intensity transformations and spatial filtering
DESCRIPTION
Intensity Transformations and Spatial Filtering. Basics of Intensity Transformation and Spatial Filtering. Spatial Domain Process Neighborhood is rectangle, centered on ( x,y ), and much smaller in size than image. Neighborhood size is 1x1, 3x3, 5x5, etc. Intensity Transformation. - PowerPoint PPT PresentationTRANSCRIPT
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Intensity Transformations and Spatial Filtering
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Basics of Intensity Transformation and Spatial Filtering
Spatial Domain Process
Neighborhood is rectangle, centered on (x,y), and much smaller in size than image.
Neighborhood size is 1x1, 3x3, 5x5, etc.
, ,g x y T f x y Origin (0,0)
(x,y)
(M-1,0)
(0,N-1)
3x3 Neighborhood of (x,y)
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Intensity TransformationT[f(x,y)] is Intensity
Transformation, if the neighborhood size is 1x1.
Intensity Transformation can be written as follows
s = T[r],
where s = g(x,y), and r = f(x,y)
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Image Negatives s = L-1 – r
where intensity level is in the range[0, L-1]
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Log Transformations s = c Log(1+r)
Log Transformation is used to expand the value of the dark pixels while compressing the higher-level value.
It is used to compress the intensity of an image which has very large dynamic range.
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Log Transformations of Fourier Spectrum
Original Image
Fourier Spectrum
Log Transform of
Fourier SpectrumWe cannot see the Fourier spectrum,
because its dynamic range is very large.
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Power-Law (Gamma) Transformations
If <1, expand dark pixels, compress bright pixels.
If >1, compress dark pixels, expand bright pixels.
0.04 0.10
0.20 0.40
0.64
1.0
1.5
2.5 5.0
10.0
s cr
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Examples of Gamma Transformations
3.0
4.0 5.0
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Contrast StretchingIf r<r1 then
s = r*s1/r1If r1<= r<=r2 then
s = (r-r1)*(s2-s1)/(r2-r1)+s1If r>r2 then
s = (r-r2)*(255-s2)/(255-r2)+s2If r1=r2 and s1=0,s2=255, the
transform is called “Threshold Function”.
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Examples of Contrast Stretching
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Contrast Stretching in Medical Image
Window Width/Level(Center) s1=0,s2=255
width (w)=r2-r1, level (c)=(r1+r2)/2
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Histogram & PDF
h(r) = nr
where nr is the number of pixels whose intensity is r.
The Probability Density Function (PDF) h r
p rM N
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Cumulative Distribution Function (CDF)
PDF CDF
Transfer Function
r
s
0
rCDF r p r dr
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Example of Histogram and Cumulative Distribution Function (CDF)
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Low Contrast Image
The image is highly concentrated on low intensity values.
The low contrast image is the image which is highly concentrated on a narrow histogram.
HighConcentra
te
LowConcentra
te
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Histogram Equalization
The Histogram Equalization is a method which makes the histogram of the image as smooth as possible
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The PDF of the Transformed Variable
s = Transformed Variable.
= The PDF of r = The PDF of s
s T r
rp r
sp s
1
/
s r
r
drp s p r
ds
p rdT r dr
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Transformation Function of Histogram Equalization
The PDF of s
0255
r
rs T r p r dr
0255
255
1
255
r
r
r
s r
dT rds
dr drd
p r drdrp r
drp s p r
ds
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Histogram Equalization Example
Intensity # pixels
0 20
1 5
2 25
3 10
4 15
5 5
6 10
7 10
Total 100
CDF of Pr
20/100 = 0.2
(20+5)/100 = 0.25
(20+5+25)/100 = 0.5
(20+5+25+10)/100 = 0.6
(20+5+25+10+15)/100 = 0.75
(20+5+25+10+15+5)/100 = 0.8
(20+5+25+10+15+5+10)/100 = 0.9
(20+5+25+10+15+5+10+10)/100 = 1.0
1.0
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Histogram Equalization Example (cont.)
Intensity (r)
No. of Pixels(nj)
Acc Sum of Pr
Output value Quantized Output (s)
0 20 0.2 0.2x7 = 1.4 1
1 5 0.25 0.25*7 = 1.75 2
2 25 0.5 0.5*7 = 3.5 3
3 10 0.6 0.6*7 = 4.2 4
4 15 0.75 0.75*7 = 5.25 5
5 5 0.8 0.8*7 = 5.6 6
6 10 0.9 0.9*7 = 6.3 6
7 10 1.0 1.0x7 = 7 7
Total 100
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Histogram MatchingHow to transform the variable r
whose PDF is to the variable t whose PDF is .
0
0
1
255
255
r
r
t
t
s T r p r dr
G t p t dt s
t G t
rp r
tp t
r T( ) s G-1( ) t