Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 3Intensity Transformations & Spatial Filtering
Chapter 3Intensity Transformations & Spatial Filtering
Intensity transformation T maps the intensity r0 of a pixel, P, to a new intensity value s0=T(r0 )intensity value s0 T(r0 ). The mapping is performed using a transfer function
TT
Examples of two transfer functionsExamples of two transfer functions• Transformation function does not take
into the intensity of adjacent pixels.• It does not increase the number of
© 1992–2008 R. C. Gonzalez & R. E. Woods
intensity values available.
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 3Intensity Transformations & Spatial Filtering
Chapter 3Intensity Transformations & Spatial Filtering
Spatial Filter F maps the intensity Ii of a pixel, Pi, to a new intensity value based on its neighborhoodbased on its neighborhood.
TT
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 3Intensity Transformations & Spatial Filtering
Chapter 3Intensity Transformations & Spatial Filtering
Example of Intensity Transformations.• They map L intensity values
a new L intensity value s0
T
0
© 1992–2008 R. C. Gonzalez & R. E. Woods
r0
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 3Intensity Transformations & Spatial Filtering
Chapter 3Intensity Transformations & Spatial Filtering
Intensity transformation – NegativeOriginal mammogram (left) andthe negated mammogram g g
L
s0
0 Lr0
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 3Intensity Transformations & Spatial Filtering
Chapter 3Intensity Transformations & Spatial Filtering
Intensity transformation – LogOriginal Fourier spectrum (left) andthe log transformed spectrum (right)the log transformed spectrum (right) with c = 1
)1log( rcs
s0
)g(
s0
© 1992–2008 R. C. Gonzalez & R. E. Woods
r0
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 3Intensity Transformations & Spatial Filtering
Chapter 3Intensity Transformations & Spatial Filtering
Power-Law (Gamma) Transformations
crs Where c and γ are positive constants.
cs
It also sometime written as: )( rcs
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 3Intensity Transformations & Spatial Filtering
Chapter 3Intensity Transformations & Spatial Filtering
Gamma correction Gamma correction aims to improve the
correctness of an image when display g p yon a screen. Gamma correction controls the overall
brightness of an image. Incorrect images can look either
bleached out, or too dark.
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 3Intensity Transformations & Spatial Filtering
Chapter 3Intensity Transformations & Spatial Filtering
Gamma correction
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 3Intensity Transformations & Spatial Filtering
Chapter 3Intensity Transformations & Spatial Filtering
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 3Intensity Transformations & Spatial Filtering
Chapter 3Intensity Transformations & Spatial Filtering
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 3Intensity Transformations & Spatial Filtering
Chapter 3Intensity Transformations & Spatial Filtering
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 3Intensity Transformations & Spatial Filtering
Chapter 3Intensity Transformations & Spatial Filtering
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 3Intensity Transformations & Spatial Filtering
Chapter 3Intensity Transformations & Spatial Filtering
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 3Intensity Transformations & Spatial Filtering
Chapter 3Intensity Transformations & Spatial Filtering
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 3Intensity Transformations & Spatial Filtering
Chapter 3Intensity Transformations & Spatial Filtering
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 3Intensity Transformations & Spatial Filtering
Chapter 3Intensity Transformations & Spatial Filtering
Image HistogramThe image histogram of a digital image with intensitylevels in the range [0,L-1] is a discrete function g [ , ]h(rk) =nk , where rk is the kth intensity value and nk is the number of pixels in the image with the intensity rk
Histogram NormalizationHistogram NormalizationIt is common to normalize the histogram by dividingnk by the number of pixels in the image. The normalized intensity p(r )= n /MN estimates theThe normalized intensity p(rk)= nk /MN estimates the probability of occurrence of intensity level rk in an image
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 3Intensity Transformations & Spatial Filtering
Chapter 3Intensity Transformations & Spatial Filtering
Histogram Equalization
In general we assumeIn general we assume 1. T(r) is monotonically increasing2. 0 ≤ T(r) ≤ L-1 for 0 ≤ r ≤ L-1
Let pr(r) and ps (s) be a probability density functions. If we assume pr(r) and T(r) are know, the ( )
dsdrrpsp rs )()(
Image processing interests on the following formulation, where the right side is the cumulative distribution function
r
© 1992–2008 R. C. Gonzalez & R. E. Woods
r
r dwwpLrTs0
)()1()(
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 3Intensity Transformations & Spatial Filtering
Chapter 3Intensity Transformations & Spatial Filtering
)()1()()1()(
0
rpLdwwpdrdL
drrdT
drds
r
r
r
Use the previous formulation yields a uniformUse the previous formulation yields, a uniform probability density function
11)()()(
rpdrrpsp1)()1(
)()()(
LrpL
rpds
rpspr
rrs
Histogram equalization determine the transformation that seek to produce antransformation that seek to produce an output image that has a uniform histogram.
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 3Intensity Transformations & Spatial Filtering
Chapter 3Intensity Transformations & Spatial Filtering
Example:
10;)1(
2)( 2
Lr
Lr
rp
112)()1()(
0)1()(
0
2
0
2
Lrwdw
LdwwpLrTs
otherwiseLrp
rr
r
r
1)1(2
)1(2)()(
11)()()(
12
2
1
2
00
Lr
drd
Lr
drds
Lr
dsdrrpsp
LLp
rs
r
11
2)1(
)1(2
2
Lr
LL
r
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 3Intensity Transformations & Spatial Filtering
Chapter 3Intensity Transformations & Spatial Filtering
Histogram Equalization In the discrete case
1,..,2,1,0;)1()()1()(00
LknMNLrpLrTs
k
jj
k
jjrkk
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 3Intensity Transformations & Spatial Filtering
Chapter 3Intensity Transformations & Spatial Filtering
Example:A 3bit image of size 64x64 [0, L-1] =[0,7]
33.1)(7)(7)(
1
0
0
000
rprprTs rj
jr
equalized histogram
08.3)(7)(7)(7)( 100
11
rprprprTs rrj
jr
00.7,86.6,65.6,23.6,67.5,55.4 765432 ssssss
We round the s values to get values of the equalized histogram
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 3Intensity Transformations & Spatial Filtering
Chapter 3Intensity Transformations & Spatial Filtering
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 3Intensity Transformations & Spatial Filtering
Chapter 3Intensity Transformations & Spatial Filtering
Histogram MatchingLet us assume histograms are continuous functions, and let T(r) defined as earlier and G(z) defined similarly, then
r
r dwwpLrTs0
)()1()( z
z dttpLzGs0
)()1()(
Since T(r) = G(z) then
And
Since T(r) = G(z), then)()]([ 11 sGrTGz
This shows that that an image whose intensity levels have a specific probability density g y p p y yfunction can be obtained as follow:1. Obtain pr(r) from the input image and determine the value of s (as above)2. Use a specified PDF to obtain the transformation function G(z)3. Compute the inverse transformation z=G-1(s)4. Obtain the output image by first equalizing the input image (intensity are the s values)
for each intensity (s value) perform the inverse mapping z = G-1(s) to obtain the
© 1992–2008 R. C. Gonzalez & R. E. Woods
corresponding pixel in the output image
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 3Intensity Transformations & Spatial Filtering
Chapter 3Intensity Transformations & Spatial Filtering
Example:
rLr
Lr
rp 2 10;)1(
2)(
rr
r
r
Lrwdw
LdwwpLrTs
otherwiseLp
0
2
0 112)()1()(
0)1()(
rz
z Lzdww
LdwwpLzG
LL
0 2
22
0 2 113)()1()(
11
Now we can compute the values of z by
3/1
2
3
)1(Lzs
Now we can compute the values of z by
3/12)1( sLz
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 3Intensity Transformations & Spatial Filtering
Chapter 3Intensity Transformations & Spatial Filtering
Example:A 3bit image of size 64x64 [0, L-1] =[0,7]
0 1 2 3 4 5 6 7
1 3 5 6 7 7 7 7
The s values from the previous example
1 3 5 6 7 7 7 7
00.0)(7)(7)( 0
0
00
rprpzG rj
jz
00.0)](7)([7)(7)( 10
1
01
zpzpzpzG rrj
jz
The G(z) values from the previous example0 1 2 3 4 5 6 7
0.0 0.0 0.0 1.05 2.45 4.55 5.95 7.00
( ) f p p
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 3Intensity Transformations & Spatial Filtering
Chapter 3Intensity Transformations & Spatial Filtering
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 3Intensity Transformations & Spatial Filtering
Chapter 3Intensity Transformations & Spatial Filtering
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 3Intensity Transformations & Spatial Filtering
Chapter 3Intensity Transformations & Spatial Filtering
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 3Intensity Transformations & Spatial Filtering
Chapter 3Intensity Transformations & Spatial Filtering
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 3Intensity Transformations & Spatial Filtering
Chapter 3Intensity Transformations & Spatial Filtering
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 3Intensity Transformations & Spatial Filtering
Chapter 3Intensity Transformations & Spatial Filtering
1 1
Spatial FilteringApplying a 3x3 filter ,w, on the image f
1
1
1
1),(),(),(
i jjyixfjiwyxg
A l i l (2 2b) fil hApplying a general (2ax2b) filter ,w, on the image f
a b
a
ai
b
bjjyixfjiwyxg ),(),(),(
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 3Intensity Transformations & Spatial Filtering
Chapter 3Intensity Transformations & Spatial Filtering
CorrelationIs the process of moving a filter maskover the image and computing the sumg p gof products at each location.
C l tiConvolutionmoves the reversed filter mask over the image and computing the sum of products at each locationproducts at each location.
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
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Gonzalez & Woods
Chapter 3Intensity Transformations & Spatial Filtering
Chapter 3Intensity Transformations & Spatial Filtering
Correlation and Convolution in the 2D space.
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 3Intensity Transformations & Spatial Filtering
Chapter 3Intensity Transformations & Spatial Filtering
3x3 Spatial filters, which result in a blurring effect. The blurring depends on the ration between the central value and the “boundary” values.y
The effect of applying averaging filters of size 3,5,9, 15, and 35.
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 3Intensity Transformations & Spatial Filtering
Chapter 3Intensity Transformations & Spatial Filtering
In many application we often apply multiple filters and some of them may appear y ppcontradicting from the first glance.
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 3Intensity Transformations & Spatial Filtering
Chapter 3Intensity Transformations & Spatial Filtering
An Image as a Functionld hi k b i• We could think about an image row, as a
one dimensional function, – x is the position of the pixel – y is the color of the pixel-grayscale.
• Similarly, 2D image could be treated as 3D function.
• Actually these functions are not 3D, but 2.5 D as there is one value for each x, y value va ue
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 3Intensity Transformations & Spatial Filtering
Chapter 3Intensity Transformations & Spatial Filtering
An Image as a Function• As we treat image as a function
It i ibl t t i
)()('lim)('0 x
xfxxfxfx
– It is possible to compute is derivative, but we need to exchange ∆x by 1.
I i ibl h fi
)()1(1
)()1()(' xIxIxIxIxI
)(')('lim)('' xfxxfxf • It is possible to compute the first
and the second derivative of an image.
1)(')1(')(''
lim)(0
xIxIxI
xxf
x
• Derivative help in determining local minima, local maxima, and change in derivative direction )()1(2)2(
)()1()1()2(1
xIxIxIxIxIxIxI
)()()(
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 3Intensity Transformations & Spatial Filtering
Chapter 3Intensity Transformations & Spatial Filtering
First & Second Derivatives• Let us consider a vertical cut on the
fi t t ifirst two images• The function we get are below each
images
• The first derivative
S d D i ti• Second Derivative
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 3Intensity Transformations & Spatial Filtering
Chapter 3Intensity Transformations & Spatial Filtering
Gradient OperatorThe Gradient Operator for an image f at the location (x, y) is g ( , y)
Which is often approximated by:
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & WoodsGradient OperatorChapter 3
Intensity Transformations & Spatial FilteringChapter 3
Intensity Transformations & Spatial Filtering
Gradient Operator
Gradient OperatorThe direction of the Gradient vector at the location (x, y) is
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & WoodsLaplacian OperatorChapter 3
Intensity Transformations & Spatial FilteringChapter 3
Intensity Transformations & Spatial Filtering
Laplacian Operator
Laplacian OperatorThe Laplacian Operator for an image f at the location (x, y) is
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Image FilterChapter 3
Intensity Transformations & Spatial FilteringChapter 3
Intensity Transformations & Spatial Filtering
Image Filter
Image FilterImage Filters change the appearance of an image or part of an image by altering the values of its pixels.
Image Blur
Median Filter
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 3Intensity Transformations & Spatial Filtering
Chapter 3Intensity Transformations & Spatial Filtering
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 3Intensity Transformations & Spatial Filtering
Chapter 3Intensity Transformations & Spatial Filtering
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 3Intensity Transformations & Spatial Filtering
Chapter 3Intensity Transformations & Spatial Filtering
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 3Intensity Transformations & Spatial Filtering
Chapter 3Intensity Transformations & Spatial Filtering
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 3Intensity Transformations & Spatial Filtering
Chapter 3Intensity Transformations & Spatial Filtering
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 3Intensity Transformations & Spatial Filtering
Chapter 3Intensity Transformations & Spatial Filtering
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 3Intensity Transformations & Spatial Filtering
Chapter 3Intensity Transformations & Spatial Filtering
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 3Intensity Transformations & Spatial Filtering
Chapter 3Intensity Transformations & Spatial Filtering
Set Theory
One way of defining a set A is in terms of its characteristic function . An element )(xA
x belongs to set A if and only if , where .1)( xA }1,0{:)( UxA
In such a scheme we define set operation as:
• Union as
I i
))(),(max()( xxx BABA
))()(i ()( 1)( x• Intersection as
• Complement as
S t I l i if d l if (f ll ) i li
))(),(min()( xxx BABA
)(1)( xx AA
1)( xA
1)( 1)( xBA x• Set Inclusion as if and only if (for all x) implies
• Set Equality as A = B if and only if (for all x)
1)( xA 1)( xB
)()( xx BA
BA x
x
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 3Intensity Transformations & Spatial Filtering
Chapter 3Intensity Transformations & Spatial Filtering
Fuzzy Set Theory
A fuzzy set is defined in terms of a membership function .]1,0[: UA
A characteristic function is a special case of a membership function and a regular set is a
special case of a fuzzy set.
The set operations are defined as:
• Union as
• Intersection as
))(),(max()( xxx BABA
))(),(min()( xxx BABA
• Complement as
• Set inclusion as if and only if (for all x)
S li A if if (f )
)(1)( xx AA
)()( xx BA BA x
© 1992–2008 R. C. Gonzalez & R. E. Woods
• Set Equality as A = B if and only if (for all x) )()( xx BA x
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
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Gonzalez & Woods
Chapter 3Intensity Transformations & Spatial Filtering
Chapter 3Intensity Transformations & Spatial Filtering
Illustrating The membership functions of regular and fuzzy set
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
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Gonzalez & Woods
Chapter 3Intensity Transformations & Spatial Filtering
Chapter 3Intensity Transformations & Spatial Filtering
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 3Intensity Transformations & Spatial Filtering
Chapter 3Intensity Transformations & Spatial Filtering
otherwise
cazacazazbabza
zTriangle0
/)(1/)(1
)(:
dbzbcaz
bzaazcabza
zTrapezodal/)(1
1/)(1
)(:
zaazbabza
zSigma
otherwise
1/)(1
)(:
0
bzaacaz
az
otherwise
2
0
0
2
otherwise
dbzbacaz
accbazSShapeS
0
21),,;(: 2
© 1992–2008 R. C. Gonzalez & R. E. Woods
zabcbcczSczcbcbczS
zShapeBell),2/,,(1
),2/,,()(:
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 3Intensity Transformations & Spatial Filtering
Chapter 3Intensity Transformations & Spatial Filtering
Rule-Based classification of using fuzzy sets.
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
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Gonzalez & Woods
Chapter 3Intensity Transformations & Spatial Filtering
Chapter 3Intensity Transformations & Spatial Filtering
The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again.
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 3Intensity Transformations & Spatial Filtering
Chapter 3Intensity Transformations & Spatial Filtering
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 3Intensity Transformations & Spatial Filtering
Chapter 3Intensity Transformations & Spatial Filtering
The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again.
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 3Intensity Transformations & Spatial Filtering
Chapter 3Intensity Transformations & Spatial Filtering
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 3Intensity Transformations & Spatial Filtering
Chapter 3Intensity Transformations & Spatial Filtering
Using fuzzy sets for intensitytransformation 1. Define a set of rules to change pixelg p
intensity.2. Transfer the rules into fuzzy set3. User the rules to change intensity
Example:1. If a pixel is dark, then make it darker2 If i l i th k it2. If a pixel is gray, then make it gray 3. If a pixel is bright, then make it brighter
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 3Intensity Transformations & Spatial Filtering
Chapter 3Intensity Transformations & Spatial Filtering
Using fuzzy sets for intensitytransformation 1. Define a set of rules to change pixelg p
intensity.2. Use fuzzy set to apply this rules.
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 3Intensity Transformations & Spatial Filtering
Chapter 3Intensity Transformations & Spatial Filtering
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 3Intensity Transformations & Spatial Filtering
Chapter 3Intensity Transformations & Spatial Filtering
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 3Intensity Transformations & Spatial Filtering
Chapter 3Intensity Transformations & Spatial Filtering
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 3Intensity Transformations & Spatial Filtering
Chapter 3Intensity Transformations & Spatial Filtering
Using fuzzy sets for spatial filtering
© 1992–2008 R. C. Gonzalez & R. E. Woods