Transforms a s o s

• Transforms that are commonly used are separabley pExamples: Two-dimensional DFT, DCT, DST, HadamardWe can then use 1-D transforms in computing the 2D separable transforms:separable transforms:

Take 1-D transform of the rows => Xrows(K1,K2)Take 1-D transform of columns of Xrows(K1,K2) => X(K1,K2)

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The 2D Discrete Fourier Transform (2D DFT)e sc ete ou e a s o ( )

• The DFT consists of samples of the DTFT at( )21,ωωXp ( )21,

( )

( )2

1,,0;211

1

11 1

−==

K

NKNK

K L

π

πω

⇒ periodicity in space (or time) domain (and in frequency domain)

( )1,,0;222

2

22 2

−== NKNK

K Lπω

⇒ sequence x(n1,n2) extended periodically with horizontal period of N1and vertical period of N2

x(n1,n2+ N2) = x(n1,n2)x(n1 + N1,n2) = x(n1,n2)

But, we restrict ourselves to the main period to recover the original signal

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2D DFT

. ( ) ( )21DFT

21 ,, KKXnnx ⎯⎯→←

( ) ( ) 10;,,1

0

1

0

22

2121

1

1

2

2

22

21

1

1

−≤≤= ∑∑−

=

−

=

−−

ii

N

n

N

n

nNKjn

NKj

NKeennxKKXππ

( ) ( ) 10;,1,1

0

1

0

22

2121

21

1

1

2

2

22

21

1

1

−≤≤= ∑ ∑−

=

−

=ii

N

K

N

K

nNKjn

NKj

NneeKKXNN

nnxππ

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2D DFT

• Unitary DFT (Symmetric Form): y ( y )

( ) ( ) 10;,1,1

0

1

0

22

2121

21

1

1

2

2

22

21

1

1

−≤≤= ∑∑−

=

−

=

−−

ii

N

n

N

n

nNKjn

NKj

NKeennxNN

KKXππ

( ) ( ) 10;,1,1

0

1

0

22

2121

21

1

1

2

2

22

21

1

1

−≤≤= ∑ ∑−

=

−

=ii

N

K

N

K

nNKjn

NKj

NneeKKXNN

nnxππ

( )

( ) ( )

22

2121

22

21

1

1

21

1, eeNN

KKn

NKjn

NKj

nn =Φ−−

ππ

( ) ( )21 21KK nn ΦΦ= ⇒ separable

( ) 2,1;12

==Φ−

ieN

Ki

i

i

i

nNKj

in

π

where

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( )Ni

i

2D DFT

Note: ( ) 10;2,2, 1121

21

−≤≤⎟⎟⎞

⎜⎜⎛

=NKKKXKKX ππ( )

10;,,

222121 −≤≤⎟⎟

⎠⎜⎜⎝ NKNN

XKKX c

samples of period between [0 2π) of DTFT

DTFT periodic of period 2π

2D DFT of an image:

[0,2π) of DTFT

N1×N22π (N2) ×1052π (N2)

π2D DFT

πmagnitude

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0 2π (N1)π

2D DFT

Note: ( ) 10;2,2, 1121

21

−≤≤⎟⎟⎞

⎜⎜⎛

=NKKKXKKX ππ( )

10;,,

222121 −≤≤⎟⎟

⎠⎜⎜⎝ NKNN

XKKX c

samples of period between [0 2π) of DTFT

DTFT periodic of period 2π

2D DFT of an image:

[0,2π) of DTFT

N1×N22π (N2) ×1052π (N2)

π2D DFT

πmagnitude

2π2π

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0 2π (N1)π

2D DFT

For convenience, use the shorthand notation

aN

jaN

Nj

N eWeWππ 22

−−=⇒=

2D DFT Separable ⇒ can use only 1D DFTs

( ) ( )KXnx ↔

( ) ( ) 1,,0;1 1

0−== ∑

−

=

NKWnxN

KXN

n

KnN K

( ) ( ) 1,,0;1 1

0−== ∑

−

=

− NnWKXN

nxN

K

KnN K

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2D DFT

• Matrix Representation:

where Fi are Ni×Ni unitary matrices and i=1,221

ˆ XFFX T=

1 ⎪⎫⎪⎧

Ki

But: F symmetric ⇒

2,1;1,0;1=−≤≤

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

= iNnKWN iii

nKN

ii

ii

iF

TFF =

ni

But: Fi symmetric ⇒Fi unitary ⇒

⇒

ii FF =**1i

Tii FFF ==−

⎨⎧ =

*21

ˆ XFFX T

Note:

⎩⎨

= *2

*1 XFFX

1,0;12121 −≤≤

⎭⎬⎫

⎩⎨⎧===⇒== NnKW

NNNN Kn

NFFF

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⎭⎩ N

2D DFT

• With the DFT, linear shifts and linear convolutions are replaced by circular shifts and circular convolutions

• Circularity is denoted using modulo operator Notation: (n)N = n modulo N( )N

If r = (n)N then 0 ≤ r ≤ N-1Note: (-n)N = (N-n)N

Circular shift:

Circular convolution

( ) ( )( ) ( )2122211 ,, 22

2

11

11KKXWWmnmnx Km

NKm

NNN ↔−−

same as linear convolution but shifting corresponds to circular shift and flipping corresponds to circular reverse (-n)N = N-n

( ) ( ) ( ) ( )2121212121 ,,,, KKHKKXnnhnnx NN ↔×

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2D DFT

• Conjugate symmetry for real images

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

( )21 21

*21

21*

2121

,,

,,real,

NN KKXKKX

nnxnnxnnx

−−=⇔

=⇔

( ) ( ) ( )( )21 2211

*21 ,, NN KNKNXKKX −−=⇔

pick the >0 index lying in period 0,…,Ni-1

⎞⎛⎞⎛

22

211

1 2,

2KNKKNK −→−→Replace

2,1;102

,22

,2 2

21

1*2

21

1

=−≤≤

⎟⎠⎞

⎜⎝⎛ ++=⎟

⎠⎞

⎜⎝⎛ −−⇔

iNK

KNKNXKNKNX

ii

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,;ii

2D DFT⇒ Conjugate symmetry about ⎟⎟

⎠

⎞⎜⎜⎝

⎛22,

21 NN

N2-1

⇒ ( )2,1 KKXhalf of the coefficients are redundant and need not be stored

⎟⎠⎞

⎜⎝⎛

2,

221 NN

The DFT is a fast transformDFT separable ⇒ Transformation can be performed using (N1+N2) 1D DFTs for an N1×N2 image

N1-10

DFTs for an N1×N2 imageEach 1D DFT of size N requires complex multiplications and adds (CMADs) using an FFT implementation⇒ DFT can be performed in

NN2log

2

( ) tilIf 2 NNONNN ⇒

( )2122121221

DFT columnsmatrix

121

2

DFT rowsmatrix

222

1 loglog2

log2

log2

21

NNNNONNNNNNNNNN

NN

⇒=+44 3442144 344 21

operations

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( )operationslog If 22

21 NNONNN ⇒==

The 2D Discrete Cosine Transform (2D DCT)e sc ete Cos e a s o ( C )

• Advantages:Real transform (for real images)Real transform (for real images) Coefficients nearly uncorrelated Excellent Energy Compaction

• There are different types of DCTs but here we are interested in• There are different types of DCTs, but here we are interested in a unitary transform type, where the transformation matrices are such that: C1=C2 ⇒ Type IV

( ) ( ) ⎞⎛⎞⎛1 1N N

( ) ( ) ( ) ( ) ( ) ( )∑∑−

=

−

=⎟⎟⎠

⎞⎜⎜⎝

⎛ +⎟⎟⎠

⎞⎜⎜⎝

⎛ +=

1

0

1

02

2

21

1

122112121DCT

1

1

2

22

12cos2

12cos,,N

n

N

nK

NnK

NnKCKCnnxKKX ππ

01 ⎫⎧ K( ) 2,1;10;

0;2

0;=−≤≤

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

≠

=

= iNKK

N

KN

KC ii

i

ii

iiwhere

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⎪⎭⎪⎩ Ni

2D DCTC

( ) ( ) ( ) ( ) ( ) ( )∑ ∑−

=

−

=⎟⎟⎠

⎞⎜⎜⎝

⎛ +⎟⎟⎠

⎞⎜⎜⎝

⎛ +=

1

0

1

02

2

21

1

1221121DCT21

1

1

2

22

12cos2

12cos,,N

K

N

KK

NnK

NnKCKCKKXnnx ππ

Inverse and Forward DCT are the same DCT separable ⇒ can use 1D transforms

• 2D DCT of an imagegN1×N2

×102π (N2)

2D DCT

magnitude

Note: A real transform only goes half way around the Unit Circle 0 π (N1)

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y g y

2D DCTC

( ) ( ) ( ) ( ) ( ) ( )∑ ∑−

=

−

=⎟⎟⎠

⎞⎜⎜⎝

⎛ +⎟⎟⎠

⎞⎜⎜⎝

⎛ +=

1

0

1

02

2

21

1

1221121DCT21

1

1

2

22

12cos2

12cos,,N

K

N

KK

NnK

NnKCKCKKXnnx ππ

Inverse and Forward DCT are the same ⇒ DCT separable ⇒ can use 1D transforms

• 2D DCT of an imagegN1×N2

×102π (N2)

2D DCT

magnitude

Note: A real transform only goes half way around the Unit Circle 0 π (N1)

ππ

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y g y

2D DCTC

• Disadvantage:The DCT does not have nice convolution and modulation properties.

• DCT can be related to DFT and computed via FFTs ⇒ DCT related to DFT of symmetric extension of a sequence

⇒ ( ) image. afor neededoperationslog 2121221 NNNNNNO ×

⇒ Fast transform.

( ) gpg 2121221

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2D DCTC

• Matrix Representation

1001 ⎫⎧ NK

( ){ }KnCii ,=C

( ) ( ) 2,1;

1011

;2

12cos2

10;0;1

, =

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

≤≤−≤≤

⎟⎟⎠

⎞⎜⎜⎝

⎛ +

−≤≤=

= i

NnNK

KNn

N

NnKN

KnCi

ii

i πwhere

Ci unitary and real ⇒ Ci orthogonal

102 ⎪⎭⎪⎩ −≤≤⎟⎠

⎜⎝ NnNN iii

Ci real and orthogonal ⇒ C-1 = CT and C = C*

TT

T

**

21

ˆˆ

ˆ

CXCCXCX

XCCX =

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2121 CXCCXCX ==

2D DCTC

Note:if N1 = N2 = N ⇒ C1 = C2 = C orthogonal

T XCCX̂ =

• Interpretation of DCT

TCXCX ˆ=

pDFT ⇒ periodic extension of sequencesDCT ⇒ symmetric and periodic extension of sequences (main period symmetric) y )

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2D DCTC

• Consider an image

g h I

d e f

N2-1

n2x(n1, n2)

Symmetrically extend image about N1-1 and N2-1 axes

a b cN1-1

n10

n2

2N2-12

y(n1, n2) = x(n1, n2) + x(2N1-1-n1, 2N2-1-n2) + x(2N1-1-n1, n2) + x(n1, 2N2-1-n2)

a b c

d e f

g h i

c b a

f e d

i h gg h i

d e f

a b cn1

i h g

f e d

c b a

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2N1-1n10

2D DCTC

Now, take DFT of y(n1, n2) → note that this is a 2N1×2N2 point DFT⇒ DCT can be expressed in terms of DFT⇒ Can use FFT to make a fast DCT implementation with

operations for an N1×N2 image( )21221 log NNNNO• Compare DFT to DCT

N1×N2π (N2)×102N1×N22π (N2)2π (N2)

×105

π

00 2 ( )

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0 π (N1)0 2π (N1)π

2D DST (Discrete Sine Transform)S ( sc ete S e a s o )

• Definition:

( ) ( ) ( )( ) ( )( )∑∑−

=

−

=⎟⎟⎠

⎞⎜⎜⎝

⎛++

+⎟⎟⎠

⎞⎜⎜⎝

⎛++

+++=

1

0

1

022

211

121

2121DST

1

1

2

2

111

sin111

sin,1

21

2,ˆN

n

N

nKn

NKn

Nnnx

NNKKX ππ

• Properties similar to DCT (separable, real, fast transform) but also symmetric ⇒ faster transform

⎞⎛( ) ( )( )

( ) T

i

ii

iin

n

iN

KnN

Ki

SS =⇒Φ=

=⎟⎟⎠

⎞⎜⎜⎝

⎛+

+++

=Φ 2,1;1

11sin1

1 π

• Very good energy compaction but less than DCT

( )iK ni

SS =⇒Φ=

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2D DSTS

• Inverse and forward transforms are the same

( ) ( ) ( )( ) ( )( )∑∑−

=

−

=⎟⎟⎠

⎞⎜⎜⎝

⎛++

+⎟⎟⎠

⎞⎜⎜⎝

⎛++

+++=

1

0

1

022

211

121DST

2121

1

1

2

2

111

sin111

sin,ˆ1

21

2,N

K

N

KKn

NKn

NKKX

NNnnx ππ

• Matrix representation

2121ˆ XSSXSSX == T S symmetric

21ˆ SXSX = since unitary S1 and S2 real and symmetric

subindex corresponds to n1, K1

subindex corresponds to n2, K2

( ){ } ( ) ( ) 2,1;1,0;111sin

12, =−≤≤⎟⎟

⎠

⎞⎜⎜⎝

⎛+

++

+== iNKnK

Nn

NKnS i

iiii

πSwhere

directions directions

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2D Discrete Hadamard Transformsc ete ada a d a s o

Based on a matrix description (Typically defined in terms of a matrix)Very efficient: matrix elements (and thus elements of basis vectors) take only the binary values ±1 ⇒ well-suited for DSP ⇒can be implemented without multiplierscan be implemented without multipliers Symmetric transform

• Definition:F N N i ( ) ith N 2n ( f 2)For an N×N image x(n1,n2) with N = 2n (power of 2)

H is also real and orthogonal ⇒ H -1 = H *T = H T

NnnnnTn 2log;ˆ === XHHXHHX

Hn is also real and orthogonal ⇒ Hn-1 = Hn

T = HnT

where Hn are N×N matrices where N = 2n and usually n=1,2,3 ⇒ work on image blocks of size 2×2 4×4 8×8 respectively

nn HXHX ˆ=

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image blocks of size 2×2, 4×4, 8×8, respectively

2D Discrete Hadamard Transformsc ete ada a d a s o

Hn generated using the Kronecker product and the core matrix

⎥⎦

⎤⎢⎣

⎡−

=11

112

11H

by the following recursion

⎥⎦

⎤⎢⎣

⎡−

=⊗=⊗= −−−−

11

111111 2

1 nnnnn HH

HHHHHHH

Example:

⎦⎣ −− 112 nn HH

⎥⎤

⎢⎡ 1111

⎥⎥⎥⎥

⎦⎢⎢⎢⎢

⎣ −−−−−−

=⎥⎦

⎤⎢⎣

⎡−

=

111111111111

21

21

11

112 HH

HHH

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⎦⎣ 1111

2D Discrete Hadamard Transformsc ete ada a d a s o

• Frequency interpretationLets look at the 4×4 H2 matrix:

⎥⎤

⎢⎡

11111111

1

low frequency basis vector

hi h f b i t

Sequency = # of sign changes

0

3

⎥⎥⎥⎥

⎦⎢⎢⎢⎢

⎣ −−−−−−

=

111111111111

21

2Hhigh frequency basis vector

mid frequency basis vector

mid frequency basis vector

3

1

2

Recall that for sinusoidal signals, frequency can be defined in terms of zero crossing or sign changes The basis vectors of a Hadamard matrix can be obtained by sampling a class of functions called the Walsh functions which can take only the binary values ±1 and which form a complete orthonormal basis in L2 ⇒ Hadamard transform also called Walsh-Hadamard transform

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2D Discrete Hadamard Transformsc ete ada a d a s o

Sequency = number of zero crossings of a Walsh function = transitions (sign changes) in a basis vector of Hn

Generally, we order the basis vectors (row or columns since H=HT) in terms of increasing sequency ⇒ This is called Walsh-ordered H d d T f (if b i t d d)Hadamard Transform (if basis vectors ordered)

⎥⎥⎤

⎢⎢⎡

−− 11111111

1

⎥⎥⎥

⎦⎢⎢⎢

⎣ −−−−

=

111111111111

21

2H

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2D Discrete Hadamard Transformsc ete ada a d a s o

• Transform interpretation

Original XH

transformedH⇒

π

N i l ti d d l ti ti b t f t

π0

• No nice convolution and modulation properties, but very fast transform.

• Hadamard expressed in terms of Kronecker products and also t i b i l t d i ( )as sparse matrices ⇒ can be implemented in

additions and subtractions. ( )NNO 2log

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