discrete image processing

8/19/2019 Discrete Image Processing

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Chapter 03 Image Transforms:

Introduction to Unitary Transform, DFT,

Properties of 2-D DFT, FFT, IFFT,

a!sh transform,"adamard Transform,

Discrete Cosine Transform,

Discrete a#e!et Transform

1


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$a%0&%''( Using Fast "adamard transform, find ) *+n for *n. /1, 2, 2, 1

'g%'2%'0( Pro#e that 2-Dimensiona! DFT matri is unitary matri

1c%'2%'0( Deri#e Fast a!sh Transform f!o graph for 4/1

3a%'2%'0( 5et *n. / 2 δ + n 6 3 δ + n- + 4 δ + n- 2 + $ δ + n -3

Find 1 point DFT using FFT f!o graph

2c%'2%'0( 7i#en f * , y . / h *, y. /

find !inear con#o!ution of input image f *, y.

&( 5et + n 8e 1 point se9uence ith ) + / ', 2, 3, 1

Find FFT of the fo!!oing se9uence using ) + and not otherise

*' . p + n / * -'.n + n * 2. 9 + n / + -n 6'

$ & ;

< = >'0

' 2

3 1

2


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Discrete Image Transformations:

?part from DFT, a num8er of !inear transformations can 8e used for

image processing

Image transform :

@efers to a c!ass of unitary matrices used for representing

images

?n image can 8e epanded in terms of

a discrete set of 8asis arrays ca!!ed 8asis images

5inear Transformations:

'( Ane Dimensiona! Discrete 5inear Transformations

2( Unitary and Arthogona! Batrices

3( To dimensiona! Discrete 5inear Transformations

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Ane Dimensiona! Discrete 5inear Transformations

if is a 4 ( ' #ector N rows, 1 column

T is a 4 ( 4 matri

Then yi / ∑ ti ( ) i 0, ', 2, ( ( ( , 4-' rows

E / T (

y0 t00 t0' ( ( T0 n-' 0 *y0 / t00 0 6 t0'( ' 6 ( ( 6 t0n-'n-'.

y' t'0 t'' ( ( t' n-' '

( / ( ( ( ( (

( ( ( ( ( (

yn-' tn-'0 ( ( ( tn-' n-' n-'

E defines !inear transformation of

- It is ca!!ed !inear transformation

as it is formed 8y the first order summation

of input e!ements - ach e!ement of y,

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Unitary and Arthogona! Batrices:

y / T( represents !inear transformation

There can 8e infinite num8er of transformation matrices T,

for a8o#e e9uation.

Common!y used transformation matrices +T 8e!ongs to

a c!ass of matrices that ha#e certain properties

If T is a unitary matri, thenT G / T (conjugate transpose

matrix inverse)

T* indicate complex conjugate o matrix ! (imaginar" part negated)

? indicate transpose o a matrix #, write rows as columns or columns as rowstranspose o matrix # $ #

! indicates inverse o a matrix ! suc% t%at ! . ! $ & (identit" matrix)

& identit" matrix $ 1 ' ' ' n x n suare matrix wit% 1s in main diagonal ' 1 ' ' and eros elsew%ere

′

-'

i, j j, i11

′


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-"mmetric matrix

& transpose o matrix is eual to t%e /atrix

0! $ 0

ort%ogonal matrix

!%e matrix w%ose column and rows are

ort%ogonal unit vectors

or w%ose transpose is eual to its inverse

0!

$ 01


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matri

is unitary matri, Then

T G / T

!ying 8y 8oth sides 8y T, T ( TG / T ( T

t T ( T / I here I is identity matri

us T is a Unitary matri if T( T G / I , identity matri

(/atrix ! multiplied " conjugate, transpose o matrix ! $ &dentit"

find hether matri T is unitary matri, chec if T ( TG / I

T is unitary and it has on!y rea! e!ements,en its orthogona! matri is same as T, or TG / T

if T is unitary matri, then T / T

u!tip!ying 8oth sides 8y T: T ( T / T ( T / I identity

′ -'

′

-'

-'

′

′

′

′

-'

′

-'


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c hether the one dimensiona! DFT matri is an eamp!e

nitary transform

urier Transform F / ( f here f is the input se9uence

is the DFT matri

f 4/ 1

' ' ' '

T matri / ' -'

' ' ' -'

' -' -

matri is unitary matri if ( G / I

multiplied " complex conjugate, transpose o euals & , identit" matrix)

( G

' ' ' ' ' ' ' 1 0 0 0

-' ( ' -' - / 0 1 0 0 / 1 + I

' ' -' ' -' ' -' 0 0 1 0 -' - ' - -' 0 0 0 1

ce DFT is an eamp!e of a unitary transform

1, indicates that DFT is not norma!iHed

is input se9uence in co!umn, then '-D DFT can 8e computed 8y F / f

′

′

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To Dimensiona! Discrete 5inear Transformations:

Input image: f *4 4.

Transformed matri F * also N x N.

4-' 4-'

Fm, n / ∑ ∑ f i, *i, , m, n.

i /0 /0

here i, , m and n are thediscrete #aria8!es, of range 0 to 4-'

*i, , m, n. can 8e thought of as an 42 42 8!oc matri

ha#ing 4 ros of 4 8!ocs,

each 8!oc is 4 4 su8matri

The 8!ocs are indeed 8y m, n

and the e!ements of each 8!oc 4 4 su8matri are indeed 8y I,

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n 0 ' ( ( n /4-'

m

0 + I, + I, ( ( ( + I,

' + i, + i, ( ( ( + i,

( (

( (

m / 4-' + i, + i, ( ( ( + i,

The transformation *i, , m, n. is ca!!ed separa8!e,

- if it can 8e separated into the products of ro-ise and

co!umn-ise component functions

*i, , m, n. / Tro *i, m. ( Tco! *, n.

- Thus the transformation can 8e carried out in to steps:

- @o-ise transformation operation fo!!oed 8y

co!umn-ise transformation operation

4-' 4-'

Fm, n /∑

+ ∑

f i, Tro *i, m. Tco !*, n. i /0 /0

N xN sumatrices

1'


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Further If the to components Tco! and Tro are identica!,

- The transformation is ca!!ed symmetric, (colums $ rows )

(matrix is eual to its transpose)

*i, , m, n. / Tro *i, m. ( Tco! *, n.

4-' 4-'

Fm, n / ∑ T *i, m. + ∑ f i, Tro *, n.

i /0 /0

F / T f T

"ere Transformation T is 8oth separa8!e and symmetric

The DFT matri is 8oth symmetric a e!! as separa8!e

hence Transform can 8e o8tained 8y using:

F / T( f( T

This is standard formu!a for computing transform of a 2D signa!

′

′

11


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Find the DFT of the gi#en image

0 ' 2 '

' 2 3 2

2 3 1 3

' 2 3 2 ' ' ' '

F / T( f(T Kince f is 1 1 matri T / ' - -' T is symmetric

' -' ' -'

' -' -

T f T

' ' ' ' 0 ' 2 ' ' ' ' 'F / ' - -' ' 2 3 2 ' - -'

' -' ' -' 2 3 1 3 ' -' ' -'

' -' - ' 2 3 2 ' -' -

1 < '2 < ' ' ' '

/ -2 -2 -2 -2 ' - -' 0 0 0 0 ' -' ' -'

-2 -2 -2 -2 ' -' -

32 -< 0 -<

/ -< 0 0 0

0 0 0 0-< 0 0 0

12


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FFT Fast Fourier Transform

- ) *. , The Discrete Fourier Transform *DFT. of *. is gi#en 8y:

4-'DFT ) + /

∑

*n. 4 0 ≤ ≤ 4-'

n /0

4 / e 2π % 4 (twiddle actor)

The In#erse Fourier Transform *IDFT. of Fourier transform ) *.:

*n. is gi#en 8y:

4-' IDFT *n. / '%4 ∑ ) *. 4 0 ≤ n ≤ -'

/ 0

Direct computation of DFT re9uires a8out 42 comp!e mu!tip!ications

Fast Fourier Transform *FFT. refers to computation of DFT using fast a!gorithms:

- Taes a8out 4 !og2 4 computations

4 42 4 !og2 4 4 / num8er of samp!es in an image

32 '021 '&0 42 / 4um8er of computations for DFT

'2< '&,3


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To common!y used Fast Fourier Transform *FFT. a!gorithms:

'( DIT-FFT Decimation-in-time (7ecimate means 8ill)

2( DIF- FFT Decimation-in-fre9uency rearrange in time)

Decimation-in-time DIT-FFT

- @e-arranging the time signa!

- p!oits the symmetric and periodic properties of

the tidd!e factor 4 : * 4 / e 2

π

% 4 )

4 / - 4 Kymmetric

4 / 4 Periodic

These to properties reduce the num8er of computations

* 6 4%2 .

6 4

8 9j 2 π 8: NN $ e

N $ e ( e8 + N:2 9j 2 π 8: N j2 π N : 2N

$ e ( e 9j 2 π 8: N j π

$ N ( 1 $ N8

N

8 + N $ vvN

e 9j2 π( 8+ N): N $ N

e 9j2 π 8:N . N

e 9j2 π $ N

8

8

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- The DFT e9uation is :

4-'

) + /∑

*n. 4 / 0, ', 2, ( ( ( 4-'

n /0

here 4

/ e 2π % 4

Decimation-in-time: sp!its the input *n. into odd and e#en parts:

) + /∑

*n. 4 6 ∑ *n. 4

n e#en *0, 2, 1( . n odd *', 3, $

4%2 -' 4%2 -'

) + /∑

*2n. 42 n( 6

∑

*2n6'. 4*2n6'.(

n/0 n/0

4%2 -' 4%2 -' ) + /

∑

*2n. 4 6 ∑ *2n6'. 4 ( 4n/0 n/0

4o 4 / 4%2 N $ e $ e $N:2

2n8 $ 8

n(

n( n(

2n( 2n(

2 2 9j 2. 2 π : N j 2 π : (N :2)

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4%2 -' 4%2 -' ;ot% are

) + / ∑ *2n. 4%2 + 4 ( ∑ *2n6'. 4%2 N:2 point

n/0 n/0 7

4%2 -' 4%2 -'

5et F'*. / ∑ *2n. 4%2 F2*. / ∑ *2n6'. 4%2 n/0 n/0

) *. / F'*. 6 4 ( F2*.

) *. / F'*. 6 4 ( F2*. for / 0, ', 2( ( 4%2 -'

n n

n( n(

1


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F'*. and F2*. are periodic ith period 4%2

'*64%2. / F'*. and

2*64%2. / F2*.

4 *64%2. / - 4

e get ) *. / F'*. 6 4 F2*. / 0, ', 2 ( ( (

)* 6 4%2. / F'*. - 4 F2*.

. and F2*. re9uires *4%2.2 comp!e mu!tip!ications, each(

4%2 additiona! mu!tip!ications are re9uired for mu!tip!ying

the tota! mu!tip!ications re9uired are 2 ( *4%2.2

6 2(4%2as com ared to 42

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) * . / F' *. 6 4 F2*. *'.

)*6 4%2. / F'*.- 4 ( F2*. for / 0 to 4%2 -' *2.

For 4 / < , i!! #ary from 0, ', 2, 3 *


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The e9uations *'. and *2. can 8e shon using a signa! f!o graph

for 4/ < as fo!!os:

*0. ------ )+0

*2. ------ )+'

*1. ------ )+2

*&. ------ )+3

N:2 point

7

4 point

7

N1

N2

N

3

N'

N1

N2

N3

&nput split

=ven:odd

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Kince F'*. and F2*. are 4%2 point DFTLs,

These can 8e further sp!it into odd and e#en parts

F'*. / 7'*. 6 4%2 72*.

F'*64%1. / 7'*. - 4%2 72*. /0, ', 2 ( (, *4%1 -'.

F2*. / "'*. 6 4%2 "2*.

F2*64%1. / "'*. - 4%2 "2*.

For 4 /


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x(') ------ )+0

x(4) ------ )+'

x(2) ------ )+2

x() ------ )+3

>1(')

>2(1) N:4

?oint7

@1(')


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# 2-point DFT is dran as,

1

+1

"ence the fina! f!o graph is shon as fo!!oing

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x(') )+0

x (4) )+'

x (2) )+2

x () )+3

>1(')


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Mit-re#ersa!

The order of input data can 8e o8tained 8y re#ersing 8its of

8inary representation

000 000 *0.

00' '00 *1.

0'0 0'0 *2.

0'' ''0 *&.

'00

00' *'.'0' '0' *$.

''0 0'' *3.

''' ''' *;.

24


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Decimation in fre9uency DIF-FFT

Decimation in fre9uency imp!ies retaining the order of input se9uence

8ut

getting an output hich is shuff!ed up

In this first sp!it up input se9uence into the first 4%2 data points and

the second 4%2 data points 4-'

DFT ) + / ∑ *n. 4

n N / 0, ', 2, ( ( ( 4-'

n /0

4 / e 2 π % 4

Kp!it into First ha!f and second ha!f parts:

4%2 -' 4 -'

) + /∑

*n. 4n +∑

*n . 4n n/0 n/4%2 (n $ ' to N:2 1, n $ N:2 to N1)

Ku8stituting n / n 6 4%2

4%2 -' 4%2 -'

) + /∑

*n. 4

n +∑

*n 64%2 . 4

*n 6 4%2.

n/0 n/02


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Ko!#ing a8o#e e9uation,

4%2 -'

) +N / ∑ + *n. 6*-'. 6 *n 64%2. 4n

n/0

Kp!it ) *N. into odd% e#en partsO

4%2 -'

) +2N / ∑ + *n. 6 *n 6 4%2. 42n O =ven part

n / 0

4%2 -'

) +2N6' / ∑ + *n. *n 6 4%2 . 4 n*26'. Add part

n / 0

4%2 -'

) +2N / ∑ + *n. 6 *n 6 4%2. 4%2n O N / 0, ', ( ( *4%2 - '. *'.

n / 0

4%2 -'

) +2N6' / ∑ + *n. *n 6 4%2 . 4%2 n ( 4%2

n O N / 0, ', ( ( 4%2 - ' *2.

n / 0

g1

g2

2


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x(') N:2 ?oint

7

(4 point

7

g1(') BC'D

x(1)

x(2)

x(3)

g1(1)

g1(2)

g1(3)BC4DBC2D

BC)D

x(4)

g2(')

BC1Dx()

x()

x()

g2(1)

g2(2)

g2(3)

N:2 ?oint7

(4 point

7

BCD

BCD

BC3)

1

1

1

1

wN'

wN1

wN2

wN3

Kigna! f!o diagram:

"ere input order is retained,

8ut the output is shuff!ed

B(') + x(4)

8 $', g1(') $ x (') + x (4)

g2(') $ x (') 9 x (4)

B(') x(4)

25

9uations 3 and 1 represent to 4%2 point DFT


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9uations 3 and 1 represent to 4%2 point DFT

Kince 4/


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x(') B(')

x(1)

x(2)

x(3)

B(4)B(2)

B()

x(4) B(1)x()

x()

x()

B()

B()

B(3)

1

1

11

Fina!,


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Compute DFT of the image using DIT DFT

0 ' 2 '

' 2 3 2

2 3 1 3

' 2 3 2

Kince image is 1 1, e need to use 1-poit FFT 8utterf!y

e first or a!ong the ros, finding DFT for each ro

then a!ong the co!umns, DFT for each co!umn

(')

(2)

(1)

(3)

>(')

@(')

>(1)

@(1)

1

1

1

1

>(') $ (') + (2) $ ' + 2 $ 2

>(1) $ (') (2) $ ' 2 $ 2

@(') $ (1) + (3) $ 1 + 1 $ 2@(1) $ (1) (3) $ 1 1 $ '

4'

41

4'

41

BC'D

BC1D

BC2D

BC3D

First @o E('), (1), (2), (3)F $ E', 1, 2, 1F

BC'D $ >(') + 4' @(') $ 2 + 1.2 $ 1

BC1D $ >(1) + 41 @(1) $ 2 j.' $ 2

BC2D $ >(') 4' @(') $ 2 9 1.2 $ 0

BC3D $ >(1) 41 @(1) $ 2 j.' $ -2


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Ko FFT of the first ro ) +N / 1, -2, 0, -2

simi!ar!y FFT of remaining three ros can 8e determined:

1 -2 0 -2 FFT of 'st

ro


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Discrete Cosine Transform *DCT.

DCT uses on!y Cosine a#es

(Ane Dimensiona! DCT of a se9uence f *., 0≤

≤

4-'

4-'

F *u. / *u. ∑

f *. cos +π

*2 6'.( u % 24 O 0≤

u≤

4-'

/0

here

*0. $√

'%4O for u / 0

*u. $√

2%4 for '≤

u≤

4-'

In#erse transform IDCT

4-'

f *. /∑

*u. F *u. cos +π

*2 6'.( u % 24O 0≤

≤

4-'34


35/85

D DCT pair:

4-' 4-'

, #. / *u.( *#. ∑

∑

f *, y . cos +π

*2 6'.(u % 24 ( cos +π

*2y 6'.(# % 2

/0 y /0 For u, # / 0, ', 2, ( ( , 4-'

rse DCT transform IDCT

4-' 4-'

y. /∑

∑

*u. *#. F *u, #. cos +π

*2 6'.( u % 24( Cos +π

*2y 6'.( # % 24

u /0 # /0

For , y / 0 to 4-'

3


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The 4 4 Cosine Transform Batri C / C *u, #. defined asO

√ '%4 u /0, 0 ≤ # ≤ 4-'

C *u, # . /

√

2%4 ( Cos +π

*2# 6'.( u % 24 '≤

u≤

4-', 0≤

#≤

4-'

In this Cosine transform Batri is rea! and orthogona!

8ut 4ot Kymmetric (transpose o matrix is

not eual to t%e matrix)

(ort%ogonal matrix w%ose column and rows are ort%ogonal unit vectors or

its transpose is eual to its inverse)

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Cosine transform Batri is rea! and orthogona!

( Not -"mmetric)

C / CG (#s onl" real, conjugate is same)

C-' / C *From T(T G / I unitary matri e9uation.

C ( C / I identity matri

The DCT of a co!umn se9uence f *., 0≤

≤

4-' can 8e computed

F / C ( f *7

Kince DCT is not symmetric

2-D DCT can 8e computed 8y

F / C f C (7

3


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Find DCT of the se9uence: ', 2, 1, ;

F / C ( f

√ '%4 ( u /0, 0 ≤ # ≤ 4-'

C *u,# . /

√ 2%4 ( Cos +π *2# 6'.( u % 24 ' ≤ u ≤ 4-', 0 ≤ # ≤ 4-'

Compute C

Kince 4 / 1 C i!! 8e 4 4 / 1 1 matri

u # 0 ' 2 3

0 0($ 0($ 0($ 0($

' 0(&$3 0(2;0$ -0(2;0$ -0(&$3

2 0($ -0($ -0($ 0($

3 0(2;0$ -0(&$3 0(&3$ -0(2;0$

C is not symmetric

√ 1:N or (u$', v$' to N1) $ √ H $ I $ '.√ 2:N . Jos Cπ (2v +1). u : 2ND or (u$1, v$') $ √ 2:4 . cos π:5 $ '.3

or (u$2, v$') $ √ 2:4 .cos π: 4 $ '.

35

F / C ( f f / ', 2, 1, ;


39/85

F / C ( F( C

C f C (!ranspose)

0($ 0($ 0($ 0($ 2 1 1 2 0($ 0(&$3 0($ 0(2;0$

/ 0(&$3 0(2;0$ -0(2;0$ -0(&$3 ( 1 & < 3 ( 0($ 0(2;0$ -0($ -0(&$3

0($ -0($ -0($ 0($ 2 < '0 1 0($ -0(2;0$ -0($ 0(&3$

0(2;0$ -0(&$3 0(&3$ -0(2;0$ 3 < & 2 0($ -0(&$3 0($ - 0(2;0$

C f F

0($0($ 0($ 0($ ' ;

0(&$3 0(2;0$ -0(2;0$ -0(&$3 ( 2 / -1(1$=

0($ -0($-0($0($ 1 '

0(2;0$ -0(&$3 0(&3$ -0(2;0$ ; -0(3';0

"ence DCT of the gi#en se9uence f is ;, *-1(1$=., ', *-0(3';0.

Find the DCT of a 1 1 image 2 1 1 2

1 & < 3

2 < '0 13 < & 2

'= -0(2;0$ -< 0(&$3

F / -2(&='3 -0(21=< 2(30


40/85

The Nronecer Product

For to matrices ? and M

The Nronecer Product ? M is gi#en 8y

mu!tip!ying M ith each e!ement ai of ? and

su8stituting the mu!tip!ied matrices, aiMfor the e!ements ai of ?

Thus if / ? M, then

/ If ? / M /

? M / /

a''M a'2M ( ( a'nM

a2'M a22M ( ( a2nM ( ( ( (

an'M an2M ( annM

2 $' 3

' 23 $

2 $

' 3

2 $

' 3

2 $

' 3

2 $

' 3' x 2 x

3 x $ x

2 $ 1 '0

' 3 2 & & '$ '0 2$

3 = $ '$

4'


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mard Transform

amard transform is 8ased on the "adamard matri

ch is a s9uare array ha#ing entries of 6' and -' on!y

amard matri of order 2 is gi#en 8y:

"*2. / a!so ritten as

s and co!umns of "adamard matri are orthogona!O

thogona!ity of #ectors, the dot product of #ectors has to 8e Hero

The first ro + ' ', The second ro + ' -'

product / + ' ' ( + ' -' / +' ' ( ' / '(' 6'(-' / 0

-'

The first co!umn The second co!umn

+ ' ' / / 6

' '

' -'6 6

6 -

'

-'

'

'

'

-'

'

'

'

-'41


42/85

For unitary transformation,

Batri in#erse ?-' / ? G*transpose row to col, comjugate)

? rea! unitary matri is ca!!ed orthogona! matri(

For such matri ? -' / ? ( /atrix inverse $ transpose)

Ko for checing if a matri ith rea! #a!ues on!y, is orthogona!

the ? -' / ? or ? ( ?-' / ? ?

or ? ? / I ( identit" matrix)

Is "adamard matri orthogona!

"*2. / ? / ? / * # transpose row to col.)

?( ? / / / 2 / 2 + I

' '

' -'

' '

' -'

' '

' -'

' '

' -'

2 0

0 2

' 0

0 '

Thus "adamard matri is orthogona!, 8ut since e get constant 2, it means

that It is not norma!iHed

4ormaiHed 2 2 "adamard transform, orthogona! matri is gi#en 8y

"4*2. / '%√2 ' '

' -' 42


43/85

Is norma!iHed "adamard matri orthogona!

"4*2. / '%√2 ? / '%√2

?( ? / / / +I

Therefore

"4*2. / '%√2

is ca!!ed norma!iHed 2 2 "adamard matri

' '

' -'

' '

' -'

'%√2 '%√2

'%√2 -'%√2

'%√2 '%√2

'%√2 -'%√2

' 0

0 -'

' '

' -'

43


44/85

"adamard matrices of the order 2n can 8e recursi#e!y generated

"*2n. $ "*2. ( "*2n-'.

Therefore "*2. / "*2'. " *20.

simi!ar!y "1 / *"2. "*2.

From Nronecer Product :

/ ( "*2. / '( +"*2. '( +"*2.

'( +"*2. -'( +"*2.

' ' ' ' 0

' -' ' -' 3

' ' -' - ' '

' -' - ' 6' 2

"*1. /

-ign c%anges

Is "*1. "adamard matri is norma!iHed and orthogona!

? ( ? / +i / 1 + I 1 ' 0 0 0

0 ' 0 0

0 0 ' 0

0 0 0 '

' '

' -'

44


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" *1. is norma!iHed and orthogona!: Chec ? ( ? / '

' ' ' ' ' ' ' '

" *1. / ? ' -' ' -' ? / ' -' ' -'

' ' -' -' ' ' -' -'' -' -' ' ' -' -' '

' ' ' ' ' ' ' ' 1 0 0 0

? ( ? / ' -' ' -' ( ' -' ' -' / 0 1 0 0 / 1 + I

' ' -' -' ' ' -' -' 0 0 1 0

' -' -' ' ' -' -' ' 0 0 0 1

"*1. is orthogona!, 8ut not norma!iHed:

' 0 0 0

4orma!iHed "*1. / ' √ 1 0 ' 0 0

0 0 ' 0

0 0 0 '

4


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"ence 4orma!iHed 1 1 "adamard transform is gi#en 8y

' ' ' '

' -' ' -'

' ' -' -'' -' -' '

Kince "*2n. $ "*2. x "*2n-'.

"*


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Using e9uation: ? ( ? / + I

"*


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@adamard signal representation is similar to


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If *n. is 4 point one dimensiona! se9uence of finite #a!ued rea! num8ers

arranged in co!umn, then

the "adamard transformed se9uence is gi#en 8y :

) / T (

) +n / + " *4.( *n. here *n. is a co!umn #ector

"*4. is a 4 4 "adamard matri here 4 is the num8er of data points

The in#erse "adamard tranform is gi#en 8y:

*n. / '% 4 +" *4. ) +n

here *n. is a co!umn #ector

46


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'2th ?ugust

'


51/85

Compute the "adamard Transform of the data se9uence ', 2, 0, 3

"ere 4 /1

" *4. is a 1 1 matri

' ' ' '

' -' ' -'

' ' -' -'

' -' -' '

) +n / +" *4.( *n.

'

2

0

3

' 62 60 63

' 2 60 -3

' 6 2 -0 -3

' -2 -0 63

&

-1

0

2

)+n / /( /

" *4. ( *n. / ) +n

1

Compute the In#erse "adamard Transform of the data se9uence & 1 0 2


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Compute the In#erse "adamard Transform of the data se9uence &, -1, 0, 2

"ere 4 /1

The in#erse "adamard transform is:

*n. / '% 4 ( " *4. ( ) +n

' ' ' '

' -' ' -'

' ' -' -'

' -' -' '

&

-1

0

2

*n. / '%1

& -1 60 62

& 61 60 -2

& -1 -0 -2

& 61 -0 2

1

<

0

'2

'

2

0

3

/ '%1 /( / '%1

'% 4 "*4. ( ) +n

"ence the data se9uence is ', 2, 0, 3

2

" d d t f ti f 2 D f i 4 4


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"adamard transformation for a 2-D se9uence of siHe 4 4

If f is 4 4 image

F is transformed image

The "adamard transform is gi#en 8y : F / T( f ( T / + " *4. ( f ( " *4.

Compute "adamard transformation of the image: 2 ' 2 '

' 2 3 22 3 1 3

' 2 3 2

The image is 1 1

F / "*1. ( f ( " *1.

' ' ' ' 2 ' 2 ' ' ' ' '' -' ' -' ' 2 3 2 ' -' ' -'

F / ' ' -' -' ( 2 3 1 3 ( ' ' -' -'

' -' -' ' ' 2 3 2 ' ' -' -'

& < '2 < ' ' ' ' 31 2 -& -&

2 0 0 0 ' -' ' -' 2 2 2 23

Fast "adamard Transform


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Fast "adamard Transform

in genera! "adamard transform in#o!#es 4 *4-'. additions

For 4/2 4um8er of addition / 2(' / 2

4/1 do / 1(3 / '2

4 /< do / < ; / $1

If *n. / 1, 2, 2, 1

) +n / +" *1. ( ) *n.

) +n / / /

' ' ' ' 1 1(' 6 2(' 6 2(' 6 1(' '2

' -' ' -' 2 1('6- 2(' 6 2(' 6- 1(' 0' ' -' -' 2 1(' 62(' 6-2(' 6-1(' 0

' -' -' ' 1 1(' 6-2(' 6-2(' 61(' 1

Numer o additions $ 12, 3 per row

Kince the "adamard matri is o8tained through Nronecer product ,

it is possi8!e to reduce the num8er of additions

4

Mutterf!y diagram for Fast "adamard transform


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Mutterf!y diagram for Fast "adamard transform

Case ': 4 / 2 If the data se9uence is *n. / *0., *'.

dra the 8utterf!y diagram to compute "adamard transform

Kince 4/ 2 e re9uire "*2.

) +n / + "*2. ( *n.

)+0 / ' ' *0.

)+' / ' -' ( *'.

)+0 / + *0.(' 6 *'.('

)+' / + *0.(' - *'.('x(')

x(1)

) C'D

)C 1 D1

)*'.( *.

2 point ;utterl" or @addamard

x(') + x(1)

x(') x(1)

Mutterf!y diagram for "adamard transform


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Mutterf!y diagram for "adamard transform

Case2: 4 / 1If the data se9uence *n. is *0., *'., *2., *3.

dra the 8utterf!y diagram to compute "adamard transform

Kince 4/ 1 e re9uire "*1.

) +n / + "*1. ( ) *n.

' ' ' ' *0.

' -' ' -' *'.

)+n /' ' -' -' *2.

' -' -' ' *3.

Kince "*1. is generated using Nronecer product

)+0 *0.

)+' *'.

)+2 *2.

)+3 *3.

"*2. "*2. /

"*2. - "*2.

e partition "*1. as e!! as input


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e partition "*1. as e!! as input

7*0. *0. ' ' *0.

7*'. *'. ' -' *'.

"*0. *2.

"*'. *3.

From a8o#e:

7*0. / *0 .6 *'.

7*'. / *0. - *'.

1 additions"*0. / *2. 6 *3.

"*'. / *2. - *3.

riting ) +( in terms of 7 *(. and "*(.

)*0. / + 7*0. 6 "*0. C E x(') + x(1)F + Ex(2) +x(3)FD

)*'. / + 7*'. 6 "*'. C E x(') x(1)F + Ex(2) x(3)FD 1 additions

)*2. / + 7*0. - "*0. C E x(') + x(1)F Ex(2) +x(3)FD

)*3. / + 7*'. - "*'. C E x(') x(1)F Ex(2) x(3)FD

5et /+"*2. /

/+"*2.

D i th M tt f! di


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Draing the Mutterf!y diagram

x(') ) C'D> (')

x(1) ) C1D> (1)

x(2) ) C2D@ (')

x(3) ) C3D@ (1)

1

1

1

1

$ x(' )+ x(1)

$ x(' ) x(1)

$ x(2 )+ x(3)

$ x(2 ) x(3)

$ >(' )+ @(')

$ >(1 )+ @(1)

$ >(' )@(')

$ >(1 )@(1)

Using Mutterf!y diagram hich is a!so ca!!ed Fast "adamard Transform

The tota! num8er of additions re9uired are 1 6 1 /


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7i#en *n. 1, 2, 2 1 Find ) +n using Fast "adamard Transform

Kince 4 / 1, a 1-point 8utterf!y diagram can 8e used

>(') $ x(') + x(1) $ 4 + 2 $

>(1) $ x(') x(1) $ 4 2 $ 2

@(') $ x(2) + x(3) $ 2 + 4 $

@(1) $ x(2) x(3) $ 2 4 $ 2

BC'D $ C >(') + @(')D $ + $ 12

BC1D $ C >(1) + @(1)D $ 2 + 2 $ '

BC2D $ C >(') @(')D $ + $ '

BC3D $ C >(1) @(1)D $ 2 2 $ 4

B CnD $ E 12, ', ', 4F

x(3) ) C3D $ 4@ (1) $ 2

x(')) C'D $ 12

> (') $

x(1) ) C1D $ '> (1) $ 2

x(2) ) C2D $ '@ (') $

1

1

1

1

4

2

2

4

6

For 4 /


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)+0 *0.

)+' "*1. "*1. *'.

)+2 *2.

)+3 *3. )+1 *1.

)+$ "*1. - "*1. *$.

)+& *&.

)+; *;.

)+0 "*2. "*2. "*2. "*2. *0.

)+' *'.

)+2 "*2. -"*2. "*2. -"*2. *2.

)+3 *3.

)+1 "*2. "*2. -"*2. -"*2. *1.

)+$ *$. )+& "*2. - "*2. -"*2. "*2. *&.

)+; *;.

=

<

K

@

'


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Let = C'D $ 1 1 x (')

= C1D $ 1 1 x (1)

< C2D $ 1 1 x (2)

< C3D $ 1 1 x (3)

> C4D $ 1 1 x (4)

> CD $ 1 1 x ()

@ CD $ 1 1 x ()

< CD $ 1 1 x ()

=C'D $ x (') + x (1)

=C1D $ x (') x (1)

C1D $ x (4) x ()

@C'D $ x () + x ()

@C1D $ x () x ()

1


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x(') B C'D

x(1) B C1D

x(2) B C2D

x(3) B C3D

x(4) B C4D

x() B CD

x() B CD

x() B CD

=(')

=(1)

(1)

@(')

@(1)

&(')

&(1)

&(2)

&(3)

M(')

M(1)

M(2)

M(3)

"*1.

1

1

1

1

1

1

1

1BC'D $ &C'D + MC'D, BC4D $ &C'D 9MC'D

BC1D $ &C1D + MC1D, BCD $ &C1D 9MC1D



"*(') + @('), M(2) $ >(') @(')

j(1) $ >(1) + @(1), M(3) $ >(1) @(1)

1

1

1

1

2

Jompute


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x(') B C'D

x(1) B C1D

x(2) B C2D

x(3) B C3D

x(4) B C4D

x() B CD

x() B CD

x() B CD

=(')

=(1)

(1)

@(')

@(1)

&(')

&(1)

&(2)

&(3)

M(')

M(1)

M(2)

M(3)

"*1.

1

1

1

1

1

1

1

1BC'D $ &C'D + MC'D, BC4D $ &C'D 9MC'D



BCD $ &C3D + MC3D, BCD $ &C3D 9MC3D

"*(') + @('), M(2) $ >(') @(')

j(1) $ >(1) + @(1), M(3) $ >(1) @(1)

!otal additions:sutraction $ 24 (N log2N)as opposed to 5 x (51) $ 3

a!sh Transform:The a!sh transform matri is o8tained from "adamard matri


64/85

The a!sh transform matri is o8tained from "adamard matri

8y re-arranging the ros in increasing order of sign

' ' ' ' ' ' ' '0

' -' ' -' ' -' ' -' ;' ' -' -' ' ' -' -' 3

' -' -' ' '- ' -' ' 1

' ' ' ' -' -' -' -' '

' -' ' -' -' ' -' ' &

' '-' -' -' -' ' ' 2' -' -' ' -' ' ' -' $

"*


65/85

P!otting each of the ro of the "adamard matri gi#es us the

a!sh 8asis functions

(!%e comination o asis unctions represent t%e

continuous unctions)

a!sh transformation can 8e ca!cu!ated using the matri e9uations

) +n / *4. ( *n.

The In#erse a!sh transformation is gi#en 8y

*n. / '%4 ( + *4. ( ) *n.

a!sh 8asis functions Ke9uency Kign changes


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Compute Discrete a!sh function of the data se9uence ', 2, 0, 3


67/85

Compute Discrete a!sh function of the data se9uence ', 2, 0, 3

' ' ' ' 0

' -' ' -' 3

' ' -' - ' '

' -' - ' 6' 2

"*1. /

*1. /

' ' ' ' 0

' ' -' - ' '

' -' - ' 6' 2

' -' ' -' 3

) +n / ' ' ' ' ' ' 62 60 63 &

' ' -' - ' (. 2 / ' 62 -0 -3 / 0

' -' - ' 6' 0 ' -2 -0 63 2

' -' ' -' 3 ' -2 60 -3 -1

-ign c%ange

Gearranging sign c%anges

&n ascending order

) +n / *n. ( *n. ) +n

The "aar Transform


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- It is deri#ed from "aar Batri

"aar transform !ie most of the other transforms is

separa8!e and can 8e epressed as:

F / " f "

here f is 4 4 image

" is 4 4 transformation matri

F is the resu!ting 4 4 matri

The "aar transform " contains, the "aar 8asis functions hp9 *.

hich are defined o#er the continuous c!osed inter#a! +0, '

The "aar 8asis functions are:

h00 *. / '%√4 ( ∈ +0,'

2p%2O *9 - '.% 2p ≤

≤

*9 0($.% 2 p

hp9 *. / '%√4 2p%2O *9 0($.% 2p

≤

≤

9 % 2p

0 O otherise, ∈

+0,' 5

"aar Transformation matri


69/85

"aar Transformation matri

Case': 4 / 2

0≤

p≤

!og2 4

0≤

p≤

' log22 $ 1

Therefore p /0

Now '≤

9≤

2p

'≤

9≤

20 2' $1

Therefore 9 /'

?s h00 *. / '% √4 ( ∈ +0,'

Therefore h00 *. / '%√2 ( ∈ +0,'

20%2 O *'-'.%20 ≤

≤

*' 0($.% 20 for p /0, 9 /'hp9 *. / '% √4 2

0%2O *' 0($.% 20 ≤

≤

'% 20

0O otherise

h0' *. / '%√

2

'O 0 ≤ ≤ 0($

-'O 0($ ≤ ≤ ' 6

From a8o#e e9uations

% ( )


70/85

1:√2

'.

%''(x)

1 '. 1

%'1(x)

1:√2

1:√2

Kamp!ing the a8o#e to a#eforms ith 4/2

riting in matri form

"aar*2. / '% √2 It is same as "adamard "4*2.' ' ' -'

1:√2

11

1:√2

1:√2

'

Kimi!ar!y, "aar*1. for 4 / 1 can 8e found out


71/85

"aar*1. / '% √1

' ' ' '

' ' -' -'

√2 -√2 0 0

0 0√

2 -√

2

"aar*


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Find the "aar transform of the signa! *n. / ', 2, 0, 3

) +n / + "aar *4. ( *n.

/ '% √1 ( /'%2 /'%2

) +n / '%2 &, 0, -√

2, -3√

2

' ' ' ' ' '626063 &

' ' -' -' 2 '62-0-3 0

√2 -√2 0 0 0 √ 2- 2√26060 -√2

0 0√

2 -√

2 3 060 60 -3√

2 -3√

2

2

Find the "aar Transform of the gi#en pseudo image


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2 1 2 1

1 2 3 2

2 3 4 3

1 2 3 2

F / + "aar *4. ( f ( "aar *4.

/ "aar *4. ( f ( "aar *4.

/ '% √1 ( ( '% √1

F /

' ' ' ' 2 ' 2 ' ' ' ' '

' ' -' -' ' 2 3 2 ' ' -' -'

√

2 -√

2 0 0 2 3 1 3√

2 -√

2 0 00 0 √2 -√2 ' 2 3 2 0 0 √2 -√2


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* .

It is s!ight modification of "aar Transform

increases the speed of imp!ementation

@etains the sign changes

@ep!aces a!! non-Hero #a!ues 8y ',√

2 and 2 8y ' -

√

2 and -2 8y -'

"aar *2.B /

"aar*1.B /

"aar*loal properties

-emi>loal properties

Local properties

4

x(') )C'D

=(') #(')


75/85

x(1) )C4D

x(2) )C2D

x(3) )CD

x(4) )C1D

x() )CD

x() )C3D

x() )CD

1

1

1

1

1

1

1

1

1

Mutterf!y diagram for Bodified "aar

=(1)

>(')

>(1)

@(')

@(1)


76/85

*n. /


77/85

7enerate the pattern% 8asis images for the a!sh transform 4 /1


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*1. / ' ' ' '

' ' -' -'

' -' -' '

' -' ' -'

u%# ' ' ' ' ' ' -' -' ' -' -' ' ' -' ' -' rows

' ' ' ' ' ' -' -' ' -' -' ' ' -' ' -'

' ' ' ' ' ' -' -' ' -' -' ' ' -' ' -'

' ' ' ' ' ' -' -' ' -' -' ' ' -' ' -'' ' ' ' ' ' -' -' ' -' -' ' ' -' ' -'

' ' ' ' ' ' -' -' ' -' -' ' ' -' ' -'

' ' ' ' ' ' -' -' ' -' -' ' ' -' ' -'

-'-' -' -' -' -' ' ' ' ' ' -' -' ' -' '

-'-' -' -' -' -' ' ' ' ' ' -' -' ' -' '

' ' ' ' ' ' -' -' ' -' -' ' ' -' ' -'

-'-' -' -' -' -' ' ' ' ' ' -' -' ' -' '

' ' ' ' ' ' -' -' ' -' -' ' ' -' ' -'

-'-' -' -' -' -' ' ' ' ' ' -' -' ' -' '

' 1 2 3

'

'

'

'

'

'

-'-'

'

-'

-'

'

'

1

2

3

'' '1 '2 '3

1' 11 12 13

2' 21 22 23

Jol.

5

0 ' 2 3


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' ' ' ' ' ' -' -' ' -' -' ' ' -' ' -'

'

' '

'

'

'

-'

-'

'

-'

-'

'

'

-'

'

'

1 1 1 1

1 1 1 1 1 1 1 1

1 1 1 1

1 1 1 1 1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1 1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1 1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 111 1 1

1 11 1

1 1 1 1

1 1 1 11 1 1 1

1 1 1 1

1 1 1 1

1 1 1 11 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

'

1

2

3

6


80/85

5'


81/85

51


82/85

4D

52

/ e 2 %4


83/85

4 / e 2 π %4

4n / 1

n / e 2 π %1 n( For n / 0 , /0O 10(0 / e 2 π % 1 ( 0( 0 / '

n /', /' 1'(' / e 2 π % 1 ( '( ' / -

n /2, /' 12(' / e 2 π % 1 ( 2( ' / -'

n /3, /' 13(' / e 2 π % 1 ( 3( ' /

n /2, /2 11(' / e 2 π % 1 ( 2( 2 / '

n /3, /2 13(2/ e 2 π % 1 ( 3( 2 / -'

n /3, /3 13(3/ e 2 π % 1 ( 3( 3 / -4

' $ 1

41 $ j

42 $ 1

43 $ j

44 $ 1

53

3a % '2%20'0


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3a % '2%20'0

5et *n. / 2 δ + n 6 3 δ + n- ' + 4 δ + n- 2 + $ δ + n -3

Find 1 point DFT using FFT f!o graph


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discrete image processing

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