DFT Domain Image

7/27/2019 DFT Domain Image

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Lecture 6. DFT Domain ImageFilterin

ELEN E4830 Digital Image Processingpr ng

Zhu Liu David Gibbon

z u researc .att.comAT&T Labs - Research

cg researc .att.comAT&T Labs - Research

Note: Part of the materials in the slides are from Gonzalezs Digital

Image Processingand Prof. Yao Wangs lecture slides


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Lecture Outline

1D/2D discrete Fourier transform (DFT)

u

1D/2D unitary transform

Discrete Cosine transform

Karhunen Loeve transform

Frequency domain filters

Spring 2010 ELEN E4830 Digital Image Processing Lecture 6, Page 2


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Discrete Fourier Transform (DFT):

DTFT for Finite Duration Si nals

:becomesansformFourier tr

:1,...,1,0fordefinedonlyissignaltheIf = Nn

)1,0(,)2exp()(1

0

=

=

uunjnf(u)FN

n

u

:(DFT)transformForward

:yieldsrescalingand,110atSampling == ,...,N-,k/N, ku(u)Fu

1,...,1,0,)/2exp()(11

)(1

0

===

=

NkNknjnfN

)N

k(F

NkF

N

n

u

1

:(IDFT)transformInverse

1

N


,...,,,0=N k


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Property of 1D DFT - Periodicity

.modulore resentswhere

.0),))((()(

Nk

NkorkkFkF N


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Property of 1D DFT - Translation

)}/(2exp{)()))((( oNo NknjkFnnf

).))(((}/2exp{)( 00 NkkFNnkjnf

Special case

N is even, k0 = N/2.

).))2

((()1)((}exp{)( Nn NkFnfnjnf =



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Property of 1D DFT - Symmetry

Conjugate symmetry for real sequences** ==

|)(||)(|

NN

orkNFkF =

.,22

=

)()/2exp()(1)/)(2exp()(1)(1

*

1* kFNknjnfNnkjnfkFNN

== =

00 nn ==

F(k) F(k)

0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 8k k


N=8 N=9


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Property of 1D DFT - Convolution

Circular convolution1N

=

=0

)()))((()()(k N

khknnhn

f(((n-k))N)

f(n) h(n)

0 1 2 3 4 n 0 1 2 3 4 n nn+1

n+2n+3

n+4 kn-1

0

onvo u on eorem)()()()( kHkFnhnf



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2D Discrete Fourier Transform

Definition

= - = -, , , , , , , , , ,

is a finite length 2D sequence1 1 1 )(2

=== +M N

N

ln

M

kmj

.1,...,1,0,1,...,1,0,),(1),(

,...,,,,...,,,,,

1 1 )(2

0 0

===

+

= =

NnMmelkFnmf

MN

M N

N

ln

M

kmj

m n

Comparing to DTFT

0 0= =

lknvmuj

+ )(2

NMm n= = ,,,,

+=2/1 2/1

)(2 dudvevunm nvmuj


2/1 2/1


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Property of 2D DFT Periodicity

.0,0),))((,))(((),( NlorlMkorklkFlkF NM


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Property of 2D DFT - Convolution

Circular Convolution 1 1M N

= =

=0 0 ),()))((,))(((),(),( k l NM lkhlnkmfnmhnmf

Convolution Theorem

,,,,



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Property of 2D DFT - Separability

Separability 2D DFT can be accom lished b N- oint 1D DFT of each row,

followed by M-point 1D DFT of each column

+

==1 21 21 1 )(2 111 M

M

kmjN

N

lnjM N

N

ln

M

kmj

eenmenmlkF

Separable Images

= == = 0 00 0 m nm n NN

)()(),()()(),( lFkFlkFnfmfnmf yxyx ==

If h(m,n)=hx(m)hy(n), then 2D CC can be accomplished by N-point 1D CC of each row, followed by M point 1D CC of each


.


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Computer Implementation of 2D DFT

+

=1 1 )(2

),(1

),(M N

N

ln

M

kmj

enmflkF

Complex array structure

= =m n

Pre-compute the constants- , , , , -

e-j2ln/N for all l,n=0, 1, , N-1.

For real square image, only need to calculate

~



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Display of the Magnitude of 2D DFT

Shifting the low frequency components into the

center

).))2

((,))2

(((),()1)(,(),( )( NMnm NlMkFlkGnmfnmg == +

A B D Cl l

C D B A

F(k,l) G(k,l)

Low High

k k



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Display of the Magnitude of 2D DFT

Amplitude rescaling )),(1log(),( lkFlkG +=



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Fast Fourier Transform (FFT)

Direct computation of N-point DFT takes N2

o erations 1 1N

,...,,exp

0

===

njn

N n

,

reducing the computation from N2 to N log2(N)

Complex conjugate symmetry of Nknje /2

( )*/2/)(2/2/)(2 NknjNnkjNknjNnNkj eeee ===

eNnNkjNNnkjNknj eee /)(2/)(2/2 ++ ==


- ,


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Computation of 2D DFT

2D (MxN) point DFT can be computed in a

se arable manner:

First compute N-point FFT for each rowM N log2 (N) Then computeM-point FFT for each columnN M log2 (M)

Total computationM N (log2 (N)+ log2 (M))

IfM=N: 2N2

log2 (N)



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Calculate Linear Convolution Using DFT

1D DFT

1, 2

g(n) = f(n)*h(h) is length N = N1+N2-1.

o use , nee o ex en n an n o

length N by zero padding.

f(n) h(n) g(n)*Convolution

F(k) H(k) G(k)x

DFT DFT DFT

Multiplication



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Calculate Linear Convolution Using DFT

f(n), N1=5 fe(n), N=8Fe(k)

n n k

h(n), N2=4 he(n), N=8He(k)

n n k

* e eee


kn


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Comparison of Complexity

f(n) is length N1, h(n) is length N2 , g(n) is length

N=N +N -1 12N

=

==0

)()))((()()()(k

N khknnhnng

Each point, N2 multiplications

Overall, N * N O N2

Via FFT

N-point FFT for f(n) and h(n)

N multiplications in the DFT domain

N-point inverse FFT for F(k)*H(k)


Overall, 3NlogN + N O(NlogN)


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One Dimensional Unitary Transform

Let CN represent the N dimensional complex

s ace.

Let h0, h1, , hN-1 represent N linearlyinde endent vectors in CN.

For any vectorf CN,1N

)0(

,0

===

t

k

k

AffB == 1

.)1(

],,...,,[ 110

==

twhere N

MthhhB

fand t form a transform pair


)1( t


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Inner Product

Definition of inner product1N

==>==< ff

=>=====>=


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Definition of Unitary Transform

Forward transformN 1

*

h

>========< FH

( ) )1(4

1221213221

4

1

1210

1310

0221

,1111

1111

4

1,

*

3,23,2 jjjjjjjjjjjj

T =+++++=

>==< FH



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Property of Separable Transform

When the transform is separable, we can

.

First, do 1D transform for each row using

l,

Second, do 1D transform for each column of

.

Proof: =

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

M N

nmFnmlkT

= =

=

=1

*1 1

**

0 0

,

),()(),()()(MM N

l

m n

lmUmhnmFngmhkk

Spring 2010ELEN E4830 Digital Image Processing

Lecture 6, Page 41

== = 00 0 mm n

Computational Savings of


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Computational Savings of

Separable Transform

A MxN transform takes about MxN

and the total number of calculation isM2N2 (N4 if M=N)

If the transform is separable

1. Calculate M N- oint 1D transform, eachrequiring N2 calculations

2. Calculate N M-point 1D transform, each

requ r ng ca cu a ons3. Overall: MN2+NM2 (2N2, if M=N)



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Other Transforms

Discrete cosine transform (DCT)

Hadamard transform

Haar transform

Etc.



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Discrete Cosine Transform (DCT)

The basis vectors for 1D N-point DCT 1

=

=

=

+

= .1,...,12

0

)(,2

)12(

cos)()(Nk

N

N

kwithN

kn

knhk

DCT

11 NN

==

==

11

00

)2

cos()()()()()(

NN

nn

kN

nfknfnhkt

== == 00 )2cos()()()()()( kkk Nktkktnhnf



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Hadamard Transform

Hm is a 2mx2m matrix, and it can be defined

recursivel

=

11

21 mm

mH

HHH

10=

H

=11

11

2

11H

1111

1111

1

=

111111112

2


Constructing Signal Dependant


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Constructing Signal Dependant

Transform Bases Applications

Feature selection

es gn ng cr er a

Signal de-correlation capability(Uncorrelated

componen s can e processe ana yze more e c en y

Energy compaction capability (Use the least numberof coefficients to re resent the ori inal si nal accuratel

=

=

===

1

11

0

,)(,)(

N

N

Kk

kKK

K

k

kK hktffeiserrorionapproximatthehktf

Spring 2010 ELEN E4830 Digital Image Processing Lecture 6, Page 46=

== )(:Kk

KK kteenergyError

K h L T f (KLT)


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Karhunen Loeve Transform (KLT)

For a signal with covariance matrix Cf -

KLT achieves the minimal approximation error

ons ruc on o

Let k

and k

the k-th eigenvalue and

norma ze e genvec or o f, en

.,, ,lklkkkkf with >=


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Proof of KLT Properties

Diagonalize Cf.

0

H

[ ]

== 110

1

Nf

H

f

H

t LM C

CC

00

1

00

H

H

N

L

L

=

=

1

111100

100 N

NN

H

N

L

MOMML

M

Minimize approximation error Selecting the K coefficients associated with the K


.

Filt i i th F D i


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Filtering in the Frequency Domain

Relation between spatial and frequency domain operation:

),(),(),(),(),(),( vuFvuHvuGyxfyxhyxg ==


.,,,,,

L P Filt U i DFT Wi d


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Low-Pass Filter Using DFT Window

Filtering in DFT domain:

=, , ,

Ideal Low-Pass Filter H k l = 1 in low fre uenc ran e the four corners!

= 0 in high frequency range

Can think of H(k,l) as a mask in the DFT domain The (k,l)th coefficient is unchanged if H(k,l)=1, and is set to

zero if H(k,l)=0.

, ,

in low-frequency range, and lower values in highfre uenc ran e


L P Filt i DFT D i


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Low-Pass Filter in DFT Domain

The window size where

=

,

(2 W1+1)x(2 W2+1)

Before shifting

R1=zeros(M,1);

R1(1:W1+1,M-W1+1:M)=1;

R2=zeros N,1 ; R2(1:W2+1,N-W2+1:N)=1;

H=R1*R2;

Given G(k,l)=F(k,l)H(k,l), matlab may make small numerical errors inthe inverse transform leading to complex valued images. g(m,n) isbest reconstructed via:

g=real(ifft2(F.*H));


Example


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Example


Example


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Example


Spatial Filter Corresponding


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to Ideal LPF

10

30

40

50

10

10 20 30 40 50 60

60

20

30

40

50 IDFT

10 20 30 40 50 60

60

Ideal LPF with

W1=W2=10


High Pass Filtering


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High Pass Filtering

The high-pass filtered image can be

low pass filtered image. -

If w(k,l) is a low-pass DFT window, simply define a

hi h- ass window h k l b h k l = 1 w k l . High-pass filtering in spatial domain:

If L is a low-pass filter of size W, simply define a

high pass filter H via H(m,n) = (m,n) - L(m,n).


High Pass Filtering in DFT Domain


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High Pass Filtering in DFT Domain


Before shifting

Introduction to Wavelets


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Introduction to Wavelets

Fourier Transform analyzes the entire signalf t y

Frequenc

Short Time Fourier Transform (STFT) analyzest

Time

f(t) y

window

Freque

n


tTime

Wavelets Analysis


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Wavelets Analysis

Wavelets are obtained from the mother wavelet

t b dilations and shiftin

a is the scaling parameter b is the shifting parameter

)(

1

)(, a

bt

atba

=

Haar wavelet

t

0

-

1

10.5

1D Continuous wavelet transform (CWT)

y

= dxxxfbaF baW )()(),( ,*

Freque

n


Time

Two Band Subband Decomposition


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Two Band Subband Decomposition

H0: Lowpass filter, y0: approximation of x11

==

Haar Transform


,22

10

Wavelet Transform = Subband Tree


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Wavelet Transform Subband Tree

N N/2

N/4

N/2J

N/2J

The above structure can be applied to rows of an image first, and then columns,


forming 2D wavelet decomposition

2D Wavelet Transform for Images


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2D Wavelet Transform for Images

2D wavelet transform is accomplished by applying the 1D decomposition along rows

of an image first, and then columns.

H0 2

H0 2

H1 2In ut

RowsLL

LH

H1 2H0 2

H1 2

Image

(NxN) HL

HH


Wavelet Decomposition


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Wavelet Decomposition

LH: low pass for horizontal direction, high pass in vertical directionHL: low pass for vertical direction, high pass in horizontal direction

HH: high pass in both horizontal and vertical directions

LL

LH

HL

HH


Reading


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Reading

R. Gonzalez, Digital Image Processing,

. . . . .

A. Jain, Fundamentals of Digital Image, .


Homework (1)


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( )

1. Find the NxN point DFT of the following 2D image f(m, n), 0 m, n N:

a) f(m, n) = 1, n = 0; f(m, n) = 0; n 0. b) f(m, n) = 1, m = 0; f(m, n) = 0, m 0.

c) f(m, n) = 1, m = n; f(m, n) = 0, otherwise. d) f(m, n) = 1, m = N 1- n; f(m, n) = 0,

otherwise.

From the result, what can you say about the relation between the directionality of an

mage w a o s

2. One can use the DFT algorithms to compute the linear convolution of an image F(m,

n) with a filter H(m, n). Let the convolved image be denoted by Y(m, n).

i) Suppose the image size is 256x256 and the filter size is 11x11: What is the

required size of the DFT to obtain the convolution of these two? Explain the exact

steps to obtain the convolution result.

, , ,

obtain Z(m, n) as follows: Z = IDFT (DFT(X)DFT(H)). The DFT and IDFT in the aboveequation are both 256x256 points. For what values of (m, n) does Z(m, n) equal Y(m,

n) ?


Homework (2)


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( )

3. Perform DFT (using the fft2 function in Matlab) on several test images of your choice(sample images are available at the Matlab image toolbox directory. Display themagnitude of the DFT image with and without shifting (using shift) and logarithmicmapping, to see the effect of shifting and logarithmic mapping. Include the printouts inyour submission. Also, examine in which frequency range the DCT coefficients havelarge magnitudes and explain why. Note that you may want to work on a small portion

of your image (say 256x256 or less), to save computation time.Note that if your image is an RGB image, you should convert it to a grayscale image

" "only. If the original image is an index image, you should not apply DFT to the indeximage directly, rather you should derive the grayscale image from it, using the"ind2gray" function.

4. Write a program (in Matlab or C) which filters an image by zeroing out certain DFT.

1) performing DFT (you can use the fft2" function in Matlab);

2) zeroing out the coefficients at certain frequencies (see below);

3) performing inverse DFT to get back a filtered image. Truncate or scale the imagero erl such that its ran e is between 0 and 255.

For part 2, try the following two types of filters: (a) let F(k, l) = 0 for TN < k, l < (1 - T)N, T = 1/4, 1/8 (i.e., low-pass filtering); (b) let F (k, l) = 0 for 0 k, l TN and (1 - T)N k, l N -1, T=1/4, 1/8 (i.e., high pass filtering);

Compare the original and processed images. Comment on the function of the two


.

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