Chapter3 Image Transforms - USTChome.ustc.edu.cn/~xuedixiu/image.ustc/course/dip/DIP14...Chapter3...

Chapter3 Image Transforms

• Preview3 1G l I d i d Cl ifi i• 3.1General Introduction and Classification

• 3.2 The Fourier Transform and Propertiesh bl f• 3.3 Other Separable Image Transforms

• 3.4 Hotelling Transform

Digital Image Processing Prof.zhengkai Liu Dr.Rong Zhang

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PreviewGeneral steps of operation in frequency domain

DFT IDFT),( vuH

General steps of operation in frequency domain

Pre- Post-

),( vuF ),(),( vuHvuF

Preprocessing

Postprocessing

)(f )( yxg),( yxf ),( yxg


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Preview

Lowpass filterLowpass filter And highpass filter


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PreviewDirectly display DFT coefficientsDirectly display DFT coefficients


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PreviewDisplay DFT coefficients after log operateDisplay DFT coefficients after log operate


5


Preview Rotational properties of DFT


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Preview Examples of DFT


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Chapter3 Image TransformsChapter3 Image TransformsPreview Blurred image and its DFTPreview Blurred image and its DFT


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Preview Examples of DCT


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3.1 General Introduction and Classification3.1.1 introduction

• image transforms are the bases of image processing g g p gand analysis

• this chapter deals with two-dimensional transforms and their propertiesp p

• image transforms are used in image enhancement, g g ,restoration, reconstruction, encoding and description


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3.1 General Introduction and Classification3.1.1 classification

DFT and its properties⎧ ⎧ DFT and its propertiesDCT

Hadamard Transform

⎧ ⎧⎪ ⎪

⎧⎪ ⎪⎪⎪ ⎪⎪ ⎪Hadamard Transform

separable transformsimage transforms others Walsh Transform

Slant Transform

⎪ ⎪⎪ ⎪⎪ ⎨ ⎪⎨ ⎨⎪

⎪⎪⎪

⎪

⎪⎪ Slant Transform

Haar Transform

statistical transforms: Hotelling transform

⎪⎪⎪⎪⎪ ⎩⎩

⎪⎪⎪⎪⎩ statistical transforms: Hotelling transform⎪⎩


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3.2 Fourier Transform and Properties3.2.1 introduction

Optical PrismOp c s


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3.2 Fourier Transform and Properties3.2.1 introduction

DFT

Mathematic Prisme c s


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3.2 Fourier Transform and Properties3.2.2 definitions: 1-D CFT

The One-Dimensional Continuous Fourier Transform and its Inverse

2( ) ( ) j uxF u f x e dxπ+∞ −= ∫( ) ( )f−∞∫

2( ) ( ) j xuf x F u e duπ+∞

∞= ∫−∞∫


16

3.2 Fourier Transform and Properties3.2.2 definitions:1-D DFT

The One-Dimensional Discrete Fourier Transform and its Inverse

21N j uxπ− −

∑ 0 1 1u N=0

( ) ( )j ux

N

x

F u f x e=

=∑ 0,1, 1u N= −L

21

0

1( ) ( )N j ux

Nf x F u eN

π−

= ∑ 0,1, 1x N= −L0uN =


17

Basis function DFT N=64Digital Image Processing

Prof.zhengkai Liu Dr.Rong Zhang18

Basis function DFT N=64

3.2 Fourier Transform and Properties3.2.2 definitions (2) :1-D DFT

Other expressions:

)()()( ujIuRuF +=

[ ])()()( jFF φor

[ ])(exp)()( ujuFuF φ=

E l ’ f lEuler’s formula:

[ ] uxjuxuxj πππ 2sin2cos2exp −=−Digital Image Processing


[ ]

3.2 Fourier Transform and Properties3.2.2 definitions:spectrum, Phase angle Power spectrum

Magnitude or spectrum [ ] 21

)()()( 22 uIuRuF +=Magnitude or spectrum

Ph l h t

[ ])()()( uIuRuF +=

[ ])()(arctan)( uRuIu =φPhase angle or phase spectrum [ ])()(arctan)( uRuIu =φ

)()()()( 222Power spectrum（Spectral density)

)()()()( 222 uIuRuFuP +==


20

3.2 Fourier Transform and Properties3.2.2 definitions:1-D DFT example

If a signal is expressed as f(x)={2,3,4,4}, its DFT are:3

0(0) ( ) exp(0) (0) (1) (2) (3) 13

xF f x f f f f

=

= = + + + =∑3

0 / 2 3 / 2

0(1) ( ) exp( 2 / 4) 2 3 4 4 2j j j

xF f x j x e e e e jπ π ππ − − −

=

= − = + + + = − +∑33

0 2 3

0(2) ( ) exp( 4 / 4) 2 3 4 4 1j j j

xF f x j x e e e eπ π ππ − − −

=

= − = + + + = −∑3

/ /∑ 0 3 / 2 3 9 / 2

0(3) ( ) exp( 6 / 4) 2 3 4 4 2j j j

xF f x j x e e e e jπ π ππ − − −

=

= − = + + + = − −∑


21


And the Fourier spectra are :p

(0) 0φ =| (0) | 13F = ( )φ

(1) 0.85φ π=2 2 1/ 2| (1) | [( 2) 1 ] 5F = − + =

(3) 0 85φ π= −(3)φ π=

2 2 1/ 2| (3) | [( 2) ( 1) ] 5F = − + − =

2 1/ 2| (2) | [( 1) ] 1F = − =

(3) 0.85φ π= −| (3) | [( 2) ( 1) ] 5F = − + − =


22


Graphic illustration :

|F(u)|

p

f(x) jF(u)

u0 1 2 3

Φ(u)

x0 1 2 3

Φ(u)π

-0.85π

0 1 2 3-0.85π0 u


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Fourier Transform and Properties3.2.2 definitions:1-D DFT exampleIf a signal is expressed as f(x)={3,2,3,1,4,5,0,2}, its DFT are:

7

0(0) ( ) exp(0) (0) (1) (2) (3)

xF f x f f f f

=

= = + + +∑(4) (5) (6) (7) 20f f f f+ + + + =

70 / 4 / 2 3 / 4(1) ( ) exp( 2 / 8) 3 2 3 1 4j j j jF f x j x e e e e eπ π π ππ − − − −= − = + + + +∑

0

5 / 4 3 / 2 7 / 4

(1) ( ) exp( 2 / 8) 3 2 3 1 4

5 0 2 2.4142 0.1716x

j j j

F f x j x e e e e e

e e e jπ π π

π=

− − −

= = + + + +

+ + + = − −

∑

70 / 2 3 / 2 2

0(2) ( ) exp( 4 / 8) 3 2 3 1 4j j j j

xF f x j x e e e e eπ π π ππ − − − −

=

= − = + + + +∑


245 / 2 3 7 / 25 0 2 4 4j j je e e jπ π π− − −+ + + = −

70 3 / 4 3 / 2 9 / 4 3

0

(3) ( ) exp( 6 / 8) 3 2 3 1 4j j j jF f x j x e e e e eπ π π ππ − − − −= − = + + + +∑0

15 / 4 9 / 2 21 / 45 0 2 0.4142 5.8284x

j j je e e jπ π π

=

− − −+ + + = +7

0 2 3 4

0

5 6 7

(4) ( ) exp( 8 / 8) 3 2 3 1 4

5 0 2 0

j j j j

x

j j j

F f x j x e e e e e

e e e

π π π π

π π π

π − − − −

=

− − −

= − = + + + +

+ + + =

∑5 0 2 0e e e+ + + =

70 5 / 4 5 / 2 15 / 4 5

0

(5) ( ) exp( 10 / 8) 3 2 3 1 4j j j j

x

F f x j x e e e e eπ π π ππ − − − −

=

= − = + + + +∑0

25 / 4 15 / 2 35 / 45 0 2 0.4142 5.8284x

j j je e e jπ π π

=

− − −+ + + = −7

0 3 / 2 3 9 / 2 6j j j jπ π π π∑ 0 3 / 2 3 9 / 2 6

0

15 / 2 9 21 / 2

(6) ( ) exp( 12 / 8) 3 2 3 1 4

5 0 2 4 4

j j j j

x

j j j

F f x j x e e e e e

e e e j

π π π π

π π π

π − − − −

=

− − −

= − = + + + +

+ + + = +

∑5 0 2 4 4e e e j+ + + +

70 7 / 4 7 / 2 21 / 4 7

0

(7) ( ) exp( 14 / 8) 3 2 3 1 4j j j j

x

F f x j x e e e e eπ π π ππ − − − −

=

= − = + + + +∑


25

035 / 4 21 / 2 49 / 45 0 2 2.4142 0.1716

xj j je e e jπ π π− − −+ + + = − +

3.2 Fourier Transform and Properties3.2 Fourier Transform and Properties3.2.2 definitions:1-D DFT example

And the Fourier spectra are :

| (0) | 20F = (0) 0φ =| (0) | 20F =2 2 1/ 2| (1) | [( 2.4142) ( 0.1716) ] 2.4203F = − + − =

(0) 0φ =(1) 0.9774φ π= −(2) 0 25φ2 2 1/ 2| (2) | [4 ( 4) ] 5.6569F = + − =

2 2 1/ 2| (3) | [0.4142 5.8284 ] 5.8431F = + =

(2) 0.25φ π= −(3) 0.4774φ π=

| (4) | 0F =2 2 1/ 2| (5) | [0.4142 ( 5.8284) ] 5.8431F = + − =

(4)φ π=(5) 0.4774φ π= −| ( ) | [ ( ) ]

2 2 1/ 2| (6) | [4 4 ] 5.6569F = + =

( )φ(6) 0.25φ π=(7) 0 9774φ


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2 2 1/ 2| (7) | [( 2.4142) 0.1716 ] 2.4203F = − + = (7) 0.9774φ π=

3 2 Fourier Transform and Properties3.2 Fourier Transform and Properties3.2.2 definitions:1-D DFT example



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3 2 Fourier Transform and Properties3.2 Fourier Transform and Properties3.2.2 definitions:1-D DFT example



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3.2 Fourier Transform and Properties3.2.2 definitions: 2D-DFT

The Two-Dimensional Discrete Fourier Transform and its Inverse

1 12 ( / / )( , ) ( , )

M Nj ux M vy NF u v f x y e π

− −− += ∑ ∑

0,1, 1u M= −L

0 1 1v N= −L0 0

( , ) ( , )x y

F u v f x y e= =∑ ∑ 0,1, 1v N=

1 12 ( / / )

0 0

1( , ) ( , )M N

j ux M vy N

u vf x y F u v e

MNπ

− −+

= =

= ∑ ∑ 0,1, 1x M= −L

0,1, 1y N= −L


29


Magnitude or spectrum

[ ] 2122 ),(),(),( vuIvuRvuF +=

Magnitude or spectrum

Phase angle or phase spectrum

[ ]),(),(arctan),( vuRvuIvu =φPhase angle or phase spectrum

),(),(),(),( 222 vuIvuRvuFvuP +==

Power spectrum（Spectral density)

),(),(),(),( vuIvuRvuFvuP +


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Magnitude Phaseimage


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Typical images and their spectra


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3.2 Fourier Transform and Properties3.2 Fourier Transform and Properties3.2.2 definitions: 2D-IDFT

IDFT


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IDFT


34


IDFTIDFT


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3.2 Fourier Transform and Properties3.2.3 Properties: Display

•Usually the Fourier spectra are displayed as intensity functionUsually the Fourier spectra are displayed as intensity function.• many image spectra decrease rather rapidly as a function of increasing frequency

•their high-frequency terms have a tendency to become obscured when displayed in image.


36

3.2 Fourier Transform and Properties

• A useful processing technique which compensates for this difficulty consists of displaying the function

3.2.3 Properties: Display

this difficulty consists of displaying the function

1( , ) log(1 ( , ) )D u v F u v= +

Or

2

( , ) 100 ( , )

F u vD u v

⎧ +⎪= ⎨2 ( , )255 if ( , ) 100 255

D u vF u v⎨ + >⎪⎩

•instead of ( , )F u v


37

3.2 Fourier Transform and Properties3.2.3 Properties: Display

( , )F u v

2 ( , )D u v

( )D

2 ( , )

1( , )D u v


38

3.2 Fourier Transform and Properties3.2.3 Properties: separability

The DFT pair can be expressed in the separable forms:

⎤⎡⎤⎡ −− vyjuxj NN ππ 221 11

The DFT pair can be expressed in the separable forms:

⎥⎦⎤

⎢⎣⎡−

⎥⎦⎤

⎢⎣⎡−= ∑∑

== Nvyjyxf

Nuxj

NvuF

N

y

N

x

ππ 2exp),(2exp1),(1

0

1

0

∑∑−

=

−

=⎥⎦⎤

⎢⎣⎡

⎥⎦⎤

⎢⎣⎡=

1

0

1

0

2exp),(2exp1),(N

v

N

u NvyjvuF

Nuxj

Nyxf ππ


39


The principal of the separability property is that f(x,y) or F(u,v)p p p y p p y f( ,y) ( , )can be obtained in two steps by successive applications of the 1-DFourier transform or its inverse

Y

(N-1)

V

(N-1)

V

(N-1)

f(x,y)

X

( )(0 0)

列变换

乘以NF(x,v)

X

( )(0 0)

行变换F(u,v)

U

( )(0 0)(N-1)(0,0) (N-1)(0,0) (N-1)(0,0)

图3.2.2 由2步1-D变换计算2-D变换


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For a N*N image, it can be separated into 2N 1D-DFT


41

3.2 Fourier Transform and Properties3.2.3 Properties: periodicity and conjugate symmetryThe discrete Fourier transform and its inverse are periodic with

),(),(),(),( NvNuFNvuFvNuFvuF ++=+=+=Period N

The Fourier transform also exhibits conjugate symmetry since

( ) ( )F u v F u v∗

Or more interestingly,

( , ) ( , )F u v F u v= − −

( , ) ( , )F u v F u v= − −


42

|F(u)|1D periodicity and conjugate symmetry

|F(u)|

One period-N/2 N/2 N-1,N0

|F(u)|

-N/2 N/2 N 10Digital Image Processing


One periodN/2 N/2 N-10

2D periodicity and conjugate symmetry

(0 0) (0,N-1)

One period

(0,0) (0,N 1)

One period

(N-1,0)( )

(2N-1,2N-1)


44

3.2 Fourier Transform and Properties3.2.3 Properties: periodicity and conjugate symmetry

Lowfrequencies

High frequencies


45

3.2 Fourier Transform and Properties3.2.3 Properties: translation

The translation properties of the Fourier transform pair are given byp p p g y

[ ] ),()(2exp),( 0000 vvuuFNyvxujyxf −−⇔+π[ ] 0000

and

[ ]NvyuxjvuFyyxxf )(2exp),(),( 0000 +−⇔−− π


46


A hift i f( ) d t ff t th it d f it F iA shift in f(x,y) does not affect the magnitude of its Fourier transform since

[ ]0 0| ( , ) exp 2 ( ) | | ( , ) |F u v j ux vy N F u vπ− + =[ ]0 0| ( ) p ( ) | | ( ) |j y


47


Example: in the case u0=v0=N/2, it follows thatExample: in the case u0 v0 N/2, it follows that

and

( )( 1) ( / 2 / 2)x yf +( , )( 1) ( / 2, / 2)x yf x y F u N v N+− ⇔ − −


48


Example of One-Dimensional DFT shiftf(x)

F( )F(u)shift

F(u)No shift


49


Example of Two-Dimensional DFT shift

F( )F(u,v)No shift

F(u,v) shifted

(0,0) (0,N-1)

(N-1,0)


50

(2N-1,2N-1)


Example of Two-Dimensional DFT shift

F(u)F(u)No shift

F(u,v)shifted

f(x,y)


51

shifted

3.2 Fourier Transform and Properties3.2.3 Properties: rotation

If we introduce the polar coordinates

cos( )x r θ= sin( )y r θ= cos( )u ω φ= sin( )v ω φ=

Th f( ) d F( )b d i l( )f θ ( )F φThen f(x,y) and F(u,v)become and respectively( , )f r θ ( , )F ω φ

),(),( 00 θφθθ +⇔+ wFrf


52

3.2 Fourier Transform and Properties3.2.3 Properties: rotation


53

3.2 Fourier Transform and Properties3.2.3 Properties: Distributivity

It follows directly from the definition of the transform pair that,

{ } { } { }),(),(),(),( 2121 yxfFyxfFyxfyxfF +=+

y p

And, in general that,

{ } { } { }),(),(),(),( 2121 yxfyxfFyxfyxfF ⋅≠⋅


54

3.2 Fourier Transform and Properties3.2.3 Properties: scaling

It is also easy to show that for two scalar a and b

)()( vuaFyxaf ⇔

It is also easy to show that for two scalar a and b

),(),( vuaFyxaf ⇔and

⎟⎠⎞

⎜⎝⎛⇔

bv

auF

abbyaxf ,1),(


55

3.2 Fourier Transform and Properties3.2.3 Properties: average value

A widely-used definition of the average value of a 2DA widely used definition of the average value of a 2DDiscrete function is given by the expression

∑∑− −1 1~ 1 N N

∑∑= =

=0 0

2 ),(1),(x y

yxfN

yxf

b i i f i d fi i i f i ldSubstitution of u-v-0 in definition of 2D DFT yields

− −1 11 N N 1~

∑∑= =

=0 0

),(1)0,0(x y

yxfN

F )0,0(1),( FN

yxf =


56

3.2 Fourier Transform and Properties3.2.3 Properties: convolution

The convolution of two functions f(x) and g(x), denoted by

∫∞

f(x)*g(x), is defined by the integral:

dzzxgzfxgxf ∫∞−

−=∗ )()()()(

Where z is a dummy variable of integration


57

3.2 Fourier Transform and Properties3.2 Fourier Transform and Properties3.2.3 Properties: convolutionExample1: graphic illustration of convolution f(x)*g(x)

( ) ( )

Example1: graphic illustration of convolution f(x)*g(x)

f(z)

z

1

g(z)

z1/2

g(-z) g(x-z)

z z1/2 1/2

z

0 1(a)

z

0 1(b) (c) (d)

z z

0-1 -1 x0

( ) ( ) ( ) ( ) ( ) ( )f(z)g(x-z)

z

1/2110 ≤≤ x 121 ≤≤ x

1/2z

f(z)g(x-z)

1/2x

f(x)*g(x)

z

0 x 1-1 0 1x-1 x

z

(f)(e)

0 1 2

x

(g)


58

图3.2.3 1-D函数卷积示例

3.2 Fourier Transform and Properties3.2 Fourier Transform and Properties3.2.3 Properties: convolution

Example2: graphic illustration of convolution f(x,y)*g(x,y)

x

ζζ ζ ζM-1A-1

N-1

g(ζ,η)

f(ζ,η)

g(-ζ,-η)

g(x-ζ, y-η)B-1

ηηf(ζ,η)

η ηy


59

3.2 Fourier Transform and Properties3.2.3 Properties: convolutionExample2: graphic illustration of convolution f(x,y)*g(x,y)p g p f( ,y) g( ,y)

ζ ζ ζ xA+M-1(0,0)

ζ ζ ζ

η η η B+N-1

f(ζ,η) *g(x-ζ, y-η)f(ζ,η) *g(x-ζ, y-η)

y f(x,y)*g(x,y)

f(ζ,η) *g(x-ζ, y-η)


60

3.2 Fourier Transform and Properties3.2.3 Properties: convolution

If f(x y) has the Fourier transform F(u v) and g(x y)has the FourierIf f(x,y) has the Fourier transform F(u,v) and g(x,y)has the FourierTransform G(u,v), then f(x,y)*g(x,y) has the Fourier transformF(u,v)G(u,v).This result, formally stated as:( ) ( ) y

),(),(),(),( vuGvuFyxgyxf ⇔∗

And the convolution in frequency domain reduces to multiplicationIn the spatial-domain

),(),(),(),( vuGvuFyxgyxf ∗⇔


61

3.2 Fourier Transform and Properties3.2.3 Properties: convolutionDefinition of 1D-discrete convolutionDefinition of 1D discrete convolution

⎧ )(xf 10 −≤≤ Ax

⎩⎨⎧

=0

)()(

xfxfe

⎧

10 ≤≤ Ax1−≤≤ MxA

⎩⎨⎧

=0

)()(

xgxge

10 −≤≤ Bx

1−≤≤ MxB

∑−

−=∗1

)()(1)()(M

mxgmfxgxf x=0,1,…,Mx=0,1,…,M--11∑=

=∗0

)()()()(m

eeee mxgmfM

xgxf x 0,1,…,Mx 0,1,…,M 11


62

3.2 Fourier Transform and Properties3.2.3 Properties: convolutionDefinition of 2D-discrete convolutionDefinition of 2D discrete convolution

⎨⎧ ),(

)(yxf

yxf10 −≤≤ Ax andand 10 −≤≤ By

⎩⎨= 0

),( yxfe1−≤≤ MxA oror 1−≤≤ NyB

⎩⎨⎧

=0

),(),(

yxgyxge

andand

oror

0 1x C≤ ≤ −

1−≤≤ MxC

10 −≤≤ Dy

1−≤≤ NyD y

1,...,1,0 −= Mx∑∑− −

−−=∗1 1

),(),(1),(),(M N

nymxgnmfyxgyxf ∑∑= =0 0

),(),(),(),(m n

eeee nymxgnmfMN

yxgyxf1,...,1,0 −= Ny


63

question

DFT ??

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

64

The answer is:

Amplitude Phasep Phase


65

The answer is:

Amplitude Phasep Phase


66

3.3 Other Separable Transforms3.3.1 General formulations

Definition1: if X is an N-by-1 vector and T is an N-by-N matrix then:1

,0

N

i i j jj

y t x−

=

=∑ polynomial expression

0 0,0 0,1 0, 1 0Ny t t t x−⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥

L

Y TX=1 1,0 1,1 1, 1 1Ny t t t x−⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥

L

M O MM

1,0 1,1 1, 1 11 N N N N NNt t t xy − − − − −−

⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥ ⎣ ⎦⎣ ⎦⎣ ⎦ L

matrix expressionDigital Image Processing


matrix expression

3.3 Other Separable Transforms

1X T Y

3.3.1 General formulations

Definition2: inversion 1X T Y−=Definition2: inversion

If k(T) N h i i li f1 tT T− ∗= 1tTT TT I∗ −= =

If rank(T)= N then it is a linear transform

If then it is a Unitary transform

1 tT T− =If then it is a orthogonal transform 1tTT TT I−= =


68


Example: 1-D Discrete Fourier Transform (DFT)Example: 1 D Discrete Fourier Transform (DFT)

21

( ) ( )N j ux

NF u f x eπ− −

=∑ F Tf=0

( ) ( )x

F u f x e=

=∑211( ) ( )

N j uxNf Fπ−

∑

F Tf=

1f T F−

0

( ) ( ) N

u

f x F u eN =

= ∑ 1f T F=

1 tT T− ∗

It is a Unitary transform

T T=


69

3.3 Other Separable Transforms

Definition3: 2 D transformation

3.3.1 General formulations

1 1

( )M N− −

Φ∑∑

Definition3: 2-D transformation

, ,0 0

( , , , )m n i ji j

y x i j m n= =

= Φ∑∑

Can be thought of as a MN-by-MN block matrix haveM rows of M blocks, each of which is an N-by-N

( , , , )i j m nΦ

matrix


70


Definition4: if can be separated into the product ( , , , )i j m nΦ p pof rowwise and columnwise component function, that is

( , , , )j

( , , , ) ( , ) ( , )r ci j m n T i m T j nΦ =

then the transformation is called separablethen the transformation is called separable

[ ]1 1

( , ) ( , )M N

m n i j c ry x T j n T i m− −

= ∑ ∑[ ], ,0 0

( , ) ( , )m n i j c ri j

y x j n i m= =∑ ∑


71


Definition5: if two component are identical:p

( , , , ) ( , ) ( , )i j m n T i m T j nΦ =

then the transformation is called symmetric1 1M N

[ ]1 1

, ,0 0

( , ) ( , )M N

m n i ji j

y x T j n T i m− −

= =

= ∑ ∑oror

Y TXT=


72

3.3 Other Separable Transforms3.3 Other Separable Transforms3.3.1 General formulations

Example: 2-D Discrete Fourier Transform (DFT)

Separable and Symmetric Unitary transform

⎥⎦⎤

⎢⎣⎡−

⎥⎦⎤

⎢⎣⎡−= ∑∑

−

=

−

= Nvyjyxf

Nuxj

NvuF

N

y

N

x

ππ 2exp),(2exp1),(1

0

1

0

lets exp( 2 / )MW j Mπ=

exp( 2 / )NW j Nπ=

then


73

3 3 Oth S bl T f3.3 Other Separable Transforms3.3.2 Discrete Cosine Transform (DCT)3.3. sc e e Cos e s o ( C )

The 1-D DCT pair is given by the expression:1

0

(2 1)( ) ( ) ( ) cos2

N

x

x uC u a u f xN

π−

=

+⎡ ⎤= ⎢ ⎥⎣ ⎦∑ x=0,1,…N-1

1

0

(2 1)( ) ( ) ( ) cos2

N

u

x uf x a u C uN

π−

=

+⎡ ⎤= ⎢ ⎥⎣ ⎦∑

u=0,1,…N-10 2u N= ⎣ ⎦

where

⎧ 1 0( )

2 1,2, , 1

N ua u

N u N

⎧ =⎪= ⎨= −⎪⎩ L

when

when


74

, , ,⎪⎩


Basis function matrix

1/ 2 1/ 2 1/ 2⎡ ⎤⎢

L⎥

x

ucos / 2 cos3 / 2 cos(2 1) / 2cos 2 / 2 cos 6 / 2 cos 2(2 1) / 2

23 / 2 9 / 2 3(2 1) / 2

N N N NN N N N

C N N N N

π π ππ π π

⎢−⎢

⎢ −⎢

= ⎢

L

L

⎥⎥⎥⎥⎥

u

cos3 / 2 cos9 / 2 cos3(2 1) / 2cos 4 / 2 cos12 / 2 cos 4(2 1) / 2

NC N N N NN

N N N Nπ π ππ π π

= −⎢⎢ −⎢⎢

L

L

M M O M

⎥⎥⎥⎥

cos( 1) / 2 cos3( 1) / 2 cos(2 1)( 1) / 2N N N N N N Nπ π π⎢⎢ − − − −⎢⎣ ⎦

M M O M

L

⎥⎥⎥


75


The 1 DCT basis function for N=8Digital Image Processing


The 1-DCT basis function for N=8


DCT 1D N=64Digital Image Processing


DCT_1D N 64


The 2-D DCT pair is given by the expression:

1 1 (2 1) (2 1)( , ) ( ) ( ) ( , ) cos cosN N x u y vC u v a u a v f x y π π− − + +⎡ ⎤ ⎡ ⎤= ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦∑∑

0 0( , ) ( ) ( ) ( , ) cos cos

2 2x yC u v a u a v f x y

N N= =⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

∑∑

u,v=0,1,…N-11 1

0 0

(2 1) (2 1)( , ) ( ) ( ) ( , ) cos cos2 2

N N

u v

x u y vf x y a u a v C u vN N

π π− −

= =

+ +⎡ ⎤ ⎡ ⎤= ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦∑∑

x,y=0,1,…N-1


78

3.3 Other Separable Transforms3.3.2 Discrete Cosine Transform (DCT)

u

v

x

y

The 2D-DCT basis images for N=4

y


79

The 2D DCT basis images for N 4

3.3 Other Separable Transforms3.3.2 Discrete Cosine Transform (DCT)

A l f 2D DCTAn example of 2D-DCT


80

3.3 Other Separable Transforms3.3.2 Discrete Cosine Transform (DCT)Relationship between DFT and DCTp

DCTDCT

DFTDFT


81

3.3 Other Separable Transforms3.3.2 Discrete Cosine Transform (DCT)Relationship between DFT and DCTp

Shifted DFT

DFTDCT


82

3 3 Oth S bl T f3.3 Other Separable Transforms3.3.2 Walsh transform3.3. W s s o

1 1n−

When N=2n, the kernel function is:1( , )g x uN

= 1( ) ( )

0

( 1) i n ib x b u

i

− −

=

−∏

1

11( ) ( )1( ) ( ) ( 1) i i

nNb x b uW f

−−

∑ ∏

the discrete Walsh transform of a function f(x), denote by W(u), is:

1( ) ( )

0 0

( ) ( ) ( 1) i n ib x b u

x i

W u f xN

− −

= =

= −∑ ∏

Wh b ( ) i h k h bi i h bi i fWhere bk(z) is the kth bit in the binary representation of z.Eg: n=3, z=6 (110 in binary), we have that

b (z)=0 b (z)=1 and b (z)=1Digital Image Processing


b0(z) 0, b1 (z) 1, and b2(z) 1

3 3 Oth S bl T f3.3 Other Separable Transforms3.3.2 Walsh transform： 1-D transform3.3.2 Walsh transform： 1 D transformThe values of g(x,u) are list in below

u\x 0 1 2 3 4 5 6 7x

0 + + + + + + + +

1 + + + +

u

1 + + + + - - - -

2 + + - - + + - -

3 + + - - - - + +

4 + - + - + - + -

5 + - + - - + - +

6 + - - + + - - +

7 + - - + - + + -


84

3 3 Oth S bl T f3.3 Other Separable Transforms3.3.2 Walsh transform： 1-D transform3.3.2 Walsh transform： 1 D transform

Inverse kernel and transform:

1

1( ) ( )( , ) ( 1) i n i

nb x b uh x u − −

−

= −∏0i=∏

11 nN −−1( ) ( )

0 0

( ) ( ) ( 1) i n ib x b u

u i

f x W u − −

= =

= −∑ ∏


85

3 3 Oth S bl T f3.3 Other Separable Transforms3.3.2 Walsh transform： 1-D matrix expression N=83.3.2 Walsh transform： 1 D matrix expression N 8

0 0

1 1

1 1 1 1 1 1 1 11 1 1 1 1 1 1 1

w fw f⎡ ⎤ ⎡ ⎤⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥− − − −⎢ ⎥ ⎢ ⎥⎢ ⎥

2 2

3 3

4 4

1 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 1

w fw fw f

⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥− − − −⎢ ⎥ ⎢ ⎥⎢ ⎥− − − −⎢ ⎥ ⎢ ⎥⎢ ⎥=⎢ ⎥ ⎢ ⎥⎢ ⎥− − − −⎢ ⎥ ⎢ ⎥⎢ ⎥

5 5

6 6

77

1 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 1

w fw f

fw

⎢ ⎥ ⎢ ⎥⎢ ⎥− − − −⎢ ⎥ ⎢ ⎥⎢ ⎥

⎢ ⎥ ⎢⎢ ⎥− − − −⎢ ⎥ ⎢⎢ ⎥

− − − −⎢ ⎥⎢ ⎥ ⎢⎣ ⎦ ⎣ ⎦⎣ ⎦

⎥⎥⎥77 f⎢ ⎥⎢ ⎥ ⎢⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎥


86

3 3 Oth S bl T f3.3 Other Separable Transforms3.3.2 Walsh transform： 1-D transform3.3.2 Walsh transform： 1 D transform The values of g(x,u) are list in below for N=8

u\x 0 1 2 3 4 5 6 7 Sequency of the x

0 + + + + + + + +

1 + + + +

01

rowu

1 + + + + - - - -

2 + + - - + + - -

3 + + - - - - + +

132

4 + - + - + - + -

5 + - + - - + - +

76

6 + - - + + - - +

7 + - - + - + + -

45


87

3 3 Oth S bl T f3.3 Other Separable Transforms3.3.2 Walsh transform： 1-D ordered Walsh transform3.3.2 Walsh transform： 1 D ordered Walsh transform The values of g(x,u) are list in below for N=8


0 + + + + + + + +

1 + + + +

01

rowu

1 + + + + - - - -

2 + + - - - - + +

3 + + - - + + - -

123

4 + - - + + - _ +

5 + - - + - + + -

45

6 + - + - - + - +

7 + - + - - + + -

67


88

3 3 Oth S bl T f3.3 Other Separable Transforms3.3.2 Walsh transform： 2-D transform3.3.2 Walsh transform： 2 D transform The direct and inverse kernel functions are expressed as:

1 1

1[ ( ) ( ) ( ) ( )]1( ) ( 1) i n i i n i

nb x b u b y b vg x y u v − − − −

−+= −∏

0

( , , , ) ( 1)i

g x y u vN =

= ∏1 1

1[ ( ) ( ) ( ) ( )]

0

1( , , , ) ( 1) i n i i n i

nb x b u b y b v

i

h x y u vN

− − − −

−+

=

= −∏0i=

And the direct and inverse transforms are:

1 1

11 1[ ( ) ( ) ( ) ( )]

0 0 0

1( , ) ( , ) ( 1) i n i i n i

nN Nb x b u b y b v

x y i

W u v f x yN

− − − −

−− −+

= = =

= −∑∑ ∏11 11 nN N −− −

∑∑ 1 1[ ( ) ( ) ( ) ( )]

0 0 0

1( , ) ( , ) ( 1) i n i i n ib x b u b y b v

u v i

f x y W u vN

− − − −+

= = =

= −∑∑ ∏


89


The direct and inverse kernel functions areSeparable and SymmetricSeparable and Symmetric

1 1 1 1( , , , ) ( , ) ( , ) ( , ) ( , )g x y u v g x u g y v h x u h y v= =

So it can be implemented in two steps


90


u

vv

x

The Walsh transform basis images for N=4y


91

g

3 3 Oth S bl T f3.3 Other Separable Transforms3.3.2 Hadamard transform: 1-D transform 3.3. d d s o : s oWhen N=2n, the kernel function is:

1n−

0

( ) ( )1( , ) ( 1)i i

i

b x b u

g x uN

=∑

= −

1 D Hadamard transform of a function f(x) denote by H(u) is:

Where the summation in the exponent is performed in modulo 2

1-D Hadamard transform of a function f(x), denote by H(u), is:

11 ( ) ( )1

n

i iN b x b u−

− ∑0

( ) ( )

0

1( ) ( )( 1)i i

i

x

H u f xN

=

=

∑= −∑


92

3 3 Oth S bl T f3.3 Other Separable Transforms3.3.2 Hadamard transform: 1-D inverse transform 3.3. d d s o : ve se s o

Inverse kernel and transform:1

0

( ) ( )

( , ) ( 1)

n

i ii

b x b u

h x u

−

=∑

= −

1

0

1 ( ) ( )

( ) ( )( 1)

n

i ii

N b x b u

f x H u

−

=

− ∑= −∑

0

( ) ( )( 1)u

f x H u=

= −∑


93

3 3 Oth S bl T f3.3 Other Separable Transforms3.3.4 hadamard transform： 1-D matrix expression N=8

1 1 1 1 1 1 1 1h f⎡ ⎤ ⎡ ⎤⎡ ⎤

3.3.4 hadamard transform： 1 D matrix expression N 8

0 0

1 1

2 2

1 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 1

h fh fh fh f

⎡ ⎤ ⎡ ⎤⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥− − − −⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥− − − −⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥3 3

4 4

5 5

1 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 1

h fh fh fh

⎢ ⎥ ⎢ ⎥⎢ ⎥− − − −⎢ ⎥ ⎢ ⎥⎢ ⎥=⎢ ⎥ ⎢ ⎥⎢ ⎥− − − −⎢ ⎥ ⎢ ⎥⎢ ⎥

− − − −⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢⎢ ⎥ ⎥

6 6

77

1 1 1 1 1 1 1 11 1 1 1 1 1 1 1

h ffh

⎢ ⎥ ⎢⎢ ⎥− − − −⎢ ⎥ ⎢⎢ ⎥

− − − −⎢ ⎥⎢ ⎥ ⎢⎣ ⎦ ⎣ ⎦⎣ ⎦

⎥⎥⎥


94

3 3 Oth S bl T f3.3 Other Separable Transforms3.3.2 Hadamard transform： 1-D transform3.3.2 Hadamard transform： 1 D transform The values of g(x,u) are list in below for N=8


0 + + + + + + + +

1 + + + +

07

rowu

1 + - + - + - + -

2 + + - - + + - -

3 + - - + + - - +

734

4 + + + + - - _ -

5 + - + - - + - +

16

6 + + - - - - + +

7 + - - + - + + -

25


95

3 3 Oth S bl T f3.3 Other Separable Transforms3.3.2 Hadamard transform： 1-D transform3.3.2 Hadamard transform： 1 D transform

The 1 D Hadamard transform basis function for N=8Digital Image Processing


The 1-D Hadamard transform basis function for N=8


h b h k l i iAnother way for generating kernel matrix

2

1 111 12

H⎡ ⎤

= ⎢ ⎥⎣ ⎦

For the two-by-two case, the kernel matrix is

1 12 −⎣ ⎦

And for successively larger N, these can be generated fromThe block matrix from

N NH HH H H

⎡ ⎤= = ⊗⎢ ⎥

The block matrix from

2 2N NN N

H H HH H

= = ⊗⎢ ⎥−⎣ ⎦


97


Another way for generating kernel matrix

For examples1 1 1 1 1 1 1 11 1 1 1 1 1 1 1⎡ ⎤⎢ ⎥

4

1 1 1 11 1 1 111 1 1 14

H

⎡ ⎤⎢ ⎥− −⎢ ⎥=⎢ ⎥− − 8

1 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11H

⎢ ⎥− − − −⎢ ⎥⎢ ⎥− − − −⎢ ⎥− − − −⎢ ⎥= ⎢ ⎥1 1 1 14

1 1 1 1⎢ ⎥⎢ ⎥− −⎣ ⎦

8 1 1 1 1 1 1 1 181 1 1 1 1 1 1 11 1 1 1 1 1 1 1

H ⎢ ⎥− − − −⎢ ⎥

− − − −⎢ ⎥⎢ ⎥− − − −⎢ ⎥1 1 1 1 1 1 1 1⎢ ⎥

− − − −⎢ ⎥⎣ ⎦


98

3 3 Oth S bl T f3.3 Other Separable Transforms3.3.4 Hadamard transform： 1-D ordered Hadamard transform3.3.4 Hadamard transform： 1 D ordered Hadamard transform

Lets kernel function:1

( ) ( )

01( ) ( 1)

nb x p ui i

ig x u

−

=∑

= 0( , ) ( 1) ig x uN

= −

0 1( ) ( )np u b u−= ⎫⎪where

1 1 2( ) ( ) ( )n np u b u b u− −⎪= + ⎪⎬⎪⎪

M

where

1 1 0( ) ( ) ( )np u b u b u−⎪= + ⎭

And inverse kernel function:1

( ) ( )

0( , ) ( 1)

nb x p ui i

ih x u

−

=∑

= −


99

3 3 Oth S bl T f3.3 Other Separable Transforms3.3.4 Hadamard transform： 1-D ordered Hadamard transform3.3.4 Hadamard transform： 1 D ordered Hadamard transform The values of g(x,u) are list in below for N=8

u\x 0 1 2 3 4 5 6 7 Sequency of the S ith th

0 + + + + + + + +

1 + + + +

01

rowSame with theOrdered

Walsh transform

1 + + + + - - - -

2 + + - - - - + +

3 + + - - + + - -

123

4 + - - + + - _ +

5 + - - + - + + -

45

6 + - + - - + - +

7 + - + - - + + -

67


100

3 3 Oth S bl T f3.3 Other Separable Transforms3.3.4 Hadamard transform： 1-D ordered Hadamard transform3.3.4 Hadamard transform： 1 D ordered Hadamard transform

Then the ordered Hadamard transform pair is

1( ) ( )

0

11( ) ( )( 1)

nb x p ui i

i

N

H u f x

−

=

− ∑= −∑

0( ) ( )( 1)

xH u f x

N =∑

1( ) ( )

nb

−

∑ ( ) ( )

0

1

( ) ( )( 1)b x p ui i

i

N

f x H u =

− ∑= −∑

0u=


101

3.3 Other Separable Transformsp3.3.4 Hadamard transform： 2-D transform

The kernel functions of 2-D Hadamard are:1

[ ( ) ( ) ( ) ( )]

01( , , , ) ( 1)

nb x b u b y b vi i i i

ih x y u vN

−+

=∑

= −1

[ ( ) ( ) ( ) ( )]

01( , , , ) ( 1)


ig x y u vN

−+

=∑

= −

N

NBoth the direct and inverse kernel function are Separable and Symmetric because

1 1 1 1( , , , ) ( , ) ( , ) ( , ) ( , )g x y u v g x u g y v h x u h y v= =

Separable and Symmetric, because


102


And the 2-D Hadamard transform is defined as:

1[ ( ) ( ) ( ) ( )]

0

1 11( , ) ( , )( 1)


i

N N

H u v f x yN

−+

=

− − ∑= −∑∑

0 0x yN = =

1[ ( ) ( ) ( ) ( )]1 11

nb x b u b y b vi i i iN N

−+− − ∑

0

1 1

0 0

1( , ) ( , )( 1) i

N N

x yf x y H u v

N=

= =

∑= −∑∑


103


uu

v

x

The Hadamard transform basis images for N=4

y


104

The Hadamard transform basis images for N=4

3.3 Other Separable Transformsp3.3.4 Hadamard transform： 2-D ordered transform

The direct and inverse kernel functions of 2-D ordered Hadamard are same as: 1 [ ( ) ( ) ( ) ( )]

1n b x p u b y p vi i i i− +∑

A d h 2 D d d H d d f i d fi d

01( , , , ) ( , , , ) ( 1) ig x y u v h x y u vN

=∑

= = −

And the 2-D ordered Hadamard transform is defined as:1 [ ( ) ( ) ( ) ( )]

1 11n b x p u b y p vi i i iN N− +

− − ∑∑∑ 0

0 0

1( , ) ( , )( 1) i

x y

H u v f x yN

=

= =

∑= −∑∑

1 [ ( ) ( ) ( ) ( )]n b x p u b y p vi i i i− +[ ( ) ( ) ( ) ( )]

0

1 1

0 0

1( , ) ( , )( 1)b x p u b y p vi i i i

i

N N

x y

f x y H u vN

+

=

− −

= =

∑= −∑∑


105

0 0x y

3.3 Other Separable Transformsp3.3.4 Hadamard transform： 2-D ordered transform

uu

v

x

Basis image of 2-D ordered Hadamard transform y


106

3.3 Other Separable Transformsp3.3.5 Haar transform： definitions1 Haar function1. Haar function

The Haar functions hk(z) are defined on the interval [0,1], and k=0 1 N-1 N=2n Let theInteger be specified0 1k N≤ ≤ −k 0,1…N 1, N 2 . Let theInteger be specified ( uniquely) by two other integers, p and q, as

0 1k N≤ ≤ −

2 1pk q= + −2 1k q+

Where 2p is the largest power of 2 such that and q-1 is Th i d

2 p k≤The remainder

For example, k=1=20+1-1, p=0, q=1k , p 0, qk=23=24+8-1, p=4, q=8k=100=26+37-1, p=6, q=37


107

, p , q

3.3 Other Separable Transformsp3.3.5 Haar transform： definitions1 Haar function1. Haar function

The Haar functions are defined by

0 00( ) ( ) 1h z h z N= = [0,1]z∈

1 2122 2

2 p pqqp z −−⎧ ≤ <

⎪ 2 21 22

2 2

1( ) ( ) 2

0p p

q qpk pqh z h z z

N−

⎪⎪= = − ≤ <⎨⎪⎪⎩ others0⎪⎩ others


108

3.3 Other Separable Transformsp3.3.5 Haar transform： definitions2 Haar transform matrix2. Haar transform matrix

0 0 0(0 / ) (1/ ) ( 1/ )h N h N h N N−⎡ ⎤L0 0 0

1 1 1

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

h N h N h N Nh N h N h N N

H

⎡ ⎤⎢ ⎥−⎢ ⎥=⎢ ⎥⎢ ⎥

L

M M O M

1 1 1(0 / ) (1/ ) ( 1/ )N N Nh N h N h N N− − −

⎢ ⎥⎢ ⎥−⎢ ⎥⎣ ⎦K


109


For example: N=20 00

1(0 / 2) (0 / 2)h h= =

0 02

1 1

(0 2) (1 2)(0 2) (1 2)

h hH

h h⎡ ⎤

= ⎢ ⎥⎣ ⎦

0 00(0 / 2) (0 / 2)2

h h

0 001(1/ 2) (1/ 2)2

h h= =1 1(0 2) (1 2)h h⎣ ⎦ 2

11 01

1(0 / 2) (0 / 2)2

h h= =k=0 p=0, q=0

1 011(1/ 2) (1/ 2)2

h h= = −k=1 p=0, q=1

2

1 111 12

H⎡ ⎤

= ⎢ ⎥−⎣ ⎦


110


For example: N=4 For example: N=8p1 1 1 11 1 1 11H

⎡ ⎤⎢ ⎥− −⎢ ⎥

⎡ ⎤4 2 2 0 02

0 0 2 2

H ⎢ ⎥= ⎢ ⎥−⎢ ⎥⎢ ⎥−⎣ ⎦

1 1 1 1 1 1 1 11 1 1 1 1 1 1 1

2 2 2 2 0 0 0 0

⎡ ⎤⎢ ⎥− − − −⎢ ⎥⎢ ⎥− −⎢ ⎥

8

2 2 2 2 0 0 0 01 0 0 0 0 2 2 2 28 2 2 0 0 0 0 0 0

H⎢ ⎥⎢ ⎥− −= ⎢ ⎥−⎢ ⎥⎢ ⎥0 0 2 2 0 0 0 0

0 0 0 0 2 2 0 00 0 0 0 0 0 2 2

⎢ ⎥−⎢ ⎥

−⎢ ⎥⎢ ⎥−⎢ ⎥⎣ ⎦


111

0 0 0 0 0 0 2 2⎢ ⎥⎣ ⎦

3.3 Other Separable Transformsp3.3.5 Haar transform： definitions2. Haar transform matrix. s o

h f b i f i fDigital Image Processing


The 1-D Haar transform basis function for N=8

3.3 Other Separable Transformsp3.3.5 Haar transform： definitions3 Haar transform3. Haar transform

G HF=Direct transform:f

0 01 1 1 1 1 1 1 1g f⎡ ⎤⎡ ⎤ ⎡ ⎤

⎢ ⎥⎢ ⎥ ⎢ ⎥

1F H G−=Inverse transform:

1 1

2 2

1 1 1 1 1 1 1 1

2 2 2 2 0 0 0 01 0 0 0 0 2 2 2 2

g fg fg f

⎢ ⎥⎢ ⎥ ⎢ ⎥− − − −⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥− −⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥3 3

4 4

5 5

1 0 0 0 0 2 2 2 28 2 2 0 0 0 0 0 0

0 0 2 2 0 0 0 0

g fg fg f

⎢ ⎥− −⎢ ⎥ ⎢ ⎥= ⎢ ⎥⎢ ⎥ ⎢ ⎥−⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥−⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥

6 6

7 7

0 0 0 0 2 2 0 00 0 0 0 0 0 2 2

g fg f

⎢ ⎥⎢ ⎥ ⎢ ⎥−⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦⎢ ⎥⎣ ⎦


113

3.3 Other Separable Transformsp3.3.5 Haar transform： properties3 Haar transform3. Haar transform

u

v

x

h f b i i f

y


114The Haar transform basis images for N=4

3.3 Other Separable Transformsp3.3.5 Haar transform： mallat algorithms

h(-n) 2 2 h(n)

+

jka

1jka +

j

g(-n) 2 2 g(n)

+

1jkd +

jka

LL1

LH1

HL1

HH1

HL2

HL3

LH2

HH2

LH3 HH3


115

3.3 Other Separable Transformsp3.3.5 Haar transform： mallat algorithms


116

3.3 Other Separable Transformsp3.3.6 Slant transform： definitions

2

1 111 12

S⎡ ⎤

= ⎢ ⎥−⎣ ⎦Unitary kernel matrix starting with:

1 0 1 0⎡ ⎤⎢ ⎥⎡ ⎤

M

M

And iterating it according to the schema:

/ 2

1 0 1 00 0

0N N N N Na b a b S

⎢ ⎥⎡ ⎤ ⎢ ⎥⎢ ⎥− ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥

MM M M

M

L L L L L L L M

( / 2) 2 ( / 2) 20 012 0 1 0 1

0 0

N N

N

I IS

− −⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥= ⎢ ⎥⎢ ⎥ ⎢ ⎥−⎢ ⎥ ⎢ ⎥⎢ ⎥

M M M M

L L L L L L L L L L

MM M M

/ 2

( / 2) 2 ( / 2) 2

0 0

00 0

N N N N

N

N N

b a b aS

I I− −

⎢ ⎥⎢ ⎥− ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥− ⎢ ⎥⎢ ⎥⎣ ⎦⎢ ⎥

M M MM

L L L L L L L M

M M M M


117

( / 2) 2 ( / 2) 2N N⎢ ⎥⎣ ⎦⎢ ⎥⎢ ⎥⎣ ⎦M


Where I is the identity matrix of order N/2-2 and

1 22

2

34( 1)N

NaN

⎡ ⎤= ⎢ ⎥−⎣ ⎦

1 22

2

44( 1)N

NbN

⎡ ⎤−= ⎢ ⎥−⎣ ⎦

For example: N=4⎡ ⎤

4

1 1 1 1

3/ 5 1/ 5 1/ 5 3/ 51S

⎡ ⎤⎢ ⎥

− −⎢ ⎥= ⎢ ⎥4 1 1 1 141/ 5 3/ 5 3/ 5 1/ 5

⎢ ⎥− − −⎢ ⎥⎢ ⎥− −⎣ ⎦


118


For example: N=8

1 1 1 1 1 1 1 1⎡ ⎤1 1 1 1 1 1 1 1

7 / 21 5 / 21 3/ 21 1/ 21 1/ 21 3/ 21 5 / 21 7 / 211 1 1 1 1 1 1 1

− − − −− − − −

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

8

1/ 5 3/ 5 3/ 5 1/ 5 1/ 5 3/ 5 3/ 5 1/ 518 3/ 5 1/ 5 1/ 5 3/ 5 3/ 5 1/ 5 1/ 5 3/ 5

7 / 105 1/ 105 9 / 105 17 / 105 17 / 105 9 / 105 1/ 105 7 / 105

S− − − −

=− − − −

⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥7 / 105 1/ 105 9 / 105 17 / 105 17 / 105 9 / 105 1/ 105 7 / 105

1 1 1 1 1 1 1 1

1/ 5 3/ 5 3/ 5 1/ 5 1/ 5 3/ 5 3/ 5

− − − −− − − −

− − − − 1/ 5

⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦⎣ ⎦


119

3.3 Other Separable Transformsp3.3.6 Slant transform： 1-D basis functions

The 1-D Slant transform basis functions for N=8Digital Image Processing


The 1 D Slant transform basis functions for N 8

3.3 Other Separable Transformsp3.3.6 Slant transform： 2-D basis images

u

v

x

y

The 2-D Slant transform basis functions for N=4

y


121

3.4 Hotelling Transformsg3.3.7 Hotelling transform： definitions

KLT: Karhunen-Loeve Transfrom

Th kth i i i t b d t

KLT: Karhunen Loeve TransfromPCA:Principal Components Analysis

L

The kth image in a image set can be expressed as a vector:

k=0,1, …M-10 1 1 TN

k k k kx x x x −⎡ ⎤= ⎣ ⎦L

T

The covariance matrix of the x vector is defined as

{ }

{( )( ) }Tx x xC E x m x m= − −

where{ }xm E x=

w e e

is the mean vector, E is the expected valueDigital Image Processing


, p

3.4 Hotelling Transforms3.3.7 Hotelling transform： definitions

g

11 M −

∑

They can be approximated from the samples by using the relations

0x k

k

m xM =

= ∑

and

11 ( )( )M

Ti iC x m x m

−

= − −∑

and

0

1

( )( )

1

x i x i xk

MT T

k k x x

C x m x mM

x x m mM

=

−

= −

∑

∑0kM =∑


123

3.4 Hotelling Transforms3.3.7 Hotelling transform： definitions

l Calculated N eigenvalues and

g

let Calculated N eigenvalues andarranged as 0 1 1Nλ λ λ −≥ ≥ ≥L

l [ ]

0xC Iλ− =

C l l d ilet [ ] 0x i iC I Tλ− = Calculated N eigenvectors Ti and arranged as

⎡ ⎤0

1

T

T

TT

A

⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥M

1T

NT −

⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

M

Hotelling transform ( )xY A X m= −

Inverse Hotelling transform TX A Y m+Digital Image Processing


Inverse Hotelling transform xX A Y m= +

3.4 Hotelling Transforms3.3.7 Hotelling transform： properties

Relationship between the eigenvaluse and eigenvectors:

g

x i i iC T Tλ=

Relationship between the eigenvaluse and eigenvectors:

[ ] 0x i iC I Tλ− =

0

1

T

T

TT

⎡ ⎤⎢ ⎥⎢ ⎥ T T

xC A A= ∧

0 0λ⎡ ⎤⎢ ⎥

1

1T

N

TA

T −

⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

M

1λ⎢ ⎥⎢ ⎥∧ =⎢ ⎥⎢ ⎥

Owhere

10 Nλ −

⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦


125


g

{ } { } { }( )y x xm E y E Ax Am AE x Am= = − = −mean vector of y

0m =

The covariance matrix of the Y vector is given by

0ym =

{ }{( )( ) }

( )( )

Ty y y

T

C E Y m Y m

E AX A AX A

= − −

{ }{ }( )( )

( )( )

Tx x

T T

E AX Am AX Am

E A X m X m A

= − −

= − −{ }{ }( )( )

( )( )

x x

T Tx x

E A X m X m A

AE X m X m A= − −


126


TC AC A

g

Ty x

T

C AC A

AA

=

= ∧TAA I=

C is a diagonal matrix with elements equal to

AA ∧= ∧

0λ⎡ ⎤

Cy is a diagonal matrix with elements equal to the eigenvalues of Cx, that is

1

2

0

yC

λλ

⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥

O

0 Nλ⎢ ⎥⎢ ⎥⎣ ⎦

That’s means elements of Y are uncorrelated


127

3.4 Hotelling Transforms3.3.7 Hotelling transform： inverse transform

Si H t lli t f i th l

g

TxX A Y m= +

Since Hotelling transform is orthogonal , so

1 TA A− = and xmIf we form A from K eigenvectors corresponding to the largest eigenvalues as A the recovered vector will be

ˆ TK xX A Y m= +

largest eigenvalues as AK, the recovered vector will be

K xIt can be shown that the mean square error, ems, between Xand is given by the expressionX̂

1 1 1

0 0

N k N

ms j j jj j j K

e λ λ λ− − −

= = =

= − =∑ ∑ ∑


128

3.4 Hotelling Transforms3.3.7 Hotelling transform： example

Gi h l f 2 di i h b l

g

Given the samples of 2-dimension vectors shown as below, calculate its Hotelling transform. N=2, M=27

(1,1) (1,2) (2,1) (2,2) (2,3)(3,1) (3,2) (3,3) (4,2) (4,3) (4 4) (4 5) (4 6) (5 3) (5 4)

x1

(4,4) (4,5) (4,6) (5,3) (5,4) (5,5) (5,6) (5,7) (6,4) (6,5) (6 6) (6 7) (6 8) (7 5) (7 6)(6,6) (6,7) (6.8) (7,5) (7,6) (7,7) (7,8)

0 1 2 3 4 5 6 7

x0


129


g

Let 0 1

Tk k kx x x⎡ ⎤= ⎣ ⎦ k=0,1, …26

251 [ ]25

0

1 4.444 4.296327x k

km x

=

= =∑

251 ( )( )TC ∑0

1 ( )( )27

3 4103 3 2479

Tx i x i x

k

C x m x m=

= − −

⎡ ⎤

∑3.4103 3.2479

3.2479 4.7550⎡ ⎤

= ⎢ ⎥⎣ ⎦


130


l 3 4103 3 2479λ

g

let 0xC Iλ− = 3.4103 3.24790

3.2479 4.7550λ

λ−

=−

0 7.3993λ = 1 0.7659λ =

[ ] 0C I Tλ ⎡ ⎤ ⎡ ⎤let [ ] 0x i iC I Tλ− =0

0.63140.7755

T ⎡ ⎤= ⎢ ⎥⎣ ⎦

1

0.77550.6314

T−⎡ ⎤

= ⎢ ⎥⎣ ⎦

0.6314 0.77550.7755 0.6314

A⎡ ⎤

= ⎢ ⎥−⎣ ⎦

( )xy A x m= −


131


g

y =( -4.7309 0.5899) ( -3.9555 1.2212) ( -4.0996 -0.1856) ( -3.3241 0.4458)

( -2.5486 1.0771) ( -3.4682 -0.9611) ( -2.6927 -0.3297) ( -1.9172 0.3017)

( -2.0613 -1.1052) ( -1.2859 -0.4738) ( -0.5104 0.1576) ( 0.2651 0.7890)

my = 1.0e-015 *

-0.6908 -0.1069

( 1.0406 1.4203) ( -0.6545 -1.2493) ( 0.1210 -0.6179) ( 0.8965 0.0135)

( 1.6719 0.6449) ( 2.4474 1.2762) ( 0.7524 -1.3934) ( 1.5279 -0.7620

( 2.3033 -0.1306) ( 3.0788 0.5008) ( 3.8543 1.1322) ( 2.1592 -1.5375)

( 2 9347 0 9061) ( 3 7102 0 2747) ( 4 4857 0 3567)

cy =

7.3993 0.0000

0.0000 0.7659( 2.9347 -0.9061) ( 3.7102 -0.2747) ( 4.4857 0.3567)

corry =x1 y1y0

1.0000 0.0000

0.0000 1.0000

y1

0 1 2 3 4 5 6 7 x0


132

x0


2

g

x2 y2

x1y1


133

3.4 Hotelling Transforms3.3.7 Hotelling transform： inverse transform

g

x2 y2y

x1 1KLT

x1 y1


134

Summary

Basic function

Sinusoidal transforms

(a) Discrete Fourier Transform cos sinie iθ θ θ= +( )

(b) Discrete Cosine Transform cosθi θ(c) Discrete Sine Transform

(d) Hartly Transform

sinθ

cos sinθ θ+cos sinθ θ+


135

Summary

Rectangular wave transforms

(a) Hadamard Transform( )

(b) Walsh Transform

(c) Slant Transform

(d) Haar Transform


136

Summary

Eigenvector-based transforms

(a) Hotelling Transform (K-LT)( ) g ( )

(b) SVD Transform


137

Homework

(1) 计算下图的 DFT, DCT, Hadamard 变换和Haar 变换变换

0 1 1 00 1 1 00 1 1 00 1 1 00 1 1 0

(2) Page 71(章毓晋) 3 21: 设有一组64*64的图像(2) Page 71(章毓晋) 3.21: 设有一组64*64的图像,它们的协方差矩阵式单位矩阵.如果只使用一半的原始特征值计算重建图像,那么原始图像和重建图原始特征值计算重建图像,那么原始图像和重建图像间的均方误差是多少?


138

The End


139

Chapter3 Image Transforms - USTChome.ustc.edu.cn/~xuedixiu/image.ustc/course/dip/DIP14...Chapter3...

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