Gray-l l t f ti d ti l filt ilevel transformation and ...users.encs.concordia.ca/~weiping/DIP W12/New Chapter 3.pdf · Chapter 3 Intensity Transformations & Spatial Filtering Gray-l

Chapter 3 Intensity Transformations & Spatial FilteringChapter 3 Intensity Transformations & Spatial Filtering

G l l t f ti d ti l filt iGray-level transformation and spatial filtering

© 2002 R. C. Gonzalez & R. E. Woods


)],([),( yxfTyxg

Gray-level transformation:

)(

)],([),(

rTs

yxfTyxg



Frequently UsedUsed TransformationsFrequently UsedUsed TransformationsFrequently UsedUsed TransformationsFrequently UsedUsed Transformations

1 Image negative1. Image negative

2. Logarithm:

2. Power law:

)1log( rcs

)( rcs



Negative TransformationNegative TransformationNegative TransformationNegative Transformation



Log Transformation in Frequency DomainLog Transformation in Frequency DomainLog Transformation in Frequency DomainLog Transformation in Frequency Domain



P L F ilP L F ilPower Law FamilyPower Law Family



Correction of Response of CRT MonitorCorrection of Response of CRT MonitorCorrection of Response of CRT MonitorCorrection of Response of CRT Monitor

Response of CRT:

Gamma correction:

5.2rs

4.0rs





Power Law TransformationPower Law TransformationPower Law TransformationPower Law Transformation



Piecewise Linear TransformPiecewise Linear Transformecew se ea a s oecew se ea a s o



Gray-Level SlicingGray-Level SlicingGray Level SlicingGray Level Slicing



Gray Level SlicingGray Level SlicingGray-Level SlicingGray-Level Slicing



Bit Plane RepresentationBit Plane Representationpp

Bit 7: ‘0’: gray-level 0 ~ 127 ; ‘1’: gray-level 128 ~ 255

Bit 6: ‘0’: 0 ~ 63, 128 ~ 191; ‘1’: 64 ~ 127, 192 ~ 255

:

Bit 0: ‘0’: levels 2, 4, … 254; ‘1’: 1, 3, 5, … 255


Chapter 3 Intensity Transformations & Spatial FilteringChapter 3 Intensity Transformations & Spatial FilteringBit Plane RepresentationBit Plane Representation



Bit plane reconstructed imageBit plane reconstructed imagep gp g


Chapter 3 Intensity Transformations & Spatial FilteringChapter 3 Intensity Transformations & Spatial Filtering8 Bit-Plane Images8 Bit-Plane Images


Chapter 3 Intensity Transformations & Spatial FilteringChapter 3 Intensity Transformations & Spatial FilteringHistogram of ImageHistogram of Image

Assume T(r) satisfies 2 conditions:

(a) T(r) is monotonically increasing function in [0, L-1].

(b) .10for1)(0 LrLrT



Equalization AlgorithmEqualization Algorithmq gq gAssume :)(rpr p.d.f. of gray-level;

:)(rTs gray-level transformation. The p d f of transformed image:The p.d.f. of transformed image:

)(1||)()(sTrrs ds

drrpsp

From dpLrTsr

r0

)()1()( . r0

Then

)()1()()1()(0

rpLdpdrdL

drrdT

drds

r

r

r

,

11

|)(|)1(1)()()(

LrpLrp

dsdrrpsp

rrrs

Discrete case:

)1,...1,0()( LkMNnrp k

kr


k

jjj

k

jrkk n

MNLrpLrTs

00

1)()1()(


Histogram EqualizationHistogram EqualizationHistogram EqualizationHistogram Equalization



Example 3 5Example 3 5Example 3.5Example 3.5



Similarly, s2=4.55, s3=5.67, s4=6.23, s5=6.65, s6=6.86, s7=7.00

Next, the new intensity levels are rounded to the nearest integers:

Similarly, s2=4.55, s3=5.67, s4=6.23, s5=6.65, s6=6.86, s7=7.00

Next, the new intensity levels are rounded to the nearest integers:

s0=1, s1=3, s2=5, s3=6, s4=6, s5=6, s6=7, s7=7 s0=1, s1=3, s2=5, s3=6, s4=6, s5=6, s6=7, s7=7



T f ( d f f l l)T f ( d f f l l)Transforms (c.d.f. of gray-level)Transforms (c.d.f. of gray-level)



Gray-Level MappingGray-Level Mapping



Image & HistogramImage & HistogramImage & HistogramImage & Histogram



Histogram matchingHistogram matchingHistogram matchingHistogram matching



Histogram MatchingHistogram MatchingHistogram MatchingHistogram MatchingGiven p.d.f. )(rpr , dpLrTs

r

r0

)()1()(

For desired p.d.f. )(zp z , let

dttpLzGsz

z0

)()1()(

Then )]([][ 11 rTGsGz . Discrete case:

1. 1)()1()(

00

nMNLrpLrTs

k

jjj

k

jrkk

)1,...1,0( Lk

2. For a value of q , compute )()1()(0

i

q

izq zpLzG

such that )1,...1,0()( LkszG kq

where )( iz zp is the i-th value of the specified histogram.

3 ][)]([ 11 sGrTGz )110( Lq


3. ][)]([ kkq sGrTGz , )1,...1,0( Lq

That is qsG

krT

k zsr kk ][)( 1


Example 3 8Example 3 8Example 3.8Example 3.8

00.0)(7)(0

00

j

jz zpzG

00.0)()(7)(7)( 10

1

01

zpzpzpzG zzjj

z

SimilarlySimilarly,

45.2)(,05.1)(,00.0)( 432 zGzGzG

00.7)(,95.5)(,55.4)( 765 zGzGzG )(,)(,)( 765

Rounding to integer,

1)(0)(0)(0)( zGzGzGzG 1)(,0)(,0)(,0)( 3210 zGzGzGzG

7)(,6)(,5)(,2)( 7654 zGzGzGzG



Histogram EqualizationHistogram EqualizationHistogram EqualizationHistogram Equalization













Local histogram equalizationLocal histogram equalizationLocal histogram equalizationLocal histogram equalization



Enhancement using Histogram StatisticsEnhancement using Histogram StatisticsEnhancement using Histogram StatisticsEnhancement using Histogram StatisticsThe n-th order moment ,

)()()(1

i

Ln

in rpmrr

, )()(1

i

L

i rprrEm

0i

,0i

m and 22 )( rr can be used for local enhancement.

It is common practice to estimate mean and variance: 1 11 M N 1 11 M N

1

0

1

0

),(1 M

x

N

y

yxfMN

m ,

1

0

1

0

22 ]),([1 M

x

N

yr myxf

MN

Given a small area yxS centered at (x,y),Given a small area yx , centered at (x,y),

1

0

221

0

)()(,)(,,,,,

L

iiSSiS

L

iiSiS rpmrrprm

yxyxyxyxyx

An enhancement algorithm:

otherwiseyxf

DkDkMkmyxfEyxg GSGGS yxyx

)(,),(

),( 210 ,,


otherwiseyxf ),(

:GM global mean; :GD global standard deviation.


Example using local histogram statisticsExample using local histogram statistics



Spatial filteringSpatial filtering

a b

tysxftswyxg ),(),(),( as bt

,12 am .12 bn



1 D Correlation and convolution1-D Correlation and convolution



2 D Correlation and convolution2-D Correlation and convolution



3x3 filter mask



3x3 smoothing filter



Image smoothing using smoothing filters of different sizes



Image smoothing and thresholding



Salt-and pepper noise reduction using 3x3 filter masks



Sharpening spatial filtersSharpening spatial filters1st order derivative:

2nd order derivative:

1st order derivative:Produces thicker edges;Generally has a stronger response to a gray-level step;Generally has a stronger response to a gray level step;

2nd order derivative:Stronger response to fine details thin lines isolated points;Stronger response to fine details, thin lines, isolated points;Yields a double response at step edges


For most applications, 2nd order derivative is better.


Sharpening spatial filtersSharpening spatial filters

Laplacian operator,

ff 22 y

fxfyxf

2

2

2

22 ),(

where

),(2),1(),1(2

2

yxfyxfyxfxf

2

),(2)1,()1,(2

2

yxfyxfyxfyf

),(2)1,(

)1,(),1(),1(),(2

yxfyxf

yxfyxfyxfyxf



1st and 2nd derivatives of 1 D image1st and 2nd derivatives of 1-D image



Filter masks to implement the LaplacianFilter masks to implement the Laplacian



Image sharpening using LaplacianImage sharpening using Laplacian



Unsharpening and highboost filteringUnsharpening and highboost filtering

),(),(),( yxfyxfyxg m a sk )(*)()( kf ),(*),(),( yxgkyxfyxg ma sk





Image sharpening using 1st derivative Gradient

Gradient,

Image sharpening using 1st derivative - Gradient

yfxf

y

x

GG

yxf ),(

Or

21

22),( yx GGyxf ,

||||),( yx GGyxf

If using 3x3 mask,g ,

|)2()2(|

|)2()2(|),(

741963

321987

zzzzzz

zzzzzzyxf



i i l i i diDigital approximation to gradient



Edge enhancement using gradientEdge enhancement using gradient



Image enhancement using derivativesImage enhancement using derivatives



Spatial domain image enhancement mixed methodSpatial domain image enhancement – mixed method



F tFuzzy set

})(,{ ZzzzA A

) }1.0,2 9(. . . ,) ,8.0,2 2() ,9.0,2 1() ,1,2 0(. . . ,) ,1,2() ,1,1{ (A



Fuzzy set operationEmpty set: A fuzzy set is empty iff 0)( zA for all .Zz

Fuzzy set operation

Equality: A=B iff )()( zz BA for all .Zz Complement: A is complement of A if )(1)( zz AA for all .Zz Subset: A is a subset of B iff )()( zz BA for all .Zz U iUnion: BAU has )](),(max[)( zzz BAU for all .Zz Intersection:


Intersection: BAI has )](),(min[)( zzz BAI for all .Zz


Set operation exampleSet operation example



Commonly used membership functionsCommonly used membership functions



Fuzzy set application in color sensation



A li ti f b hi f tiApplication of membership function

R1: IF the color is green, THEN the fruit is verdant. OROR

R2: IF the color is yellow, THEN the fruit is half-mature.OR

R3: IF the color is red THEN the fruit is mature


R3: IF the color is red, THEN the fruit is mature.


Fuzzy set operation

)}()(i {)(


)}(),(min{),(3 vzvz matred


Fuzzy set operation

)},(),(min{)( 0303 vzzvQ red

)},(),(min{)()},(),(min{)(

0303

0302

vzzvQvzzvQ

green

yellow


)},(),({)( 0303Q green


Membership functions for color representationMembership functions for color representation

K

vQv )(


321 QORQORQQ . Center of gravity:

K

v

v

vQ

vQvv

1

1

)(

)(

0


i l i i h iFuzzy system implementation with more inputs



l b d hFuzzy, rule-based contrast enhancement

If a pixel is dark, then make it darker.If i l i h k iIf a pixel is gray, then make it gray.If a pixel is bright, then make it brighter.

)()()( 0000

vzvzvzv bbrightggreyddark


)()()( 0000 zzz

vbrightgreydark


C h i hiContrast enhancement using histogram equalization and fuzzy system



HistogramsHistograms



S i l fil i i fSpatial filtering using fuzzy sets





Boundary detection using fuzzy rulesBoundary detection using fuzzy rules



Fuzzy spatial filteringFuzzy spatial filtering


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Gray-l l t f ti d ti l filt ilevel transformation and ...users.encs.concordia.ca/~weiping/DIP W12/New Chapter 3.pdf · Chapter 3 Intensity Transformations & Spatial Filtering Gray-l