well with pepper noise.

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Mean Filters
Image with pepper noise Probability .04
Result of harmonic filter Mask size 3
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Mean Filters
Image with salt noise Probability.04
Result of harmonic filter Mask size 3
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Mean Filters

• Yp mean filter
• Remove salt noise for negative values of P.
• Remove pepper noise for positive values of P.

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Adaptive Filters

• An adaptive filter alters its basic behavior as
  the image is processed.
• It may act like a mean filter on some parts of
  the image and a median filter on other parts of
  the image.
• The typical character used to determine the
  filter behavior are the local image
  characteristics.
• Measured by local gray-level statistics.

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Adaptive Filters

• The minimum mean-square error (MMSE) filter is a
  good example of an adaptive filter.
• sn2 noise variance.
• sl2 local variance (in the window).
• ml local mean (average in the window).

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Adaptive Filters

• MMSE filter exhibits varying behavior based on
  local image statistics
• No noise ? variance 0 ? equation returns
  original image.
• Regions with fairly constant value (no
  edge/details) ? noise variance local variance
  ? equation reduces to mean filter.
• Regions with high details (edges) ? local
  variance gtgt noise variance ? equation returns
  values close to original image.

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Adaptive Filters

• In general, MMSE filter modifies the image based
  on the noise to local variance ratio.
• High ratio implies the existence of noise in the
  window, and therefore the filter returns
  primarily the local average to reduce the noise.
• Low ratio implies high local detail, therefore
  the filter returns more of the original
  unfiltered image to preserve the detail.

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Adaptive Filters

• By being able to adapt itself to the local image
  statistics, the MMSE filter can preserve the
  details while at the same time remove the noise.
• MMSE filter works best with gaussian or uniform
  noise, and can perform better compared to the
  other filters discussed before.

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Adaptive Filters
Image with gaussian noise Variance300, mean 0
Original Image
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Adaptive Filters
Result of MMSE Mask size 5
Result of MMSE Mask size 3
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Adaptive Filters
Result of MMSE Mask size 9
Result of MMSE Mask size 7