4.4 Imaggpge Sharpening - home.ustc.edu.cnhome.ustc.edu.cn/~xuedixiu/image.ustc/course/dip/DIP14...4.4 Imaggpge Sharpening 4.4.3 High-pass filter: Homomorphic filter An image f(x y)

4.4 Image Sharpeningg p g

• Introduction • First-order derivative (in spatial domain)• Second-order derivative (in spatial domain)• High-pass filter in frequency domain

Digital Image Processing Prof.Zhengkai Liu Dr.Rong Zhang

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4.4 Image Sharpening4.4.1 Introduction

g p g

P t d t t dPurposes:highlight fine detaileasy to detect edges

Request: insensitive to noise


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4.4 Image Sharpening4.4.1 Introduction: Sharpening filters

g p g

R b t(1) Fi t d d i ti Robert(1) First-order derivative :Prewitt

KirschRobinsonSobel Kirsch

(2) Second-order derivative : Laplacian

Robinson

LoG

(3) High-pass filter in frequency domain :Id hi h filt (IHPF)

LoG

Idea high-pass filter (IHPF)Butterworth high-pass filter (BHPF)Gaussian high-pass filter (GHPF)Gaussian high-pass filter (GHPF)


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4.4 Image Sharpeningg p g4.4.2 first-order derivative : foundation

The derivatives of a digital function are defined in terms of differences e de v ves o d g u c o e de ed e s o d e e ces

(1) zero in flat segmentFor first derivative it must be

(1) zero in flat segment(2) nonzero at the onset of a gray-level step or ramp(3) nonzero along ramp

For second derivative it must be

(3) nonzero along ramp

(1) zero in flat segment(2) nonzero at the onset and end of a gray-level step or ramp(2) nonzero at the onset and end of a gray-level step or ramp(3) zero along ramp of constant slope


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4.4 Image Sharpeningg p g4.4.2 first-order derivative : foundationA basic definition of the first-order derivative of a 1-D function f(x) is the difference

( 1) ( )f f x f xx∂

= + −∂

Similarly, we define a second-order derivative as the difference2

[ ( 1) ( )] [ ( ) ( 1)]f f f f f∂2 [ ( 1) ( )] [ ( ) ( 1)]

( 1) ( 1) 2 ( )

f f x f x f x f xx

f x f x f x

= + − − − −∂

= + + − −


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4.4 Image Sharpeningg p g4.4.2 first-order derivative : foundationFirst-order derivatives of a digital image are based on various

Tf f⎡ ⎤∂ ∂

g gapproximations of the 2-D gradient. It is defined as the vector:

f ffx y

⎡ ⎤∂ ∂∇ = ⎢ ⎥∂ ∂⎣ ⎦

11222f ff

x y

⎡ ⎤⎛ ⎞∂ ∂⎛ ⎞∇ = +⎢ ⎥⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠⎢ ⎥⎝ ⎠⎣ ⎦

f ffx y

⎛ ⎞∂ ∂⎛ ⎞∇ ≈ + ⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠Magnitude:

f f⎧ ⎫∂ ∂⎪Direction: 1( , ) tan yGx yα − ⎛ ⎞

= ⎜ ⎟max ,f ff

x y⎧ ⎫∂ ∂⎪∇ ≈ ⎨ ⎬∂ ∂⎪ ⎭⎩

( , )x

yG⎜ ⎟

⎝ ⎠


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h j( 1, ) ( , )x

fG f x y f x yx∂

= = + −∂

where y, j

( , 1) ( , )yfG f x y f x yy∂

= = + −∂ x, i

These equations can be implemented using masks

,

1 01 0xG−

=1 1

0 0yG−

=1 0 0 0


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Spatial filter: the sign of filter coefficientsSpatial filter: the sign of filter coefficients

F(u) f(x)

Positiv and

u0

x

0

andnegative

(c) (d)


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4.4 Image Sharpeningg p g4.4.2 first-order derivative : display

( , )f x y ( , )g x y f= ∇

0( )g x y f∇(1) ( , )g x y f= ∇(1)( , ) ( , )g x y f x y f= + ∇(2)

( , )g x yLinear scaling (3)<TbL f⎧ ∇

⎨ T

( , ) othewise

b

t

L fg x y

L⎧ ∇

= ⎨⎩

(4)


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4.4 Image Sharpeningg p g4.4.2 first-order derivative : display


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4.4 Image Sharpeningg p g4.4.2 first-order derivative : gradient operators

Roberts operator 1 0 0 1obe s ope o 1 00 1xG =

−0 11 0yG =−

Prewitt operator1 0 11 0 1G

−= −

1 1 10 0 0G 1 0 1

1 0 1yG

−0 0 01 1 1

xG =− − −

Sobel operator1 0 12 0 2G

−= −

1 2 10 0 0G = 2 0 2

1 0 1yG =

−0 0 01 2 1

xG =− − −


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Roberts operatorobe s ope o


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Prewitt operatorew ope o


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Sobel operatorSobe ope o


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Sobel operatorSobe ope o

i i l S b loriginal Sobel Binary display


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Robinson

0

1 0 12 0 2G

−= − 1

0 1 21 0 1G

− −= − 2

1 2 10 0 0G− − −

= 3

2 1 01 0 1G

− −= −

0 2 0 21 0 1

G−

1 1 0 12 1 0

G 2 0 0 01 2 1

G 3

0 1 2

6

1 2 10 0 0G =

1 0 12 0 2G−

= − 5

0 1 21 0 1G = − 7

2 1 01 0 1G = −6

1 2 1− − −4 2 0 2

1 0 1G = −

−5

2 1 0− −7

0 1 2− −

GG2G3

G0

G1G3

G4}{

0,1...7max ii

f G=

∇ ≈


18G5

G6

G7


Kirsch

0

5 3 35 0 3G

− −= − 1

3 3 35 0 3G− − −

= − 2

3 3 33 0 3G− − −

= − − 3

3 3 33 0 5G− − −

= −0 5 0 3

5 3 3G

− −1 5 0 3

5 5 3G

−2 3 0 3

5 5 5G 3

3 5 5−

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5 5 53 0 3G = − −

3 3 53 0 5G− −

= − 5

3 5 53 0 5G−

= − 7

5 5 35 0 3G

−= −6

3 3 3− − −4 3 0 5

3 3 5G = −

− −5

3 3 3− − −7

3 3 3− − −

GG2G3

G0

G1G3

G4}{

0,1...7max ii

f G=

∇ ≈


19G5

G6

G7

4.4 Image Sharpeningg p g4.4.3 second-order derivative : definitions

Th d d d i i i d fi d2 2

2 f ff ∂ ∂∇ = +

The second-order derivative is defined as

2 2fx y

∇ = +∂ ∂

where 2

( 1 ) ( 1 ) 2 ( )f f x y f x y f x y∂+ +w e e

2 ( 1, ) ( 1, ) 2 ( , )f f x y f x y f x yx

= + + − −∂2

( 1) ( 1) 2 ( )f f x y f x y f x y∂= + +

2 [ ( 1 ) ( 1 )f f x y f x y∇ = + + +namely

2 ( , 1) ( , 1) 2 ( , )f x y f x y f x yy

= + + − −∂

[ ( 1, ) ( 1, )( , 1) ( , 1) 4 ( , )]

f f x y f x yf x y f x y f x y∇ = + + − +

+ + − −

namely


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4.4 Image Sharpeningg p g4.4.3 second-order derivative : Laplacian

It is well known as Laplacian operator. It can be implementedp p pas a mask

0 1 01 4 1G

−= − −

0 1 0−


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4.4 Image Sharpeningg p g4.4.3 second-order derivative : Laplacian

2Experiment result 2( , ) ( , ) ( , )g x y f x y f x y= + ∇


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4.4 Image Sharpeningg p g4.4.3 second-order derivative : LoG

S hi fiSmoothing first 2

22( )r

h r e σ−

= −

2 2r x y= +

Laplacian sharpeningp p g2

22 2

2 24( )

rrh r e σσσ

−⎡ ⎤−∇ = − ⎢ ⎥

⎣ ⎦⎣ ⎦


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original Sobel LoG

Thresholded LoG

Zero cross


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4.4 Image Sharpeningg p g4.4.3 High-pass filter: Ideal highpass filter (IHPF)

formula

00 ( , )( , )

1 ( )if D u v D

H u vif D D

≤⎧= ⎨⎩

formula

0

( , )1 ( , )if D u v D⎨ >⎩


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4.4 Image Sharpening

cutoff frequencies set at radii values of 15、30、80

g p g4.4.3 High-pass filter: Ideal highpass filter (IHPF)


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4.4 Image Sharpeningg p g4.4.3 High-pass filter: Ideal highpass filter (IHPF)cutoff frequencies set at radii values of 15、30、80


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4.4 Image Sharpeningg p g4.4.3 High-pass filter: Butterworth highpass filter (BHPF)

1f l2

0

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

D D u v=

+formula


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4.4 Image Sharpeningg p g4.4.3 High-pass filter: Butterworth highpass filter (BHPF)cutoff frequencies set at radii values of 15、30、80


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4.4 Image Sharpening

cutoff frequencies set at radii values of 15、30、80.

g p g4.4.3 High-pass filter: Butterworth highpass filter (BHPF)


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4.4 Image Sharpeningg p g4.4.3 High-pass filter: Gaussian highpass filter (GHPF)

f l 2 20( , ) / 2( , ) 1 D u v DH u v e−= −

formula


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4.4 Image Sharpeningg p g4.4.3 High-pass filter: Gaussian highpass filter (GHPF)cutoff frequencies set at radii values of 15、30、80.


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4.4 Image Sharpeningg p g4.4.3 High-pass filter: Homomorphic filter

An image f(x y) can be expressed as the product ofAn image f(x,y) can be expressed as the product of illumination and reflectance components

( , ) ( , ) ( , )f x y i x y r x y=

let

( , ) ln ( , ) ln ( , ) ln ( , )z x y f x y i x y r x y= = +


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Experimental resultp


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4.5 Color Image Enhancementg

I t d ti• Pseudo color image processing

ll l h

• Introduction

• Full color enhancement

• Noise in color image


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4.5 Color Image Enhancementg4.5.1 Introduction: review

1666, Sir Isaac Newton, optical prism

Digital Image Processing Prof. Zhengkai Liu Dr. Rong Zhang

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Primary colors of lightRotate slowly

Color wheel


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y


[ ] [ ] [ ]C X R Y G Z B= + + Hue 色调[ ] [ ] [ ]C X R Y G Z B+ + Hue 色调

Saturation饱和度Eg:Red 1[R]+0[G]+0[B]

Intensity辉度

Chroma 色度

Red=1[R]+0[G]+0[B]Yellow=1[R]+1[G]+0[B]

Cyan=0[R]+1[G]+1[B] Chroma 色度Cyan 0[R] 1[G] 1[B]Magenta=1[R]+0[G]+1[B]

Black=0[R]+0[G]+0[B]White=1[R]+1[G]+1[B]


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If the tristimulus values are denoted, X,Y and Z, then h i h i ffi i d fi d

(2.3.1)Xx =

the trichromatic coefficients are defined as:

(2 3 2)

xX Y Z+ +

Yy (2.3.2)yX Y Z

=+ +

Z(2.3.3)(2 3 4)

ZzX Y Z

=+ +

1x y z+ + =Digital Image Processing

Prof. Zhengkai Liu Dr. Rong Zhang42

(2.3.4)1x y z+ + =

4.5 Color Image Enhancement

Color Fundament: CIE chromaticity diagram

g4.5.1 Introduction: review

Color Fundament: CIE chromaticity diagram


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Color Model: RGB model




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Color Model: safe colors of RGB model



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Color Model: CMY model



⎡ ⎤ ⎡ ⎤ ⎡ ⎤11

C RM G⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥= −⎢ ⎥ ⎢ ⎥ ⎢ ⎥

1Y B⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦


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Color Model: HSI model



• Hue: associated with the dominant wavelength in a mixture of light waves

• Saturation: refer to the relative purity or the amount of white light mixed with hue.

• Intensity: brightness


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Color H S IColor H S IRed 0 1 0.5

Green 120 1 0.5G ee 0 0.5Blue 240 1 0.5

White 0 0 1Black 0 0 0


53


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R G BR G B

H S IDigital Image Processing


H S I


The answer is:


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Color Model: converting colors from RGB to HSI


Color Model: converting colors from RGB to HSI1 ( )I R G B= + +

(2.3.5)( )

331 [ i ( )]S R G B31 [min( , , )]

( )S R G B

R G B= −

+ +

[( ) ( )] / 2R G R B⎧ ⎫+

(2.3.6)

2 1/ 2

[( ) ( )] / 2arccos[( ) ( )( )]

R G R BR G R B G B

θ⎧ ⎫− + −

= ⎨ ⎬− + − −⎩ ⎭ (2 3 7)

360H

θθ

⎧= ⎨ −⎩

(2.3.7)

others

B G≤


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⎩


Color Model: converting colors from HSI to RGB



RG Sector: 0 120o oH≤ <

(2 3 8)(2.3.8)(1 )B I S= −cosS H⎡ ⎤

(2.3.9)cos1

cos(60 )S HR I

H⎡ ⎤

= +⎢ ⎥−⎣ ⎦。

(2.3.10)

cos(60 )H⎣ ⎦3 ( )G I B R= − +


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( )( )





GB Sector: 120 240o oH≤ <

(2.3.11)(1 )R I S= − ( )(1 )R I S=

( )120S H⎡ ⎤。 (2.3.12)( )cos 1201

cos(180 )S H

G IH

⎡ ⎤−⎢ ⎥= +

−⎢ ⎥⎣ ⎦

。

。

(2.3.13)⎢ ⎥⎣ ⎦

3 ( )B I R G= − +Digital Image Processing






BR Sector: 240 360o oH≤ <

(2.3.14)(1 )G I S= −

(2 3 15)cos( 240 )1 S HB I⎡ ⎤−+⎢ ⎥

。

(2.3.15)1cos(300 )

B IH

= +⎢ ⎥−⎣ ⎦。

(2.3.16)3 ( )R I G B= − +


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4.5 Color Image Enhancementg4.5.2 Pseudo color image processing : Intensity Slicingformula

( , ) if ( , )k kf x y c f x y V= ∈


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4.5 Color Image Enhancementg4.5.2 Pseudo color image processing : Intensity Slicingexample


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4.5 Color Image Enhancementg4.5.2 Pseudo color image processing : Intensity Slicingexample


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4.5 Color Image Enhancementg4.5.2 Pseudo color image processing : Gray level to color

In general we perform three independent transformationsIn general, we perform three independent transformations on the gray level of any input pixel

1


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4.5 Color Image Enhancementg4.5.2 Pseudo color image processing : Gray level to colorFor example

t ftransforms


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Combine several monochrome images into a single color compositeCombine several monochrome images into a single color composite


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band1 band2 band3 band1 2 3 false colorba d band2 band3 band1,2,3 false color

band5 band4 band3 band5,4,3 false colorDigital Image Processing

Prof.Zhengkai Liu Dr.Rong Zhang70

, ,



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4.5 Color Image Enhancementg4.5.3 Full color enhancement : Fundamentals

A full color image can be expressed asA full color image can be expressed as

( , ) ( , )Rc x y R x y⎡ ⎤ ⎡ ⎤( , ) ( , )( , ) ( , ) ( , )

( , ) ( , )

R

G

B

c x y R x yc x y c x y G x y

c x y B x y

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥= =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

A pixel in color image interpreted as vector

R

G

B

c Rc c G

c B

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥= =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦B⎣ ⎦ ⎣ ⎦


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4.5 Color Image Enhancementg4.5.3 Full color enhancement : Fundamentals

Mask operation in color imageMask operation in color image


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4.5 Color Image Enhancementg4.5.3 Full color enhancement : Color transformations

Adjusting the intensity of an image:j g y g

in RGB color space three components must be transformed:

1, 2,3 0 1i is kr i k= = < <

in HSI color space only intensity components is modified:s kr3 3 s krs r==1 1

2 2

s rs r==


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2 2

4.5 Color Image Enhancementg4.5.3 Full color enhancement : Color transformations

I HS

S HI


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4.5 Color Image Enhancementg4.5.3 Full color enhancement : Color Complement

in RGB color space three components must be transformed:in RGB color space three components must be transformed:

1 1 2 3 0 1s r i k= − = < <1 1, 2,3 0 1i is r i k= − = < <in HSI color space only intensity components is modified:

1 1( +0.5) s r=

2 2s r=

3 31s r= −


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4.5 Color Image Enhancementg4.5.3 Full color enhancement : Color Complement


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4.5 Color Image Enhancementg4.5.3 Full color enhancement : color corrections

In RGB and CMY(K) space, map all tree (or four) ( ) p , p ( )color components with the same transform function


78


examplep


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example


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example


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4.5 Color Image Enhancementg4.5.3 Full color enhancement : Histogram processingIn the HSI color space, only the intensity component is modified


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4.5 Color Image Enhancementg4.5.3 Full color enhancement : Histogram processing

originalI Histogram RGB Histogram


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original EqualizationRGB HistogramEqualization

4.5 Color Image Enhancementg4.5.3 Full color enhancement : Smoothing and Sharping


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4.5 Color Image Enhancementg4.5. 6 Noise in color image :Noise in RGB channels


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4.5 Color Image Enhancementg4.5. 6 Noise in color image :Noise in HSI channels


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Questions?4.20 在一条自动装配线上，有三类形状相同的工件，为了检测

方便，将工件用不同颜色标注，现只有1个单色摄象机，请

Questions?

方便，将工件用不同颜色标注，现只有1个单色摄象机，请提出一种用这个摄象机检测3种颜色的方法。

4.21设有一个能输出RGB模拟信号的彩色摄象机，一个能将这些模拟信号转化为以（1/30）s的视频速度输出RGB或HIS图象的数字化器 3块能以视频速度接受图象的帧缓存卡以及个的数字化器，3块能以视频速度接受图象的帧缓存卡，以及一个能以视频速度计算直方图的硬件。所有这些都可以与1台微机组合在一起，现要解决以下问题：生产线上有一系列形状相同但合在起，现要解决以下问题：生产线上有系列形状相同但颜色不同的工件，他们按红、黄、绿、蓝的次序排列。请借助以上硬件设计一个图象处理软件系统将工件的的颜色检测出来。这里假设工件移动速度相当慢，所以可忽略由此产生的图象模糊问题，请画出系统的流程图并对每个模块及所选处理技术进行讨论


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行讨论。

Homework：

Page 99 (章): 4.2 为什么一般情况下对离散图象的直方图均衡并不能产生完全为什么般情况下对离散图象的方图均衡并不能产完全

平坦的直方图？

设已用直方图均衡化技术对幅图象进行了增强试证明再4.3 设已用直方图均衡化技术对一幅图象进行了增强，试证明再用这个方法对所得结果增强并不会改变其结果

4.13讨论用于空间滤波的平滑滤波器和锐化滤波器的相同点、不同点以及联系。同点以及联系

4.17有一种常用的图象增强技术是将高频增强和直方图均衡化结合起来以达到使边缘锐化的反差增强效果以上两个操作的先合起来以达到使边缘锐化的反差增强效果，以上两个操作的先后次序对增强结果有影响吗？为什么？


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The End


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Documents

4.4 Imaggpge Sharpening - home.ustc.edu.cnhome.ustc.edu.cn/~xuedixiu/image.ustc/course/dip/DIP14...4.4 Imaggpge Sharpening 4.4.3 High-pass filter: Homomorphic filter An image f(x y)