digital image processing image enhancement part iv
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Digital Image Processing
Image Enhancement
Part IV
Nonlinear Filtering
• Nonlinear Filters– Cannot be expressed as convolution– Cannot be expressed as frequency shaping
• “Nonlinear” Means Everything (other than linear)– Need to be more specific– Often heuristic– We will study some “nice” ones
Impulsive (Salt & Pepper) Noise
),(
0
255
),(
nmX
nmY
with probability pa
with probability pb
with probability 1 - pa - pb
noisy pixels
clean pixels
X: noise-free image, Y: noisy image
• Definition– Each pixel in an image has a probability pa or pb of
being contaminated by a white dot (salt) or a black dot (pepper)
add salt & pepper noise
Order-Statistics Filtersnonlinear spatial filters response is based on ordering (ranking) the pixels contained in the image area encompassed by the filterreplacing of the center pixel with the value determined by the ranking result
Median FilteringMedian filter
replaces the pixel value by the median value in the neighborhoodThe principal function is to force distinct gray level points to be more like their neighbors.excellent noise-reduction capabilities with less blurring than linear smoothing filterseffective for impulse noise (salt-and-pepper noise)
Min: Set the pixel value to the minimum in the neighbourhood
Max: Set the pixel value to the maximum in the neighbourhoodcf.) max filter → R = max {zk | k = 1,2,…,9}
min filter → R = min {zk | k = 1,2,…,9}
Order-Statistics Filters
– Given a set of numbers
Denote the OS as
such that
• Applying Median Filtersto Images– Use sliding windows
(similar to spatial linear filters)– Typical windows:
3x3, 5x5, 7x7, other shapes ……
},,,{ 1221 Mxxx x
},,,{ )12()2()1( MOS xxx x
)12()1()2()1( MM xxxx
)1(1221 },,,{ MM xxxxMedian middle value
max value
min value
Median Filters
original noisy (pa = pb = 0.1)
median filtered 3x3 window median filtered 5x5 windowFrom MATLAB sample
images
Iterative Median Filters
From [Gonzalez & Woods]
• Idea: repeatedly apply median filters
1 time
2 times
3 times
Switching Median Filters
From [Wang & Zhang]
• Motivation– Regular median filters change both “bad” and “good”
pixels
• Idea– Detect/classify “bad” and “good” pixels– Filter “bad” pixels only
original noisy (pa = pb = 0.1)
regular 5x5 median filtered switching 5x5 median filtered
Switching Median Filters
From MATLAB sample images
Order Statistics (OS) Filters
• Recall Order Statistics:
For
OS
such that
• OS filter: General Form
• Special Cases
},,,{ 1221 Mxxx x
)12(1221 },,,{ MM xxxxMax
)1(1221 },,,{ MM xxxxMedian
},,,{ )12()2()1( MOS xxx x
)12()1()2()1( MM xxxx
)1(1221 },,,{ xxxxMin M
)(
12
11221 },,,{ i
M
iiM xwxxxOS
1
12
1
M
iiw
}0,,0,0,1{}{ iw
}1,,0,0,0{}{ iw
}0,,1,0,0{}{ iw
where
(M+1)-th
Order Statistics (OS) Filters
• Note: An OS Filter is Uniquely Defined by {wi}
• Example 1:
• Example 2:
)2()1()(1221 25.05.025.0},,,{ MMMM xxxxxxOS
}0,,4/1,2/1,4/1,,0{}{ iw
(M+1)-thM-th (M+2)-th
then
12
1)(1221 12
1},,,{
M
iiM x
MxxxOS
)12/(}1,,1,1{}{ Mwi
}12,,1,,{}12,,1,{ )( MixMeanMixMean ii
then
Examples
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• A 4x4 grayscale image is given by
1) Filter the image with a 3x3 median filter, after zero-padding at the image borders
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zero-paddingmedianfiltering
impulse?
impulse?
Examples
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2) Filter the image with a 3x3 median filter, after replicate-padding at the image borders
replicate-padding
medianfiltering
impulse cleaned!
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Examples3) Filter the image with a 3x3 OS filter, after
replicate-padding at the image borders. The weighting factors of the OS filter are given by
{wi | i = 1, …, 9} = {0, 0, 0, ¼, ½, ¼, 0, 0, 0}
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replicate-padding
OSfiltering
Gradient the term gradient is used for a gradual blend of colour which can
be considered as an even gradation from low to high values At each image point, the gradient vector points in the direction of
largest possible intensity increase, the length of the gradient vector corresponds to the rate of
change in that direction. Two types of gradients, with blue arrows to indicate the direction
of the gradient
Sobel operatorsRepresents a rather inaccurate approximation of the image gradientThe operator calculates the gradient of the image intensity at each pointGiving the direction of the largest possible increase from light to dark and the rate of change in that direction The result therefore shows how "abruptly" or "smoothly" the image changes at that point
Sobel Example
Sobel ExampleGrayscaleimage of a brick wall & a bike rack
scale image of a brick wall& a bike rack
Combining Spatial Enhancement Methods
Successful image enhancement is typically not achieved using a single operationRather we combine a range of techniques in order to achieve a final resultThis example will focus on enhancing the bone scan to the right
Combining Spatial Enhancement Methods
Laplacian filter of bone scan (a)
Sharpened version of bone scan achieved by subtracting (a) and (b) Sobel filter of bone
scan (a)
(a)
(b)
(c)
(d)
Combining Spatial Enhancement Methods
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The product of (c) and (e) which will be used as a mask
Sharpened image which is sum of (a) and (f)
Result of applying a power-law trans. to (g)
(e)
(f)
(g)
(h)
Image (d) smoothed with a 5*5 averaging filter
Combining Spatial Enhancement Methods Compare the original and final images
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20
02
)
assignments Chapter 3 1, 6, 10, 12, 18, 22, 23.
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