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Transcript of 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
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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
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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!
′
′
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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
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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.
''' ''' *;.
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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. $√
2%4 for '≤
u≤
4-'
In#erse transform IDCT
4-'
f *. /∑
*u. F *u. cos +π
*2 6'.( u % 24O 0≤
≤
4-'34
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D DCT pair:
4-' 4-'
, #. / *u.( *#. ∑
∑
f *, y . cos +π
*2 6'.(u % 24 ( cos +π
*2y 6'.(# % 2
/0 y /0 For u, # / 0, ', 2, ( ( , 4-'
rse DCT transform IDCT
4-' 4-'
y. /∑
∑
*u. *#. F *u, #. cos +π
*2 6'.( u % 24( Cos +π
*2y 6'.( # % 24
u /0 # /0
For , y / 0 to 4-'
3
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The 4 4 Cosine Transform Batri C / C *u, #. defined asO
√ '%4 u /0, 0 ≤ # ≤ 4-'
C *u, # . /
√
2%4 ( Cos +π
*2# 6'.( u % 24 '≤
u≤
4-', 0≤
#≤
4-'
In this Cosine transform Batri is rea! and orthogona!
8ut 4ot Kymmetric (transpose o matrix is
not eual to t%e matrix)
(ort%ogonal matrix w%ose column and rows are ort%ogonal unit vectors or
its transpose is eual to its inverse)
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Cosine transform Batri is rea! and orthogona!
( Not -"mmetric)
C / CG (#s onl" real, conjugate is same)
C-' / C *From T(T G / I unitary matri e9uation.
C ( C / I identity matri
The DCT of a co!umn se9uence f *., 0≤
≤
4-' can 8e computed
F / C ( f *7
Kince DCT is not symmetric
2-D DCT can 8e computed 8y
F / C f C (7
3
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Find DCT of the se9uence: ', 2, 1, ;
F / C ( f
√ '%4 ( u /0, 0 ≤ # ≤ 4-'
C *u,# . /
√ 2%4 ( Cos +π *2# 6'.( u % 24 ' ≤ u ≤ 4-', 0 ≤ # ≤ 4-'
Compute C
Kince 4 / 1 C i!! 8e 4 4 / 1 1 matri
u # 0 ' 2 3
0 0($ 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, ;
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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
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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
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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
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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
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"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
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'2th ?ugust
'
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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
BC2D $ &C2D + MC2D, BCD $ &C2D 9MC2D
BC3D $ &C3D + MC3D, BCD $ &C3D 9MC3D
"*(') + @('), 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
BC1D $ &C1D + MC1D, BCD $ &C1D 9MC1D
BC2D $ &C2D + MC2D, BCD $ &C2D 9MC2D
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
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The a!sh transform matri is o8tained from "adamard matri
8y re-arranging the ros in increasing order of sign
' ' ' ' ' ' ' '0
' -' ' -' ' -' ' -' ;' ' -' -' ' ' -' -' 3
' -' -' ' '- ' -' ' 1
' ' ' ' -' -' -' -' '
' -' ' -' -' ' -' ' &
' '-' -' -' -' ' ' 2' -' -' ' -' ' ' -' $
"*
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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
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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
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"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
% ( )
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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
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"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
=(') #(')
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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)
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*n. /
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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
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5'
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51
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4D
52
/ e 2 %4
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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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8/19/2019 Discrete Image Processing
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