Chapter4 Image Enhancement - USTChome.ustc.edu.cn/~xuedixiu/image.ustc/course/dip/DIP14-ch4-1.pdf · • 4.4 Enhancement by Frequency Filtering (image sharpening) • 4 5 Color Enhancement4.5

Chapter4 Image Enhancement

• Preview• 4 1 General introduction and Classification4.1 General introduction and Classification• 4.2 Enhancement by Spatial Transforming(contrast enhancement)• 4 3 Enhancement by Spatial Filtering (image smoothing)4.3 Enhancement by Spatial Filtering (image smoothing)• 4.4 Enhancement by Frequency Filtering (image sharpening)• 4 5 Color Enhancement• 4.5 Color Enhancement• Summary

Digital Image Processing Prof.Zhengkai Liu Dr.Rong Zhang

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4 1 G l I t d ti d Cl ifi ti4.1 General Introduction and Classification4 1 1 Purposes4.1.1 Purposes

• improve the visual effectsimprove the visual effects• easy to edge extracting

4.1.2 Methods

• spatial domain: point operations, local operationsp p p , p• frequency domain: DFT Filter IDFT


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4.1 General Introduction and Classification4.1.3 contents • contrast enhancement: linear transform• contrast enhancement: linear transform

non-linear transform,histogram equalizationg qhistogram matching local enhancement

i hi i k• image smoothing: averaging mask, order-statistics filterlowpass filterlowpass filter.

• image sharpening: derivatives, highpass filterg p

• color image enhancement: pseudo color processing, full color processing


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4.2 Contrast Enhancement4.2.1. Introduction: Histogram

Histogram gives an estimate of the probability ofHistogram gives an estimate of the probability of the occurrence of gray levels

( ) / 0,1, , 1k kp s n n k L= = −

Where s is the the kth gray level and the n is the number ofWhere sk is the the kth gray level and the nk is the number ofPixels in the image having gray level sk

tlapparently1

( ) 1L

kp s−

=∑0

kk=∑


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H i t l i l l lHorizontal axis: gray level valuesVertical axis: probability of gray level values


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Hi t d i litHistograms and images quality

Dark image


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Bright image


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Low-contrast image


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Double-peaks image


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Equalized image


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4.2 Contrast Enhancement4.2.1. Introduction: Classification

• Direct gray level transformationsg y• Histogram processing• Operations among images


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4.2 Contrast Enhancement4.2.2 Direct gray level transformations: Linear transformations

( )s T r ar b= = + [ , ] [ , ]r A B s C D∈ ∈

D C BC AD

Expression:

D C BC ADs rB A B A− −

= +− −

Formulation:

255

Example:C 0C=0 D=255


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4.2 Contrast Enhancement

example: A= 69 B=213 C=0 D=255

4.2.2 Direct gray level transformations : Linear transformations

1 7 122 2s r= −example: A= 69, B=213, C=0, D=255 1.7 122.2s r=


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4.2 Contrast Enhancement4.2.2 Direct gray level transformations : Piecewise-linear transformations

[0, )c r r aa

⎧ ∈⎪Formulation:

[ , )

ad cs r c r a bb a

⎪⎪

−⎪= + ∈⎨ −⎪255 ( ) [ , 255)255

b ad r b d r bb

⎪−⎪ − + ∈⎪ −⎩

cd

⎩


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4.2 Contrast Enhancement4.2.2 Direct gray level transformations : Piecewise-linear transformations

( , ) (10,50), ( , ) (210,150)a c b d= =

Original image


15( , ) (50,10), ( , ) (150,210)a c b d= =

4.2 Contrast Enhancement4.2.2 Direct gray level transformations : image negatives

1s L r= − −Formulation:


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4.2 Contrast Enhancement4.2.2 Direct gray level transformations : Log transformations

l (1 )log(1 )s c r= +Formulation:

i t t d it ic is a constant and it is assumed that 0r ≥


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4.2 Contrast Enhancement4.2.2 Direct gray level transformations : Log transformations

Example: display of DFT spectrum

Direct display After log operation


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4.2 Contrast Enhancement4.2.2 Direct gray level transformations : Power transformations

s cr γFormulation: s cr γ=Formulation:

Where and are positive constants.γc


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G tiGamma correction


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4.2 Contrast Enhancement4.2.2 Direct gray level transformations : Gray-lever slicing


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4.2 Contrast Enhancement4.2.3 Histogram processing : Histogram equalization

If a transform has the form:( )s T r= 0 1r≤ ≤

If a transform has the form:

We hope the Probability Density Function (PDF) of s isWe hope the Probability Density Function (PDF) of s is

( ) 1sp s = Inverse transform exist( )sp

satisfies the following conditions:( )T r

ve se s o e sand, 1 2 1 2if then r r s s< <

0 1r≤ ≤(a) is single-valued and monotonically in the interval (b) for 0 1r≤ ≤0 1s≤ ≤

s has the same range as the r


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C ti diti( ) ( )s r

drp s p rds

=( )s T r=Continuous condition

( )( )

rp rdsdr p s

⎫= ⎪⎬ ( )

rs p x dx= ∫( )

( ) 1s

s

dr p sp s

⎬⎪= ⎭

0( )rs p x dx∫

namely0

( ) ( )r

rT r p x dx= ∫0∫


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4.2 Contrast Enhancement

Di t diti

4.2.3 Histogram processing : Histogram equalization

Discrete condition

k k n

0 0( ) ( ) 0,1, 2,..., 1

k kj

k k r jj j

ns T r p r k L

n= =

= = = = −∑ ∑

I iIn practice

( ) ( 1) ( )k

k ks T r L p r= = − ∑0

( ) ( 1) ( )k k r jj

s T r L p r=∑


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Example:page 78p e:p ge 78

Suppose that a 64*64, 3bits image has the gray-level distributionas show in the first row the calculate steps are:

7,...,1,0, =ksk

序号

运算步骤和结果

1 列出原始图灰度级 0 1 2 3 4 5 6 7

as show in the first row, the calculate steps are:

k

kn2 统计原始直方图各灰度级象素 790 1023 850 656 329 245 122 81

3 计算原始直方图 0.19 0.25 0.21 0.16 0.08 0.06 0.03 0.02

]5.0)1int[( +−= kk tNt

)( kk ts → 10 → 31→ 52 → 64,3 → 77,6,5 →

4 计算累计直方图 0.19 0.44 0.65 0.81 0.89 0.95 0.98 1.00

5 取整 1 3 5 6 6 7 7 7

6 确定映射对应关系10 → 31→

kn7 统计新直方图各灰度级象素 790 1023 850 985 448

8 用计算新直方图 0.19 0.25 0.21 0.24 0.11


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0.3 0 25 0 40.6

0.8

1

0.44

0.650.81

0.890.95 0.98 1

Histograms

0.1

0.20.19

0.250.21

0.160.08

0.060.03 0 02

0 1 2 3 4 5 6 7

0.20

0.40.19

0 1 2 3 4 5 6 70

0.02

(a)

(b)

0 3 0 25

transformation function(a)

0.1

0.2

0.3

0.19

0.250.21

0.24

0.11original histogram

00 1 2 3 4 5 6 7

(c)

equalized histogramDigital Image Processing

Prof.Zhengkai Liu Dr.Rong Zhang28

equalized histogram


Experimental resultspe e esu s


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Experimental results


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question: page 99 4 2

Why the discrete histogram

question: page 99, 4.2

y gequalization technique does not yield a flat histogram?


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4.2 Contrast Enhancement4.2.3 Histogram processing : Histogram matching (specification)


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Hi t li ti

( ) ( ) 0,1, 2 1k

k k r js T r p r k L= = = −∑

Histogram equalization： r s

0j=

k

Histogram equalization： z v

0

( ) ( ) 0,1, 2 1k

k k z ii

v G z p z k L=

= = = −∑

1 1 1

k kv s=let then r z

1 1 1( ) ( ) ( ( ))k k k kz G v G s G T r− − −= = =


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PzG(zk)

zv, ,v

z,,Pv T(rk)


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Example:page 80Suppose that a 64*64, 3bits image has the gray-level distribution as show in the first row, and the 5th row is the

p e:p ge 80

序运算步骤和结果

,specified gray-level distribution. The calculate steps are:

7,...,1,0, =ksk

kn

序号

运算步骤和结果

1 列出原始图灰度级 0 1 2 3 4 5 6 7

2 统计原始直方图各灰度级象素 790 1023 850 656 329 245 122 81

3 计算原始直方图 0 19 0 25 0 21 0 16 0 08 0 06 0 03 0 02

4096,/)( == nnnup kku

3 计算原始直方图 0.19 0.25 0.21 0.16 0.08 0.06 0.03 0.02

4 计算原始累计直方图 0.19 0.44 0.65 0.81 0.89 0.95 0.98 1.00

5 规定直方图 0 0 0 0.2 0 0.6 0 0.2

6 计算规定累计直方图 0 0 0 0 2 0 2 0 8 0 8 1 0

31,0 → 54,3,2 → 77,6,5 →

6 计算规定累计直方图 0 0 0 0.2 0.2 0.8 0.8 1.0

7s SML映射 3 3 5 5 5 7 7 7

8s 确定映射对应关系

9s 变换后直方图 0 0 0 0 44 0 0 45 0 0 11


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9s 变换后直方图 0 0 0 0.44 0 0.45 0 0.11


0.80 6

0.6

0.2

0.40.2

0.6

0.2

0.2

0.3

0.19

0.250.21

0 16

0 1 2 3 4 5 6 7

(b)

0

0 1 2 3 4 5 6 7

0

0.1

0.16

0.080.06

0.030.02

specified histogram

0.6

0.8

0 44 0.450 1 2 3 4 5 6 7

(a)

i i l hi t 0 5

0.2

0

0.4

0.44

0.11

original histogram 0 1 2 3 4 5 6 7

(c)

resulting histogramDigital Image Processing


resulting histogram



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Example

i i l equalized matchedoriginal equalized matched


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4.2 Contrast Enhancement4.2.2 Direct gray level transformations : Bit-plane slicing

b b b b b b b b1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 0

eg: 255b7 b6 b5 b4 b3 b2 b1 b0

1 1 1 1 1 1 1 0

1 0 0 0 0 0 0 1

254

129B7

B6

B5

一个8bit的字节位面7

B5B4

B3

B2

.

..

.B2B1

B0....

位面0


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4.2 Contrast Enhancement4.2.2 Direct gray level transformations : Bit-plane slicing

b7 b6

b b bb5 b4 b3

b2 b1 b0


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4.3 Image Smoothingg g

• Introduction • Smoothing linear filters• Order- statistic filters• Low-pass filter in frequency domain


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4.3 Image Smoothing4.3.1 Introduction

g g

P i iPurposes: removing noiseRequest: keeping edges and details

Noise Edges Details Gray levels change greatl in neighborhoodgreatly in neighborhood

Having the same texture in a regionHaving the same texture in a region

Having the similar grads directions in a segment

Random in gray levels and spatial locationRandom in gray levels and spatial location

Salt-pepper noiseGaussian noise


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Gaussian noise

4.3 Image Smoothing4.3.1 Introduction: Noise models

g g

PDF of Gaussian noise:2 2( ) / 21( ) zp z e μ σ− −=PDF of Gaussian noise: ( )

2p z e

πσ=


47


g g

Gaussian noise ( , ) ( , ) ( , )g x y f x y x yη= +G uss o se ( ) ( ) ( )g y f y yη


48


g g

PDF of salt-pepper noise: aP z a=⎧⎪PDF of salt-pepper noise:

( )0

bp z P z botherwise

⎪= =⎨⎪⎩⎩


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g g

salt-pepper noise:


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4.3 Image Smoothing4.3.1 Introduction: Smoothing filters

g g

i hb i(1) S thi li filt neighbor averaging(1) Smoothing linear filters:out range pixel smoothingMaximum homogeneity smoothing

(2)Order- statistic filters: Max filters

Mid i filMin filtersMidpoint filtersMedian filtersAlpha-trimmed mean filter p

(3)Low-pass filters: Idea low-pass filter (ILPF)Butterworth low-pass filter (BLPF)Gaussian low-pass filter (GLPF)


51

4.3 Image Smoothing Maskg g4.3.2 Smoothing linear filters: neighbor averagingSpatial filter: mask, kernel, template, window

Maskcoefficients

Sp e : s , e e , e p e, w dow

w(-1,-1) w(-1,0) w(-1,1)

w(0,-1) w(0,0) w(0,-1)

w(1,-1) w(1,0) w(1,1)

maskf( 1 1)

f(x,y)f(x-1,y-1) f(z-1,y) f(x-1,y+1)

f(x y-1) f(x y) f(x y+1)f(x,y 1) f(x,y) f(x,y+1)

f(x+1,y-1) f(x+1,y) f(x+1,y+1)

Pixels undermask


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4.3 Image Smoothingg g4.3.2 Smoothing linear filters: neighbor averaging

Spatial filter: operating steps: (page 83)(1) Moving the filter mask from point to point in an image

Spatial filter: operating steps: (page 83)

(2) Multiplying the filter coefficient and the corresponding image pixels

(3) Adding all the products(3) Adding all the products (4) The sum is the response of the filter at a given point

( , ) ( , ) ( , )a b

i j bg x y w i j f x i y j= + +∑ ∑

i a j b=− =−


53


T i l kTypical mask:

1 1 1

1 1 1

1 2 1

2 4 21×

116

×1 1 1

1 1 1

2 4 2

1 2 1

9 16


54


Experimental results: different masks


55


Experimental results:different sizes


56

4.3 Image Smoothingg g4.3.2 Smoothing linear filters: out range pixel smoothing

( , ) ( , ) ( , )a b

i a j bg x y w i j f x i y j

=− =−

= + +∑ ∑formulate

i a j b

( ) ( ) - ( )g x y g x y f x y T⎧ >( , ) ( , ) - ( , )ˆ ( , )( , ) others

g x y g x y f x y Tg x y

f x y⎧ >

= ⎨⎩

Advantage: keep details in the image with salt-pepper noise


57

4.3 Image Smoothingg g4.3.2 Smoothing linear filters: out range pixel smoothing



58

4.3 Image Smoothingg g4.3.2 Smoothing linear filters: Maximum homogeneity smoothing

Advantage: keep edgesDigital Image Processing


Advantage: keep edges

4.3 Image Smoothingg g4.3.3 Order- statistic filters: Max filters

( , ) { ( , )}maxg x y f x i y j= + +formulate( , 0, 1 )maxi j= ±

o u e Advanttage:reem

oves pepperr noise


60

4.3 Image Smoothing

( , ) { ( , )}ming x y f x i y j= + +

g g4.3.3 Order- statistic filters: Min filters

formulate( , 0, 1 )

( , ) { ( , )}mini j

g y f y j= ±

o u e Advanttage:reem

oves salt nooise


61

4.3 Image Smoothingg g4.3.3 Order- statistic filters: Midpoint filter

1( , ) [ { ( , )} { ( , )}]max ming x y f x i y j f x i y j= + + + + +formulate( , 0, 1 )( , 0, 1 )

( , ) [ { ( , )} { ( , )}]2 max min

i ji jg y f y j f y j

= ±= ±o u e A

dvanttage:reemoves G

ausiaan noise


62

e

4.3 Image Smoothingg g4.3.3 Order- statistic filters: Median filter1-D: replace the signal value by the median value of neighborhood

L=5 X=[4 5 9 5 4]

Adv

[4 4 5 5 9]

vantage

[4 5 5 5 4]

e:keep edges


63

4.3 Image Smoothingg g4.3.3 Order- statistic filters: Median filter

Replaces the value of a pixel by the median of the

( , ) { ( , )}g x y f x i y jmedian= + +gray levels in the neighborhood of that pixel

( , 0, 1 )i j= ±

Shapes of 2-D filter


64

4.3 Image Smoothingg g4.3.3 Order- statistic filters: Median filter

Experiment resultExperiment result


65

4.3 Image Smoothingg g4.3.3 Order- statistic filters: alpha-trimmed mean filter

Delete the lowest and highest gray-level valuesα αDelete the lowest and highest gray level values of a sub-image, averaging the remaining pixels as output

α α

0 1 1NA A A −≤ ≤Let the pixels in a sub-image:

1( )N

g x y Aα−

= ∑Then:

( , )2 i

ig x y A

N αα =

=− ∑


66

4.3 Image Smoothingg g4.3.4 Low-pass filters: Idea low-pass filter (ILPF)

formulate ( , ) ( , )* ( , )g x y h x y f x y=

D0: cutoff frequency( , ) ( , ) ( , )G u v H u v F u v=

01 ( , )( )

if D u v DH u v

≤⎧= ⎨where

0

( , )0 ( , )

H u vif D u v D

= ⎨ >⎩

2/122 ])2/()2/[(),( NvMuvuD −+−=


67

4.3 Image Smoothingg g4.3.4 Low-pass filters: Idea low-pass filter(ILPF)


68


Properties: total image powerope es: o ge powe

2003 Digital Image Processing Prof.Zhengkai Liu Dr.Rong Zhang

69


Properties: blurring and ringing

h(x,y)


70

4.3 Image Smoothingg g4.3.4 Low-pass filters: Idea low-pass filter (ILPF)Experiment result cutoff frequencies set at radii values of 5、30、80Experiment result cutoff frequencies set at radii values of 5、30、80


71

4.3 Image Smoothingg g4.3.4 Low-pass filters: Butterworth low-pass filter

12

0

1( , )1 [ ( , ) / ] nH u v

D u v D=

+formulate

h2/122 ])2/()2/[(),( NvMuvuD −+−=

where

])()[()(


72


H( ) 0 5 h D( ) DH(u,v)=0.5 when D(u,v)=D0


73


Properties: blurring and ringing


74

4.3 Image Smoothingg g4.3.4 Low-pass filters: Butterworth low-pass filterExperiment result cutoff frequencies set at radii values of 5、30、80Experiment result cutoff frequencies set at radii values of 5、30、80


75

4.3 Image Smoothingg g4.3.4 Low-pass filters: Gaussian low-pass filter

2 20( , ) / 2( ) D u v DH u v e−=

formulate

( , )H u v e=

h2/122 ])2/()2/[(),( NvMuvuD −+−=

where

])()[()(


76


Inverse Fourier transform of the Gaussian lowpass filter l i G ialso is Gaussian


77


Applications: machine perceptionApplications: machine perception


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4.3 Image Smoothing4 3 4 L filt G i l filt4.3.4 Low-pass filters: Gaussian low-pass filter

Applications: printing and publishing industry


79

4.3 Image Smoothing

Experiment result cutoff frequencies set at radii values of 5、30、80

g g4.3.4 Low-pass filters: Gaussian low-pass filterExperiment result cutoff frequencies set at radii values of 5、30、80


80

4.3 Image Smoothingg g4.3.4 Low-pass filters: Comparisons (cutoff 5)

Ideal filter Butterworth filter Gaussian filterOriginDigital Image Processing


Ideal filter Butterworth filter Gaussian filterOrigin

4.3 Image Smoothingg g4.3.4 Low-pass filters: Comparison (cutoff 30)

Ideal filter Butterworth filter Gaussian filterOrigin Digital Image Processing


g

4.3 Image Smoothingg g4.3.4 Low-pass filters: Comparison (cutoff 80)

Ideal filter Butterworth filter Gaussian filterOriginDigital Image Processing


Origin

The End


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Documents

Chapter4 Image Enhancement - USTChome.ustc.edu.cn/~xuedixiu/image.ustc/course/dip/DIP14-ch4-1.pdf · • 4.4 Enhancement by Frequency Filtering (image sharpening) • 4 5 Color Enhancement4.5