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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 .
1. Given an N x N window W, the pixel values can be ordered , as
follows
I1<I2<….IN2
where I1….I3…IN2 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
Median filtering
In order to perform median filtering in a neghborhod of a
pixel [i.j]:
1. Sort the pixels into ascending order by gray level.
2. Select the value of the middle pixel as the new value for
pixel [i.j]
Mean Filter size =7 x 7
Median Filter size =7 x 7
This filters are excellent for impulse type of noise
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 3×3
square neighborhood is used here --- larger neighborhoods will produce
more severe smoothing.
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 filter’s neighborhood
will tend to be eliminated
Median filtering - Applications
The image has been corrupted with higher levels (i.e. p=5%
that a bit is flipped) of salt and pepper noise
Median filtering - Applications
After smoothing with a 3×3
filter, most of the noise has
been eliminated
If we smooth the noisy image with
a larger median filter, e.g. 7×7, all
the noisy pixels disappear, as
shown in this image
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 3×3 median
filter over the image three
times in order to remove
all the noise with less loss
of detail
The maximum filter selects the largest value within of pixel
values, whereas the minimum filter selects the smallest value.
Minimum filtering
( mask size =3 x 3)
Minimum filtering causes the darker
regions of an image to swell in size and
dominate the lighter regions ( mask size
=7 x 7)
Result from Maximum filtering
with mask (3 x 3)
Result from Maximum filtering
with mask (7 x 7)
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.
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.
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.
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%).
Conservative smoothing - Applications
After mean
filtering, the
image is still
noisy, as shown in
(a)
After median
filtering, all
noise is
suppressed, as
shown in (b)
Conservative smoothing
produces an image which still
contains some noise in places
where the pixel neighborhoods
were contaminated by more
than one intensity spike.
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.
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  I1  I 2  I 3  ....  I N 2
The midpoint filter is
most useful for gaussian
( I 1  I N 2 ) / 2 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
2 I1  I 2  I 3  ....  I
1
 2
N


N2
Ii
Alpha – trimmed mean = N 2
i  1
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.
Adaptive Filter
This filter compute local grey level statistics within the neighborhood
of a pixel and base their behavior on this information. For example:

 

g ( x, y)  f ( x, y)  2
f
(
x
,
y
)

f
(
x
,
y
)

 ( x, y) 
2
n
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)
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:
(X01+X02+X03+ X04+X05+X06+ X07+X08+X09+ 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 + ··· + (X10Xavg)2 )/9
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 Ri>Rj for all ji that
point is more likely to contain a line whose orientation (and width)
corresponds to that of kernel i.
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.
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.
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.
More examples - Comparing Line detector
and Canny operator
applying the Canny
operator, we obtain
More examples - Comparing Line detector
and Canny operator
Applying the line
detector yields
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.)
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
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
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
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