1

Image Transform

•Representation of the signal

•Basis Function

•Extraction of information from the signal

2

Role of Transform

White Light? VIBGYOR!

TransformTime domain

Frequency domain

3

Transform

Time domain

Frequency domain

Forward Transform

Inverse Transform

Basis Function

Finality of Co-efficients Orthogonality

4

Basis Function

5

N

0iiicx

x Input signal

Transform coefficient

Basis function

ici

i

NL

0iicx̂

ii ,xc

6

Different Image Trasforms

Fourier TransformWalsh TransformHadamard TransformDiscrete Cosine TransformKL TransformWavelet Transform

7

Fourier Transform

French Mathematical Physicist

Jean Baptiste Joseph Fourier

Idea in 1809

Flow of heat through objects

1-D Discrete Fourier Transform

]n[x ]k[X

8

Forward TransformAnalysis

]k[X ]n[xSynthesis

Inverse Transform

1-D DFT (cont..)

9

1N

0n

knN2j

1Nto0k,e]n[x]k[X

x[n] represents the signal

X[k] represents the spectrum

2-D DFT

10

][nx ),( nmf

][kX ],[ lkF

1-D signal 2-D signal

1-D Spectrum 2-D Spectrum

2-D DFT

11

1

0

N

n][kX

1

0

N

m

1

0

N

n),( nmf mk

Nj

e2

nlN

je

2),( lkF

][nxkn

Nj

e2

2-D DFT

12

Image

Magnitude

Phase

Forward Transform

13

Importance of Phase

Read Image ‘A’ Read Image ‘B’

Fourier Transform Fourier Transform

Magnitude Phase Magnitude Phase

Inverse Fourier Transform Inverse Fourier Transform

Reconstructed Image ‘A1’ Reconstructed Image ‘B1’

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Importance of PhaseOriginalimage OriginalImage

Cameraman image after phase reversal Lenna image after phase reversal

Application of Fourier Transform in Image Processing

15

Image Filtering

h(m,n)f(m,n) g(m,n)

f(m,n) Input image

h(m,n) Impulse response of the filter

g(m,n) Output image

g(m,n) = f(m,n) * h(m,n)

Application of Fourier Transform in Image Processing

16

f(m,n) DFT F(k,l) H(k,l) IDFTg(m,n)

F(k,l) Spectrum of input signal

H(k,l) Filter functiong(m,n) Output image

Convolution in time domain = Multiplication in frequency domain

G(k,l)

Walsh Transform

Nlog m 2

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1m

0i

)k(b)n(b i1mi)1(N1)k,n

sh basis function:

n time index

k frequency index

N Order

Walsh

Walsh basis for N=4

Nlog m 2

18

24log m 2

ep 1: To compute ‘m’

ep 2: Substituting N=4 and m=2 in g(n,k)

1m

0i

)k(b)n(b i1mi)1(N1)k,n(g

12

0

)(12)()1(41),(

i

kibnibkng

1

)()( 1)1(41),( kbnb iikng

Walsh basis for N=4

19

ep 3: Substituting n=0 and k=0 in g(n,k)

1

0

)()( 1)1(41),(

i

kbnb iikng

1

0

)0(1)0()1(41)0,0(

iibibg

Walsh basis for N=4

20

)0()0()0()0( 0110 )1()1(41)0,0( bbbbg

p 4: Substituting i = 0 and i = 1 in g(0,0)

)0(0b Zeroth bit position in the binary value of zero

)0(1b First bit position in the binary value of zero

0 0 0

Zeroth bit positionFirst bit position

binary representation

Walsh basis for N=4

21

Step 5: Substitute and in g(0,0)0)0(0 b 0)0(1 b

41)1()1(

41)0,0( 00 g

To compute g(2,1)

1

0

)()( 1)1(41),(

i

kbnb iikng

1

0

)1()2( 1)1(41)1,2(

i

bb iig

Walsh basis for N=4

22

Substituting i=0 and i=1 in the expression of g(2,1)

1

0

)1(1)2()1(41)1,2(

iibibg

)1()2()1()2( 0110 )1()1(41)1,2( bbbbg

)2(0b Zeroth bit position in the binary value of 2

)1(1b First bit position in the binary value of 1

)2(1b First bit position in the binary value of 2

h b h b l f

Walsh basis for N=4

23

0

12

3

0

01

ecimal number Binary representation

Zeroth bit position

First bit position

)2(0b = 0)1(1b = 0)2(1b = 1)1(0b = 1

)1(0b)2(1b)1(1b)2(0b )1()1(41)1,2(g

41)1()1(

41)1,2(g 10

0

0

1

11

Walsh basis for N=4

24

41

41

41

41

41

41

41

41

41

41

41

41

41

41

41

41

)k,n(

No sign change

One sign change

Three sign changes

Two sign changes

Walsh Basis for N=4

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41

41

41

41

41

41

41

41

41

41

41

41

41

41

41

41

)k,n(

n=1 and k=2

n=1 0 1

k=2 1 0

k=2 0 1

original

reversal

0 1

0 1No. of overlap is odd

No. of overlap of one is odd then sign is negative

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NN

NNN2 HH

HHH

HADAMARD TRANSFORM

11

112H

Jacques Salomon Hadamard

ISCRETE COSINE TRANSFORM

27

Nln

Nkmnmflklk

N

m

N

n 212cos

212cos),()()(],

1

0

1

0

02

01

)(kif

N

kifNk

02

01

)(lif

N

lifNl