Intensity Transformations and Spatial Filtering
Jan 02, 2016
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.
, ,g x y T f x y Origin (0,0)
(x,y)
(M-1,0)
(0,N-1)
3x3 Neighborhood of (x,y)
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)
Image Negatives s = L-1 – r
where intensity level is in the range[0, L-1]
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.
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.
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
Examples of Gamma Transformations
3.0
4.0 5.0
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”.
Examples of Contrast Stretching
Contrast Stretching in Medical Image
Window Width/Level(Center) s1=0,s2=255
width (w)=r2-r1, level (c)=(r1+r2)/2
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
Cumulative Distribution Function (CDF)
PDF CDF
Transfer Function
r
s
0
rCDF r p r dr
Example of Histogram and Cumulative Distribution Function (CDF)
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
Histogram Equalization
The Histogram Equalization is a method which makes the histogram of the image as smooth as possible
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
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
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
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
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