Rank filtering

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Rank filtering

• Order filter are implemented by arranging the
  neighborhood pixels in order from smallest to
  largest gray level value and using this ordering
  to select the correct value. The placement of
  the value within this ordered set is referred as
  the rank .
• Given an N x N window W, the pixel values can be
  ordered , as follows
• I1ltI2lt.IN2 where I1.I3IN2 are intensity
  values.
• 2. Select a value from a particular position in
  the list to use as the new value for the pixel.

From noise image Rank filtering mask (7 x 7 )
rank 4
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Median filtering

• In order to perform median filtering in a
  neghborhod of a pixel i.j
• Sort the pixels into ascending order by gray
  level.
• Select the value of the middle pixel as the new
  value for pixel i.j

Median Filter size 7 x 7
Mean Filter size 7 x 7
This filters are excellent for impulse type of
noise
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Median filtering
Figure 1 Calculating the median value of a pixel
neighborhood. As can be seen, the central pixel
value of 150 is rather unrepresentative of the
surrounding pixels and is replaced with the
median value 124. A 33 square neighborhood is
used here --- larger neighborhoods will produce
more severe smoothing.
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Advantages of median filter
By calculating the median value of a neighborhood
rather than the mean filter, the median filter
has two main advantages over the mean filter
1.The median is a more robust average than the
mean and so a single very unrepresentative pixel
in a neighborhood will not affect the median
value significantly. 2. Since the median
value must actually be the value of one of the
pixels in the neighborhood, the median filter
does not create new unrealistic pixel values when
the filter straddles a edge. For this reason the
median filter is much better at preserving sharp
edges than the mean filter.
Disadvantages of median filter
1. One of the major problems with the median
filter is that it is relatively expensive and
complex to compute. To find the median it is
necessary to sort all the values in the
neighborhood into numerical order and this is
relatively slow, even with fast sorting
algorithms such as quick sort.
2. Any structure that occupies less than half of
the filters neighborhood will tend to be
eliminated
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Median filtering - Applications
The image has been corrupted with higher levels
(i.e. p5 that a bit is flipped) of salt and
pepper noise
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Median filtering - Applications
If we smooth the noisy image with a larger median
filter, e.g. 77, all the noisy pixels disappear,
as shown in this image
After smoothing with a 33 filter, most of the
noise has been eliminated
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Median filtering - Applications
Note that the image is beginning to look a bit
blotchy', as graylevel regions are mapped
together. Alternatively, we can pass a 33 median
filter over the image three times in order to
remove all the noise with less loss of detail
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The maximum filter selects the largest value
within of pixel values, whereas the minimum
filter selects the smallest value.
Minimum filtering causes the darker regions of an
image to swell in size and dominate the lighter
regions ( mask size 7 x 7)
Minimum filtering ( mask size 3 x 3)
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Result from Maximum filtering with mask (7 x 7)
Result from Maximum filtering with mask (3 x 3)
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Conservative smoothing - How It Works
Like most noise filters, conservative smoothing
operates on the assumption that noise has a high
spatial frequency and, therefore, can be
attenuated by a local operation which makes each
pixel's intensity roughly consistent with those
of its nearest neighbors. However, whereas mean
filtering accomplishes this by averaging local
intensities and median filtering by a non-linear
rank selection technique, conservative smoothing
simply ensures that each pixel's intensity is
bounded within the range of intensities defined
by its neighbors. This is accomplished by a
procedure which first finds the minimum and
maximum intensity values of all the pixels within
a windowed region around the pixel in question.
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Conservative smoothing - How It Works
If the intensity of the central pixel lies within
the intensity range spread of its neighbors, it
is passed on to the output image unchanged.
However, if the central pixel intensity is
greater than the maximum value, it is set equal
to the maximum value if the central pixel
intensity is less than the minimum value, it is
set equal to the minimum value. Figure
illustrates this idea.
Figure Conservatively smoothing a local pixel
neighborhood. The central pixel of this figure
contains an intensity spike (intensity value
150). In this case, conservative smoothing
replaces it with the maximum intensity value
(127) selected amongst those of its 8 nearest
neighbors.
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Conservative smoothing - How It Works
If we compare the result of conservative
smoothing on the image segment of Figure 1 with
the result obtained by mean filtering and median
filtering, we see that it produces a more subtle
effect than both the former (whose central pixel
value would become 125) and the latter (124).
Furthermore, conservative smoothing is less
corrupting at image edges than either of these
noise suppression filters.
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Conservative smoothing - Applications
The real utility of conservative smoothing (and
median filtering) is in suppressing salt and
pepper, or impulse, noise. A linear filter cannot
totally eliminate impulse noise, as a single
pixel which acts as an intensity spike can
contribute significantly to the weighted average
of the filter. Non-linear filters can be robust
to this type of noise because single outlier
pixel intensities can be eliminated entirely.
Conservative smoothing works well for low levels
of salt and pepper noise. However, when the image
has been corrupted such that more than 1 pixel in
the local neighborhood has been effected,
conservative smoothing is less successful. For
example, smoothing the image which has been
corrupted by 1 salt and pepper noise (i.e. bits
have been flipped with probability 1).
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Conservative smoothing - Applications
Conservative smoothing produces an image which
still contains some noise in places where the
pixel neighborhoods were contaminated by more
than one intensity spike.
After median filtering, all noise is suppressed,
as shown in (b)
After mean filtering, the image is still noisy,
as shown in (a)
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Conservative smoothing - Applications
However, no image detail has been lost e.g.
notice how conservative smoothing is the only
operator which preserved the reflection in the
subject's eye.
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Hybrid filters
They are hybrid because they rely on ordering the
pixel values , but they are then calculated by an
averaging process.
The midpoint filter is the average of the maximum
and minimum within the window , as follows
Ordered set
The midpoint filter is most useful for gaussian
and uniform noise .
Midpoint
The alpha trimmed mean filter is the average of
the pixel values within the window, but with some
of the endpoint ranked values excluded.
Ordered set
Alpha trimmed mean
Where ? is the number of pixel values removed
from each end of the list , and can range from 0
to (N2-1)/2 . When ?0, no values are removed
from the list and the filter behaves as a mean
filter. If ?(N2-1)/2 , the equation becomes a
median filter.
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Adaptive Filter
This filter compute local grey level statistics
within the neighborhood of a pixel and base their
behavior on this information. For example
where ?2n is an estimate of noise variance,
?2(x,y) is the grey level variance computed for
the neighborhood centered on x,y and f_(x,y) is
the mean grey level in the neighborhood. In
homogeneus regions of an image, noise will be the
sole cause of variations in grey level (?2n
?2(x,y)) and g(x,y) f_(x,y)
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Average and Variance Value
The most common method is the average or mean. To
obtain an average value, add up all your data
values and divide by the number of data items. If
X01 is the length of your first maple leave, X02
the length of your second maple leave, etc., then
the average maple leaf length is (X01X02X03
X04X05X06 X07X08X09 X10)/10 Xavg
The most common way to describe the range of
variation is standard deviation (usually denoted
by the Greek letter sigma   ). The standard
deviation is simply the square root of the
variance
The result is the variance take its square root
to get the standard deviation. variance (
(X01-Xavg)2 (X02-Xavg)2 (X03-Xavg)2
(X10-Xavg)2 )/9
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Line Detection
The line detection operator consists of a
convolution kernel tuned to detect the presence
of lines of a particular width n, at a particular
orientation ? . Figure shows a collection of
four such kernels, which each respond to lines of
single pixel width at the particular orientation
shown
Figure Four line detection kernels which respond
maximally to horizontal, vertical and oblique
(45 and - 45 degree) single pixel wide lines.
If Ri denotes the response of kernel i, we can
apply each of these kernels across an image, and
for any particular point, if RigtRj for all j?i
that point is more likely to contain a line whose
orientation (and width) corresponds to that of
kernel i.
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Line Detection- Guidelines for Use
The result of applying the line detection
operator, using the horizontal convolution kernel
shown in Figure 1.a, is
There are two points of interest to note here

1. Notice that, because of way that the oblique
   lines (and some vertical' ends of the horizontal
   bars) are represented on a square pixel grid,
   e.g.

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Line Detection- Guidelines for Use
2. On an image such as this one, where the lines
to be detected are wider than the kernel (i.e.
the image lines are five pixels wide, while the
kernel is tuned for a single width pixel), the
line detector acts like an edge detector the
edges of the lines are found, rather than the
lines themselves.
This latter fact might cause us to naively think
that the image which gave rise to
contained a series of parallel lines rather than
single thick ones.
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Line Detection- Guidelines for Use
For example, we can skeletonize the original (so
as to obtain a representation of the original
wherein most lines are a single pixel width),
apply the horizontal line detector and then
threshold the result.
Skeletonization is a process for reducing
foreground regions in a binary image to a
skeletal remnant that largely preserves the
extent and connectivity of the original region
while throwing away most of the original
foreground pixels.
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More examples - Comparing Line detector and Canny
operator
applying the Canny operator, we obtain
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More examples - Comparing Line detector and Canny
operator
Applying the line detector yields
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By smoothing the image before line detecting, we
obtain the cleaner result
However, even with this preprocessing, the line
detector still gives a poor result compared to
the edge detector. This is true because there are
few single pixel width lines in this image, and
therefore the detector is responding to the other
high spatial frequency image features (i.e.
edges, thick lines and noise). (Note that in the
previous example, the image contained the feature
that the kernel was tuned for and therefore we
were able to threshold away the weaker kernel
response to edges.)
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Rank filters non-linear spatial filters
also called order statistic filters
sort the neighborhood pixels into
ascending order select the one of a
given rank to be the output value
Median filter select the median pixel
from the sorted list to be the output value
for an n x n neighborhood, this will be the
pixel in position floor(n2/2) 1
removes structure that occupies less than half of
the neighborhood median filter is
especially good at removing impulse noise
want the noise to occupy less than half of the
neighborhood can increase the
neighborhood size for noisier images
the shape of the neighborhood can affect amount
of damage to object edges
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Minimum and maximum filters take the
minimum or maximum pixel in the neighborhood
isn't necessary to sort - faster to just
use comparisons minimum filter -
enhances dark areas of image maximum
filter - enhances bright areas of image
Range filter (Midpoint filter) nonlinear edge
detector output pixel value is the difference
between the maximum and minimum pixel
values in a neighborhood edges are not well
localized
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Alpha-trimmed mean filter a hybrid of
linear and nonlinear approaches sort
the neighborhood pixels into ascending order
discard a given number (alpha) from each
end of the list output pixel is the
mean of the remaining pixel values
alpha 0 gives a mean filter alpha
(n2 - 1)/2 gives a median filter
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Minimal mean square error filter
adaptive filter - behavior depends on local image
properties must supply an estimate of
noise variance at each pixel, compute
the mean gray-level and gray-level variance of
neighborhood ratio of noise variance
estimate to gray-level variance in neighborhood
affects computation of output pixel
where neighborhood variance equals noise
variance, output is the mean gray-level
where neighborhood variance is large compared to
noise variance (near edges), output is
close to input pixel value