Ch3-Mathematical and Physical Backgrounds

8/10/2019 Ch3-Mathematical and Physical Backgrounds

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3-0

3.1. Overview

3.2. Linear Integral Transforms

3.3. Images as Stochastic Processes

3.4. Image Formation Physics

Chapter 3: The Image, its Mathematical

and Physical Background


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3-1

3.1. Overview

3.1.1. Linearity

(a)Additivity ( ) ( ) ( )L L L x y x y

( ) ( )L a aLx x

( ) ( ) ( )L a b aL bL x y x y

Let L : operator, mapping, function, or process

a,b: scalars

x,y: elements of a vector space

e.g., vectors, functions

(b)Homogeneity

(c)Linearity


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3-2

3.1.2. Dirac Delta Function

Heaviside function:0 0

( )

1 0

tH t

t

0

( )1

t aH t a

t a

0a

0

( ) ( ) 1

0

t a

H t a H t b a t b

t b

1( ) [ ( ) ( )]t H t H t

0( ) lim ( )t t

Pulse:

Impulse:

Dirac Delta Function:

a b

1,t dt

0,t t 0


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3-3

, , ,f x y f a b a x b y dadb

Sampling (shifting property):

, , ,f x y x a y b dxdy f a b

, 1,x y dxdy , 0 ( , )x y x y 02D delta function:

Image function:


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3-4

( ) ( ) ( ) ( ) ( ) ( ) ( )h x f x g x f g x d f x g d

3.1.3. Convolution ( )

( )f x f a a x da Assignment: Show


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3-6

Discrete case:1 1

0 0( ) ( ) ( ) ( ) ( )

, : extended , ; 1

N N

e e e en n

e e

h k f n g k n f k n g n

f g f g N A B

WraparoundIfN


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5-6

In practice,


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5-7

1 2

1 2

( , ) ( , ) ( , )

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

(1,2) ( 1, 2)

s t

p x y m s t p x s y t

m p x y m p x y

m p x y


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3-8

Properties: ,f g g f f g h f g h f g h f g f h

a f g af g f ag f g f g f g

Assignment : f g f g f g

, , ,

, , ,

f g x y f a b g x a y b dadb

f x a y b g a b dadb g f x y

2-D convolution:

Discrete: 1 1

0 0

, , ,M N

e em n

h i j g i m j n h m n


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3-9

Correlation: ( )

( ) ( ) ( ) ( ) ( ) ( )f x g x f g x d f x g d

1

0

( ) ( ) ( ) ( ),N

e e e e

n

f k g k f n g k n

1N A B


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Spatial

domain

3.2. Linear Integral Transforms

domain 1 domain 2

Transform:Frequency

domaine.g.,

Advantages: implicit explicit properties

3.2.3. Fourier Transform

Fourier series

Fourier analysis = Fourier transform


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Fourier series-- A periodic (T) functionf(x) can

be written as 0 1( ) / 2 cos sinn nnf x a a n x b n x

/ 2 /20

/ 2 /2

/ 2

/2

1 2( ) , ( )cos

2 2

, ( )sin

T T

nT T

T

n T

a f x dx a f x n xdxT T

b f x n xdxT T

3-11

( ) exp( ),nn

f x c jn x

/ 2

/ 2

1

( )exp( )

T

n Tc f x jn x dxT

In complex form

(Assignment)

where

2

0( ) [ ( )cos2 ( )sin2 ] ( )

j xf x a x b x d c e d

In continuous case,

where


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3-12

Some function is

formed by a finite

number of sinuous

functions

( ) sin (1/3)sin 2 (1/5)sin 4f x x x x

Some function requires

an infinite number of

sinuous functions tocompose

1 1 1 1( ) sin sin3 sin5 sin7 sin9

3 5 7 9f x x x x x x


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The spectrumof a periodic function, which is

composed of a finite number of sinuous functions,

is discrete consisting of components at dc, 1/T,and its multiples

For non-periodic functions, or 0T The spectrum of the function continuous


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3-14

2 cos2 sin2j te t j t

2

(cos2 sin 2 )

cos2 sin 2 ( )

j t

f t e dt f t t j t dt

f t tdt j f t tdt F

aggregates the sinuous

component of f. F frequency-

1-D:

Fourier transform (FT)

2 i tf t F f t e dt

F

-1 2 i tF f t F e d

F


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Complex spectrum Re Im( )F F j F

Amplitudespectrum

Phasespectrum

Powerspectrum

2 2Re Im ( )F F F

1

tan Im / ReF F

2 2 2

Re Im ( )P F F F

0 , 0fF f t Fdt d

1.

2. 2 2

f t dt F d

(Parsevals theory)

Properties:


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Examples:


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Discrete Fourier transform

1

0

1 ( ) 2N

n

nkF k f n exp jN N

1

0

( ) 2N

k

nkf n F k exp j

N

( ), 0, 1, , 1f n n N Input signal:

where


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Properties:

1 2 1 2af bf aF bF Linearity

Conjugate symmetry *, ( , )F u v F u v

Periodicity

, ( , )F u v F u N v

, ,F u v F M u v , ,F m n F M m n , ( , )F m n F m N n


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0 1 1{ , , , },f Nf f f 0 1 1{ , , , }NF F F F

0 1 2 3 2 1{ , , , , , , }f N Nf f f f f f

/ 2 1 0 1 / 2 1{ , , , , , , }N N NF F F F F F

0 0Let / 2 exp( 2 / ) exp( )

( ) (cos sin ) ( 1)j x x x

u N j u x N j x

e j

0( ) ( )exp( 2 / ) ( 1) ( )x

f x f x j u x N f x

0 0

0 0

( )exp( 2 / ) ( )

( ) ( )exp( 2 / )

f x j u x N F u u

f x x F u j ux N

Shifting


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Examples:

1-D signal {2 3 4 5 6 7 8 1}f

{36 - 9.6569 4 - 4 - 4 1.6569 - 4

4 1.6569 4 - 4 4 -9.6569 - 4 }

j j j

j j j

F

{2 -3 4 -5 6 -7 8 -1} f

{ 4 1.6569 4 -4 4 -9.6569 - 4

36 - 9.6569 4 - 4 - 4 1.6569 - 4 }

j j j

j j j

F

3-20

f F F

2-D image


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Rotation

Polar coordinates:

cos , sinu w v w ( , ) ( , ),f x y f r ( , ) ( , )F u v F w

0 0( , ) ( , )f r F w

cos , sinx r y r

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Convolution theorem ,f g F G f g F G

Correlation theorem

(Assignment)* *,f g F G f g F G


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Any real function can always be decomposed

into its even and odd parts, i.e.,( )ef x ( )of x

( )f x

,2 2

e of t f t f t f tf t f t ( ) ( ) ( ),e of x f x f x

( ) ( ) ( )

( ) R{ ( )} I{ ( )}e o

f x f x f x

F u F u i F u

( ) R{ ( )}

( ) I{ ( )}

e

o

f x F u

f x F u

e.g.,


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Convolution involving impulse function

( ) ( )

( ) ( )

f x g x

g x x T

( ) ( )

( ) ( ) ( ) ( )

f x g x

g x x T x x T

Copyf(x) at the location of each impulse

( ) ( )f x g x

( )g x

( )f x

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3.2.5. Sampling theory

Objective: looking at the question of how

many sampling should be takenso that no information is lost in

the sampling process

Sparsesampling

Densesampling

Continuousfunction


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f(x): band-limited

function

Sampling function

( ) ( )

( ) ( )

S x x x

x x x

Spatial domain Frequency domain

( ) ( )f x S x dxSampling

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sin[2 ( )]( ) ( )

2 ( )

sinc[2 ( )] ( )

nn

n n

n n

n

x xf x f x

x x

x x f x

Interpolation Reconstruction Formula

Whittaker-Shannon sampling theory

1

2x w 1-D:

2-D:1 1

,2 2

x yu v


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3.2.6. Discrete cosine transform

-- often used in image/video compression,

e.g.,JPEG, MPEG, FGS, H.261, H.263, JVT

1 1

0 0

1 1

0 0

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

2 2

1/ 2 0 where ( )

1 otherwise

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

2 2

wh

N N

m n

N N

u v

c u c v m nF u v f m n u v

N N N

kc k

m nf m n c u c v F u v u v

N N N

ere 0,1..., 1, 0,1,..., 1m N n N

0,1..., 1

0,1,..., 1

u N

v N


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Fourier spectrum provides all the frequencies

present in a signal but does not tell where they

are present.

Windowed Fourier transformsuffers from the

dilemma:

Small range poor frequency resolution

Large range poor localization

3.2.7. Wavelet transform

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Wavelet: wave that is only nonzero in a small region

Wave Wavelet


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Haar:

1 if 0 1/ 2

( ) 1 if 1/ 2 10 otherwise

x

x x

2

( ) sin ,x

x e xMorlet: Mexican hat: DOG, LOG

Types of wavelets:


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Operations on wavelet:

(a) Dilation:

i) Squashing ii) Expanding

(b) Translation:

i) Shift to the right ii) Shift to the left

(c) Magnitude change:

i) Amplification ii) Minification

(2 )x

( )x

( / 2)x

( 2)x ( 2)x

2 ( )x 1/ 2 ( )x


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Any function can be expressed as a sum of wavelets

of the form ( )i i ia b x c


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,

1( )s

tt

ss

Wavelet transform:decomposes a function into

a set of wavelets

: wavelets

New variables:

scale

translation

where

{0},s R R

Inverse wavelet transform:synthesize a functionfrom wavelets coefficients

,( , ) ( ) ( )sR

W s f t t dt

,( ) ( , ) ( )sR R

f t W s t d ds

t : mother wavelet


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,

1( , ) ( ) ( ),k s

x

W k s f t t N

,1( , ) ( ) ( ),k s

x

W k s f t t N

0

, ,

1 1( ) ( , ) ( ) ( , ) ( )k s k s

k j j

f t W k s t W k s tN N

3-34

Discrete wavelet transform:

, ( )k s t

, ( )k s t

: scaling

functions

: wavelet

functions

Approximation coefficients (cA)

Detail coefficients (cD)

Inverse discrete wavelet transform:


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Multiresolution Analysisviews a function at various

levels of resolution

3-35

h: step size


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Let

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SA contain all the functions on the left-hand side

andS

W those on the right-hand side

2

, ,{ (2 ) : }s

k s k s

k k

c t k c SA2

, ,{ (2 ) : }sk s k sk k

d t k d SW

Function spaces:

SA

SW

is generated by the bases

is generated by the bases

/ 2

,( ) 2 (2 ),s s

k s t t k k Z

/ 2

, ( ) 2 (2 ),s s

k s t t k k Z

i.e.,(scaling

functions)

(wavelet

functions)


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Scaling function1 0 1

( )0 otherwise

xx

/ 2( ) 2 (2 ), 0,...,2 1j j jji x x i i

/ 2( ) 2 (2 ), 0,...,2 1j j jji x x i i

1 0 1/ 2

( ) 1 1/ 2 1

0 otherwise

x

x x

Wavelet function

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Example: Haar wavelet


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Properties:

i) ii) iii)

iv)2

1 0 1 L A A A

S S

A W1 S S SA A W S SW A

3-38

Discrete signal ( ), 0, 1, , 1s i i N

is decomposed into wavelet coefficients

, ,1 ( ) ( )k s k si

d s i iN

Suppose scales and positions are based on

power of 2 (dyadic)

Approx. coefficients (cA)

Detail coefficients (cD)

, ,

1( ) ( ),k s k s

i

c s i iN

Fast Discrete Wavelet Transform:


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Inverse Discrete Wavelet Transform

, , , ,

1 1( ) ( ) ( )k s k s k s k s

k k

s t c t d tN N


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Wavelet transforms

Low pass filtering: averaging ;

High pass filtering: differencingInput data: a, b

Average:s= (a+ b) / 2 (low pass filtering)

Difference: d= as (high pass filtering)

Wavelet coefficients: (s, d).

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Inverse wavelet transforms

Addition; subtraction

Wavelet coefficients: (s, d).Addition:s + d=s+ (as)= a,

Subtraction:sd=s (as)= 2s a = b

Input data: (a, b).


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Example:

Input data 14, 22

(i) Wavelet TransformAverage: s= (1422)/2 = 18,

Difference: d = 14-18= -4

Wavelet coefficients: (18, -4).(ii) Inverse Wavelet Transform (to recover the

input data)

sd= 18+(-4) = 14,

sd= 18-(-4) = 22

Input data: (14, 22).


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2-D:


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Hadamard-Haar, Slant, Slant-Haar, Discrete sine,

Paley-Walsh, Radon, Hough

3.2.11. Other Orthogonal Image Transforms

3.2.8. Eigen Analysis

Let A be an nby n square matrix. x and are

corresponding eigenvalueand eigenvectorof A

if .

1 1

1, ( , , )

nA PDP D P AP diag

1[ ]

nP e e

: corresponding eigenvalues

Let : eigenvectors ofA

Let

Then

1, ,e en

1, , n

A x x


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3.2.9. Singular Value Decomposition

m nA : real matrix

column-orthonormal matrix m nU T

nU U Id

n nVT

nVV Id TA UWV

1 2( , , , )nW diag 1 2, , , n

(i.e.,

(i.e., ), s.t.

) and row-orthonormal matrix

The singular values ofAare the eigenvaluesof and the corresponding eigenvectors

are the columns of V

TA A

where

: non-negative singular values

of A

3-46


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3-47

1, ,e en

1, , n

1[ ]e enE

: corresponding eigenvalues

Let : eigenvectors of matrixB

Let

Then ,TB EDE 1( , , )nD diag

( )

T T T T T T T

A A UWV UWV VW U UWV 2 2 21 2( , , , )

T T T

nVW WV Vdiag V

Compare with TB EDE2

i T

A A

iTA A

(a) The eigenvalues of correspond

(b) The eigenvectors of are columns of V

to the singular values of A


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3.2.10. Principal Component Analysis (PCA)

Karhunen-Loeve(KL) or Hotelling transforms

PCA: linearly transforms a number of correlated

variables into the same number of uncorrelated

variables (principal components)

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

i nx x x i M xData vectors:

Mean vectors: { }x Em x

Covariance matrix: {( )( ) }Tx x xC E = x m x m

n by n real symmetric matrix:xC

M


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1

1,

M

x i

iM m x

1

1

1 1 1 1

1

1

1

( )( )

1( )

1[ ( ) )]

1

1

MT

x i x i x

i

MT T T T

i i i x x i x x

i

M M M MT T T T

i i x x i x x i

i i i i

MT T T T

i i x x x x x x

i

MT T

i i x x

i

C M

M

M

M

M

x m x m

x x x m m x m m

x x m m x m m x

x x m m m m m m

x x m m

Approximation:


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503-50

Let be corresponding

eigenvalues and eigenvectors of .

Assume

Construct matrix

and , 1, , ,i i i n e

1 2 n 1 2 nA e e e

xC

( )i i xA y x m

The mean of ys:

y 0m

The covariance matrix of ys:T

y xC AC A=

1

2

0 0 00 0

0 0 0

0 0

y

n

C


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( ),i i xA y x mFrom

The mean square error between

Let ,kA k n

1 1 1

n k n

i i i

i i i k

e

3-51

( )i k i xA y x m

,i iy x i iy x

1 T

i i x i xA A x y m y m

andi ix x

and

is


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Eigen faces

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Ch3-Mathematical and Physical Backgrounds