2D Systems Images are outputs of 2D systems Continuous vs. - - PDF document

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2D Systems

• Images are outputs of 2D systems
• Continuous vs. sampled (discrete) images
• 2D discrete (digital) images are sequences

which are functions of two integer arguments

• System – input-output relationship

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2D Sequence x(n1,n2)

Lim, 1990

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2D Sequence x(n1,n2) (cont’)

Lim, 1990

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2D Sequence x(n1,n2) (cont’)

Lim, 1990

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2D Sequences (cont’)

• Impulses

sequence separable ) ( ) ( ) , ( sequence step unit

• therwise

, 1 ) , ( impulse line

• therwise

1 ) ( sequence) sample (unit impulse

• therwise

1 ) , (

2 1 2 1 2 1 2 1 variable)

• ne
• f

sequence 2D indicates (T 1 1 2 1 2 1

n g n f n n x n n n n u n n n n n n

T

=    ≥ =    = =    = = = δ δ

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Impulse δ(n1,n2)

Lim, 1990

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Sequence Representation as Linear Combination of Shifted Impulses

sequence value

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Line Impulse δT(n1)

Lim, 1990

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Unit Step Sequence u(n1,n2)

Lim, 1990

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Step Sequence uT(n1)

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2D Separable Functions f(x1,x2)= f1(x1) f2(x2)

Jain, 1989

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Periodic Sequence

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Linear Systems

• Input-output relationship is called a system

if there is a unique output for any given input

• which is the principle of superposition.

[ ].

) , ( ) , (

2 1 2 1

n n x T n n y =

[ ]

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

2 1 2 2 1 1 2 1 2 2 1 1

n n by n n ay n n bx n n ax T Linearity + = + ⇔

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Linear vs. Nonlinear System

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2D Convolution (cont’)

response impulse the ) , ( and input the is ) , ( where ) , ( ) , ( ) , ( ) , ( ) , (

2 2 1 1 2 1 2 2 1 1 2 1 2 1 2 1 2 1

' 1 ' 2

k n k n h n n x k n k n h k k x n n h n n x n n y

k k

− − − − ≡ ∗ =

∑ ∑

∞ −∞ = ∞ −∞ =

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1D Convolution by Castleman

Castleman, 1996

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2D Convolution by Jain (1)

Jain, 1989

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2D Convolution by Jain (2)

Jain, 1989

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…and 2D convolution by Lim

Lim, 1990

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2D Convolution

• Applications

Deconvolution (deblurring due to deformation, motion, atmosphere) Noise removal

Estimating original signal Detecting known feature embedded in a noisy background

Feature enhancement (edges, spots)

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Applications 1: Deconvolution

Castleman, 1996

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Applications 2: Smoothing

Castleman, 1996

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Applications 3: Edge Enhancement

Castleman, 1996

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Region of Convolution

Lim, 1990

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Convolution with a Separable Sequence

Lim, 1990

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The Fourier Transform of a 2D Sequence x(m,n)

• The Fourier Transform Pair

2 1 2 1 2 1 2 2 1 2 1 2 1

)} ( exp{ ) , ( ) 2 ( 1 ) , ( ,

• )},

( exp{ ) , ( ) , (

1 2

ω ω ω ω ω ω π π ω ω π ω ω ω ω

π π ω π π ω

d d n m j X n m x n m j n m x X

m n

+ = < ≤ + − ≡

∫ ∫ ∑ ∑

− = − = ∞ −∞ = ∞ −∞ =

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Properties of the Fourier Transform by Lim

Lim, 1990

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Properties of the Fourier Transform by Castleman

Castleman, 1996

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Properties of the Fourier Transform

• The addition theorem (addition in time/spatial domain

corresponds to addition in frequency)

• The shift theorem (shifting a function causes only to phase

shift)

• The convolution theorem (convolution is equivalent to

multiplication in the other domain)

• …

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The Addition Theorem

{ } [ ]

) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (

2 2 2

s G s F dt e t g dt e t f dt e t g t f t g t f

st j st j st j

+ = + + = + Φ

∫ ∫ ∫

∞ ∞ − − ∞ ∞ − − − ∞ ∞ − π π π

Castleman, 1996

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The Fourier Transform – Example 1

Lim, 1990

low-passblur

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The Fourier Transform – Example 1

) cos( 3 1 ) cos( 3 1 3 1 6 1 6 1 6 1 6 1 3 1 )} ( exp{ ) , ( ) , (

2 1 2 2 1 1 2 1 2 1

2 1 2 1 1 2

ω ω ω ω ω ω

ω ω ω ω

+ + = + + + + = + − =

− − ∞ −∞ = ∞ −∞ =

∑ ∑

j j j j n n

e e e e n n j n n h H

Lim, 1990

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The Fourier Transform – Example 1

Lim, 1990

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The Fourier Transform – Example 2

) 2cos

• (3

) 2cos

• (3

) ( ) ( ) , (

2 1 2 2 1 1 2 1

ω ω ω ω ω ω = = H H H

Lim, 1990

the sequence is separable

• high-pass

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The Fourier Transform – Example 2

Lim, 1990

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A 2D Fourier Transform

Castleman, 1996

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Phase-Only & Magnitude- Only

Lim, 1990

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Image Synthesis

Lim, 1990

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Iterative Phase-Only

Lim, 1990

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The 2D DFT

• If g(i,k) is an NxN array, then the 2D

discrete Fourier transform pair is given by

∑∑ ∑∑

− = − = + − = − = + −

= =

1 1 ) ( 2 1 1 ) ( 2

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

N m N n N n k N m i j N i N k N k n N i m j

e n m G N k i g e k i g N n m G

π π