chapter 2 digital image transform …poseidon.csd.auth.gr/lab_publications/books/dip... · i. pitas...

I. Pitas Digital Image Processing FundamentalsDigital Image Transform Algorithms THESSALONIKI 1998

2.1

CHAPTER 2

DIGITALIMAGE TRANSFORM

ALGORITHMS

DIGITAL IMAGE PROCESSINGDIGITAL IMAGE PROCESSINGDIGITAL IMAGE PROCESSING


2.2

ContentsContentsContents

uIntroductionu2-d discrete Fourier transform

uRow-column FFT (RCFFT) algorithmuMemory problems in 2-d DFT calculationsuVector-radix FFT (VRFFT) algorithmuPolynomial transform FFT (PTFFT)

u2-d power spectrum estimationuDiscrete cosine transform (DCT)u2-d DCT


2.3

AxX =

A A− =1 *T

where x and X are the original and transformed image respectively.In most cases the transform matrices are unitary:

u The columns of A*T are the basis vectors of the transform and in 2-d transforms they correspond tobasis images.u The most popular image transforms are:/ Discrete Fourier Transform (DFT)./ Discrete Cosine Transform (DCT).

u Image transforms are represented by transform matrices A:

IntroductionIntroduction


2.4

}0,0:),{(22112121

NnNnnnR NN <≤<≤=

),(~),(~),(~22121121

NnnxnNnxnnx +=+=

where N1, N2 denote the horizontal and vertical periods.

The fundamental period of this sequence is the rectangleRN1N2 having size N1×N2:

Two-dimensional discrete Fourier transform

Let be a 2-d rectangularly periodic sequence. Its periodicity is defined as:

),(~21

nnx


2.5

Figure 1: Rectangularly periodic sequence and its fundamental period.

Two-dimensional discrete Fourier transform


2.6

+= ∑∑

−

=

−

= 2

22

1

111

1

2

2

21

21

21

22exp),(

~1),(~

1

0

1

0 N

kni

N

knikkX

NNnnx

N

k

N

k

ππ

∑ ∑−

=

−

=

−−=

1

0

1

0

1

1

2

2 2

22

1

11

2121

22exp),(~),(

~ N

n

N

n N

kni

N

kninnxkkX

ππ

u A periodic sequence can be represented by a Fourier series.u The 2-d DFS is used for the representation of a 2-d rectangularly periodic signal:

u The Fourier series coefficients are :

TwoTwo--dimensional discrete Fourier transformdimensional discrete Fourier transform


2.7

u A discrete 2-d space-limited signal can be represented by DFT as:

∑ ∑−

=

−

=

−−=

1

0

1

0

1

1

2

2 2

22

1

11

2121

22exp),(),(

N

n

N

n N

kni

N

kninnxkkX

ππ

+= ∑ ∑

−

=

−

= 2

22

1

111

1

2

2

21

21

21

22exp),(

1),(

1

0

1

0 N

kni

N

knikkX

NNnnx

N

k

N

k

ππ



2.8

2,12

exp =

−= j

NiW

jN j

πwhere:

u Another commonly used form of the DFT is:


∑ ∑−

=

−

=

=1

0

1

0

1

1

2

2

22

2

11

12121),(),(

N

n

N

n

knN

knN WWnnxkkX

22

2

11

1

1

1

2

2

21

21

21

1

0

1

0

),(1

),( knN

knN

N

k

N

k

WWkkXNN

nnx −−−

=

−

=∑ ∑=


2.9

u The DFT is related to the Fourier transform of a discrete 2-d signal which is defined as:

∑ ∑∞

−∞=

∞

−∞=

−−=1 2

2112121)exp(),(),( 2

n n

nininnxX ωωωω

∫ ∫− −

+=π

π

π

π

ωωωωωωπ 2122112121

)exp(),(4

1),(

2ddniniXnnx

2

22

1

112121

2,

2),(),(N

k

N

kXkkX πω

πωωω ===

over the unit bicircle z1 = exp(iù1), z2 = exp(iù2).

The DFT coefficients are a sampled version of the 2-d FT of a discrete sequence:



2.10

Nnn

mnmnxnny

N

NN

mod))((

)))((,))(((),(22211121

∆

=

++=

∑ ∑−

=

−

=

∆

−−=

=⊗⊗=1

0

1

0

1

1

2

2

22211121

212121

)))((,))(((),(

),(),(),(N

m

N

m

NN mnmnhmmx

nnhnnxnny

u The circular convolution of two signals can be computed by means of the circular shift of a signal.

Circular Shift

Circular Convolution



2.11

Figure 2: Circular shift of a 2-d sequence



2.12

u In most applications the linear convolution of two signals is needed. It is defined as:

∑ ∑−

=

−

=

−−=1

0

1

0

1

1

2

2

22112121),(),(),(

Q

m

Q

m

mnmnxmmhnny

where RP1P2 = [0, P1) x [0, P2) and RQ1Q2 = [0, Q1) x [0, Q2) are the regions of support of x, h.

u The region of support of the convolution is:

2,11),0[),0[ 2121=−+=×= iQPLLLR iiiLL



2.13

u The algorithm consists of the following steps:

1. Choose N1,N2 such that Ni ≥ Pi + Qi - 1, i = 1, 2.2. Pad the sequences x(n1, n2) and h(n1, n2) with zeros, so that:

−∈

∈=

2121

21

),(0

),(),(),(

21

2121

21PPNN

PP

RRnn

Rnnnnxnnx

p

−∈

∈=

2121

21

),(0

),(),(),(

21

2121

21QQNN

QQ

RRnn

Rnnnnhnnh

p



2.14

4. Calculate the DFT Yp(k1, k2), as the product of Xp(k1, k2) and Hp(k1, k2).5. Calculate yp(n1, n2) by using the inverse DFT. The result of the linear convolution is:

21212121),( ),(),(

LLpRnnnnynny ∈=


3. Calculate the DFTs of the new sequences xp(n1, n2) and hp(n1, n2).

IDFT y

DFT

DFT

××

x

h

Figure 3: Convolution calculation using the DFTs


2.15

u The DFT also supports the 2-d correlation:

),(),(),(221 121 kkYkkXkkPxy

∗=

∑ ∑−

=

−

=

++=1

0

1

02211

1

1

2

2

2121),(),(),(

N

n

N

n

mnmnynnxmmRxy

∑ ∑−

=

−

=

++=1

0

1

0

1

1

2

2

22112121),(),(),(

N

n

N

n

mnmnxnnxmmRxx

u If both sequences are real then in the frequency domain:

u The autocorrelation of an image is defined as:



2.16

)()(),()()(),(221121221121

kXkXkkXnxnxnnx =↔=

),(),(),(

),(),(),(

21221

212121

1 kkWbkkVakkX

nnwbnnvannx

⋅+⋅=↔⋅+⋅=

),(),(

)))((,))(((),(

21

22

2

11

121

22211121

kkXWWkkY

mnmnxnnykm

Nkm

N

NN

=

↔−−=

1.1. Separable sequence:Separable sequence:

Properties of the 2-d DFT

3. Circular shift:Circular shift:

2. Linearity:Linearity:



2.17

5. Complex conjugate property:Complex conjugate property:

),()))((,))(((21222111

** kkXnNnNxNN

↔−−

)))((,))(((),(22211121

*NN

kNkNXkkX −−=

4. Modulation:Modulation:

6. Reflection:Reflection:

)))((,))(((),(

),(),(

22211121

21

22

2

11

121

NNlklkXkkY

nnxWWnnyln

Nln

N

−−=↔= −−

),(),(1212

kkXnnx ↔

If x(n1, n2) is a real-valued signal:



2.18

9. Circular convolution and multiplication:Circular convolution and multiplication:),(),(),(),(

21212121kkHkkXnnhnnx ⋅↔⊗⊗

),(),(1

),(),(2121

21

2121kkHkkX

NNnnhnnx ⊗⊗↔

∑ ∑∑ ∑−

=

−

=

−

=

−

=

=1

0

1

0

21

0

1

0

2 1

1

2

2

21

21

1

1

2

2

21),(

1),(

N

k

N

k

N

n

N

n

kkXNN

nnx

7. Initial value and DC value:Initial value and DC value:

∑ ∑∑ ∑−

=

−

=

−

=

−

=

=1

0

1

0

*1

0

1

0

*1

1

2

2

2121

1

1

2

2 21

2121),(),(

1),(),(

N

k

N

k

N

n

N

n

kkYkkXNN

nnynnx

∑ ∑−

=

−

=

=1

0

1

0

1

1

2

2

21),()0,0(

N

n

N

n

nnxX∑ ∑−

=

−

=

=1

0

1

0

1

1

2

2

21

21

),(1

)0,0(N

k

N

k

kkXNN

x

8. Parseval’sParseval’s theorem:theorem:



2.19

∑−

=

=′1

0

2

2

22

22121),(),(

N

n

knNWnnXknX

∑−

=

′=1

0

1

1

11

12121),(),(

N

n

knNWknXkkX

u The simplest algorithm for the calculation of the 2-d DFT is the Row - Column FFT (RCFFT) algorithm.

u The 2-d DFT can be decomposed in N1 DFTs along rows and N2 DFTs along columns:

RowRow--column FFT algorithmcolumn FFT algorithm


2.20

22 ),(),(),(212121

kkXkkXkkXIRM

+=

= −

),(

),(tan),(

21

21

21

1

kkX

kkXkk

R

Iφ

u The transform magnitude is:

u The transform phase is:

u The magnitude of the transform provides useful information about the frequency content of an image.



2.21

)(log2

log2

log2 212

2112

1222

21 NN

NNN

NNN

NNC =+=

)(log 21221 NNNNA =

)log( 22 NkNO

u The number of complex multiplications for RCFFT is:

u If radix-2 FFT is used then the number of complex additions for RCFFT is:

u The computational complexity is of the order :



2.22

u There are memory problems in the calculation of 2-d DFT even for moderately sized images.

u A solution to this problem is the storage of the image on a hard disk in a row-wise manner.

Figure 4: (a) Storage of an image in a row-wise manner. Each image row occupies one data record. b) Access of image in a column-wise manner.

Memory problems in 2Memory problems in 2--d DFT calculationsd DFT calculations

(a) (b)


2.23

u If K signal rows (or columns) can be stored in RAM, the number of I/O operations required are:

NNNNNNNIO 32)1(232 2 +=−++=

NK

NN IO 3

2 2

+=

u The total number of I/O operations is:



2.24

u A further speed-up is obtained by combining the row transform computation with a part of the column transform computation.

u The number of I/O operations is:

knNN IO /2=



2.25

Figure 5: Radix-2 FFT algorithm for N=8.



2.26

u Another approach for the reduction of the I/O operations is to use fast matrix transposition algorithms.

u If the matrix has size N×N, where N = 2n, it can be split into four submatrices of size (N/2)×(N/2) each.

u This procedure is repeated until submatrices of size 2×2 are reached.



2.27

u This algorithm has n = log2N steps.

u The transposition can be performed in place.

u The first stage needs 2N I/O operations.

u The number of stages is log2N.

u The total number of I/O operations is:

NNN IO 2log2=



2.28

Figure 6: Three stages in the transposition of an 8×8 matrix.



2.29

u If the image is real, the complex conjugate property can be used for reducing memory requirements.

u Suppose that the dimensions N1, N2 of the image are powers of 2 and that the image is stored on a matrix A of size N1×N2.

u The first stage of the RCFFT processes the conjugate symmetry.



2.30

u The columns X´(n1, 0), X´(n1, N2/2) are real numbers.

u They are used to store the real and imaginary parts of X´(n1, k2) in place as:

[ ][ ] ),(),(Im

),(),(Re

22121

2121

kNnknX

knknX

−→′

→′

A

A10,2/1 1122 −≤≤<≤ NnNk

u This storage requires only 50% of the memory required for the storage of the complete complex signal and is performed in place.



2.31

u The column transform also satisfies the conjugate property.

u The samples X(0, 0), X(N1 / 2, 0), X(0, N2 / 2), X(N1 / 2, N2 / 2) are real and can be stored in the corresponding positions of A. The samples X(k1, 0), X(k1, N2 / 2) can be stored as:

[ ][ ][ ][ ] )2/,()2/,(Im

)2/,()2/,(Re

)0,()0,(Im

)0,()0,(Re

21121

2121

111

11

NkNNkX

NkNkX

kNkX

kkX

−→

→

−→

→

A

A

A

A

2/1 11 Nk <≤



2.32

u The real and imaginary parts of the samples X(k1, k2), 1 ≤ k1 ≤ N1-1, 1 ≤ k2 < N2/2 are stored as:

[ ][ ] )))((,))(((),(Im

),(),(Re

21 221121

2121

NN kNkNkkX

kkkkX

−−→

→

A

A

2/1,11 2211 NkNk <≤−≤≤for



2.33

Figure 7: In-place storage of the 2-d DFT of a real-valued signal. (a) Storage after row transform. (b) Storage after column transform. The cross-hatched areas denote the storage places for the imaginary part of the transform.


(a) (b)


2.34

211

2

1 2

2211

),(),(

),(),(

),(),(

2121

2121

1

0

1

02121

kkNoo

kNoe

kNeoee

N

n

N

n

knN

knN

WkkGWkkG

WkkGkkG

WWnnxkkX

+

−

=

−

=

+

++=

= ∑ ∑

u Let x(n1, n2) be a square image N×N.

u Its DFT can be split into four 2-d DFTs of size (N/2)×(N/2), by following a decimationdecimation--inin--timetime approach:

VectorVector--radix fast Fourier transform algorithmradix fast Fourier transform algorithm


2.35

22

1 2

11 212/

0

12/

0

2212100 )2,2(),( kl

N

N

l

N

l

klN WWllxkkS ∑ ∑

−

=

−

=

=

22

1 2

11 212/

0

12/

0

2212101 )12,2(),( kl

N

N

l

N

l


−

=

−

=

+=

22

1 2

11 212/

0

12/

0

2212110 )2,12(),( kl

N

N

l

N

l


−

=

−

=

+=

22

1 2

11 212/

0

12/

0

2212111 )12,12(),( kl

N

N

l

N

l


−

=

−

=

++=

Where:



2.36

Figure 8: Radix-2×2 butterfly.



2.37

Figure 9: Flow diagram of a 4×4 vector-radix FFT.



2.38

u VRFFT has log2N stages.u Each stage contains N2/4 butterflies.u Each butterfly requires 3 complex multiplications.u The total number of complex multiplications is:

NN

C 2

2

log4

3=

NNA 22 log2=

u The number of complex additions is:

u All computations of the VRFFT can be performed in place.



2.39

[ ] )(mod)()()(ˆ1

0

zPzGzXzXN

m

mkmk ∑

−

=

=

[ ] )( mod)()(ˆ1)(

1

0

zPzGzXN

zXN

k

mkkm ∑

−

=

−=

∑−

=

=1

0

),()(N

n

nm znmxzX

where P(z) is an irreducible polynomial and G(z) = 1 mod P(z)

is a polynomial N-th root of unity. The polynomial of degree N-1 is:

u A polynomial transform can be defined as:

Polynomial transform FFTPolynomial transform FFT


2.40

u Only 2N2log2N real additions are needed for the calculation of the polynomial transform.

u A polynomial transform can be employed for the fast calculation of the 2-d DFT.

)1( mod ))(()(ˆ1

0

2 += ∑−

=

NmkN

mmk zzzXzX

u The following polynomial transform is used for a 2-d FFT:



2.41

uu Polynomial transform FFT (PTFFT).Polynomial transform FFT (PTFFT).

2

2

2

1

1

021 ),()( n

N

n

nn zWnnxzX ∑

−

=

−=

∑−

=+ +=

1

0

2)12(

1

11

112)1(mod)()(ˆ

N

n

Nknnkk zzzXzX

∑−

=

=+1

0

21212

2),(),)12((N

l

lklWWlkykkkX

∑−

=+ =

1

01)12( ),()(ˆ

12

N

l

lkk zlkyzX

where y(k1,l) are the coefficients of the polynomial:



2.42

Figure 10: Polynomial transform FFT



2.43

u If a radix-2 FFT is used for the 1-d DFTs then the total number of real multiplications is:

)log4(2 22 NNC +=

A N N= +224 5( log )

u The total number of real additions is:

u A comparison between RCFFT and PTFFT shows thatPTFFT is better than RCFFT for N>16.



2.44

u The power spectrum carries important information about the 2-d signal content.

u The squared magnitude is the periodogramof the discrete signal and can be used as an estimator of its 2-d power spectrum.

21

0

1

021

21

2

2121

21

1

1

2

2

22

2

11

1),(

1

),(1

),(ˆ

∑ ∑−

=

−

=

=

=

N

n

N

n

knN

knN

xx

WWnnxNN

kkXNN

kkP

$ ( , )P k kxx 1 2

Two dimensional power spectrum estimationTwo dimensional power spectrum estimation


2.45

u The periodogram is the Fourier transform of the estimator of the autocorrelation function:

∑ ∑−= −=

++++

=1

11

2

22

),(),()12)(12(

1),(ˆ

221121*

2121

N

Nk

N

Nkxx nknkxkkx

NNnnR

),(ˆ21

nnRxx

),(ˆ21

ωωxx

P

u The periodogram is a smoothed version of the actual 2-d power spectrum. It is very poor for small images and it is a noisy power spectrum estimator.



2.46

Figure 11: (a) Test image LENNA; (b) periodogram of LENNA.



2.47

u The Bartlett estimator can reduce the noise. The 2-d signal of size N1×N2 is split into K1×K2 non-overlapping sections each having size M1×M2:

),(),( 221121 jMniMnxnnxij

++=

1,...,0,1,...,0 21 −=−= KjKi

21

0

1

021

2121

)(1

1

2

2

22

2

11

1),(

1),(ˆ ∑ ∑

−

=

−

=

=M

n

M

n

knM

knMij

ijxx WWnnx

MMkkP

u The periodogram of each section is:

where



2.48

u Blackman-Tukey 2-d power spectrum estimator: An estimator of the 2-d autocorrelation function Rxx(m1,m2) is used:

∑ ∑−

=−−

−

=

=22

01212

22

0212121

1

1

22

2

11

1

2

2

),(),(),(N

m

kmN

kmN

N

mxx

BTxx WWmmwmmRkkP

∑∑−

=

−

=

=1

0

1

021

)(

2121

1 2

),(ˆ1),(ˆ

K

i

K

j

ijxx

Bxx kkP

KKKKP

u The new spectral estimator is the average of theperiodograms of all sections:



2.49

u A separable 2-d window can be used. The triangular 1-d window or the 1-d Hamming window can be used for each of the dimensions.

u The BT estimator can be obtained by using the 2-d DFT of the windowed autocorrelation function:

[ ]),(),(),( 212121 mmRmmwDFTkkP xxBT

xx =



2.50

u Technique based on 2-d AR modeling of images.

∑∑∈

+−−=),(

212121 ),(),(),(),(ji A

nnwjninxjiannx

A is the prediction windowá(i,j) are the predictor coefficientsw(n1,n2) is a white noise random process, having variance ów

2.


Figure 12: Examples of prediction windows.


2.51

2

),(21

2

21

21

22

2

11

1),(

),(

∑ ∑∈

=

mm A

kmN

kmN

wARxx

WWmma

kkPσ

u The power spectrum can be estimated by the followingformula:



2.52

∑−

=

=1

0

)(1

)0(N

n

nxN

C

∑−

=

+=

1

0 2

)12(cos)(

2)(

N

n N

knnx

NkC

π

x nN

CN

C kn k

Nk

N

( ) ( ) ( ) cos( )

= ++

=

−

∑10

2 2 1

21

1 π

u DCT is used in the JPEG and MPEG standards.

/ Forward DCT transform:

/ Inverse DCT:

Discrete cosine transformDiscrete cosine transform


2.53

u Let x(n) be a signal. An even sequence f(n) can be produced by this signal as:

Figure 13: Creation of an even sequence

f nx n

x N n( )

( )

( )=

− − 2 1

0 1

2 1

≤ ≤ −≤ ≤ −n N

N n N



2.54

u Algorithm for the fast calculation of DCT using DFT:

1. Formation of the even sequence f(n).2. The DFT of length 2N is calculated by using a

1-d FFT of length 2N.3. Calculation of the DCT coefficients C(k) using the

relations:

10)(2

)(2/

2 −≤≤= NkforkFN

WkC

kN

where F(k) are the DFT coefficients.



2.55

u Alternative method for the fast calculation of the 1-d DCT:

+

−= ∑−

= N

kninx

NkC

N

n 2

)12(expRe)(

2)(

1

0

π

−

−= ∑

−

=

12

0

exp)(2

expRe2

)(N

n N

nkiny

N

ki

NkC

ππ

y nx n

( )( )

= 0

0 1

2 1

≤ ≤ −≤ ≤ −n N

N n N

u If x(n) is a real sequence then:

where:



2.56

u Alternative definition for the DCT pair:

C k x nn k

Nn

N

( ) ( )cos( )

=+

=

−

∑ 22 1

20

1 π

x nN

w k C kn k

Nk

N

( ) ( ) ( ) cos( )

=+

=

−

∑1 2 1

20

1 π

where: w k( )/

=

1 2

1

k

k N

=≤ ≤ −

0

1 1

u Thus, the DCT coefficients are related to the DFT coefficients by:

)()( 2/2 kFWkC k

N=



2.57

u Fast DCT algorithm:1. Formation of a sequence í(n) as:

v nx n

x N n( )

( )

( )=

− −

2

2 2 1

01

2

1

21

≤ ≤−

+

≤ ≤ −

nN

Nn N

{ })(Re2)(Re2)( 4

1

04 kVWWnvWkC k

N

N

n

nkN

kN =

= ∑−

=

{ })(Im2)( 4 kVWkNC kN−=−

2. Computation of the DFT V(k), 0 ≤ k ≤ N-1 by using an FFT algorithm of length N.3. Computation of C(k), 0 ≤ k ≤ [N/2], and C(N-k) as:



2.58

u Fast calculation of the inverse DCT: 1. Computation of V(k) as:

v nx n

x N n( )

( )

( )=

− −

2

2 2 1

01

2

1

21

≤ ≤−

+

≤ ≤ −

nN

Nn N

10 )()(2

1)( 4 −≤≤−−=

− NkkNiCkCWkV k

N

2. Computation of í(n) from V(k) by means of an inverse FFT algorithm of length N.

3. Retrieval of x(n) from í(n) by using the formula:



2.59

u A 2-d N1×N2 DCT is defined as:

2

221

0

1

0 1

112121 2

)12(cos

2)12(

cos),(4),(1

1

2

2N

kn

N

knnnxkkC

N

n

N

n

ππ ++= ∑ ∑

−

=

−

=

2

221

0

1

0 1

11212211

2121 2

)12(cos

2

)12(cos),()()(

1),(

1

1

2

2N

kn

N

knkkCkwkw

NNnnx

N

k

N

k

ππ ++= ∑∑

−

=

−

=

where:

=1

2/1)(

11kw

=1

2/1)(

22kw

11

0

11

1

−≤≤=

Nk

k11

0

22

2

−≤≤=

Nk

k

for 0 ≤ k1 ≤ N1-1, 0 ≤ k2 ≤ N2-1.

TwoTwo--dimensional discrete cosine transformdimensional discrete cosine transform


2.60

u The expansion of the signal forms a new one as:

−−−−−−

−−=

)12,12(

)12,(

),12(

),(

),(

2211

221

211

21

21

nNnNx

nNnx

nnNx

nnx

nnf

12 12

12 10

10 12

10 10

222111

22211

22111

2211

−≤≤−≤≤−≤≤−≤≤

−≤≤−≤≤−≤≤−≤≤

NnNNnN

NnNNn

NnNnN

NnNn

),(),( 212/

22/

2212

2

1

1kkFWWkkC k

NkN=

u The DCT coefficients C(k1, k2) are related to the DFT ones F(k1, k2) as:



2.61

Figure 14: (a) Original sequence x(n1,n2). (b) Expanded sequence f(n1,n2).



2.62

−−−−−−

−−=

)122,122(

)122,2(

)2,122(

)2,2(

),(

2211

221

211

21

21

nNnNx

nNnx

nnNx

nnx

nnv

12

1 1

21

12

1

21

0

21

0 12

1

21

0 2

10

222

111

2221

1

2211

1

22

11

−≤≤

+

−≤≤

+

−≤≤

+

−

≤≤

−

≤≤−≤≤

+

−

≤≤

−

≤≤

NnN

NnN

NnNN

n

NnNn

N

Nn

Nn

u Fast calculation of the 2-d DCT follows:1. Formation of a sequence í(n1,n2) as:

for:



2.63

2. The DFT V(k1,k2) is calculated by using a N1×N2 FFT algorithm.

3. The DCT coefficients are calculated as:

[ ]{ }[ ]{ }),(),(Re2

),(),(Re2),(

21142144

2214214421

1

1

1

1

2

2

2

2

2

2

1

1

kkNVWkkVWW

kNkVWkkVWWkkCk

NkN

kN

kN

kN

kN

−+=

−+=−

−



2.64

[ ]{

[ ]}),(),(

),(),(4

1),(

221211

22112144212

2

1

1

kNkCkkNCi

kNkNCkkCWWkkV kN

kN

−+−−

−−−= −−

u There is also an algorithm for the fast calculation of the inverse 2-d DCT.

1. Computation of V(k1,k2) as:

2. Calculation of í(n1,n2) by using an inverse N1×N22-d FFT algorithm.



2.65

−−−−−−

−−=

)122,122(

)122,2(

)2,122(

)2,2(

),(

2211

221

211

21

21

nNnNx

nNnx

nnNx

nnx

nnv

12

1 1

2

1

12

1

21

0

2

10 1

2

1

2

10

2

10

222

111

2221

1

2211

1

22

11

−≤≤

+

−≤≤

+

−≤≤

+

−

≤≤

−

≤≤−≤≤

+

−

≤≤

−

≤≤

NnN

NnN

NnNN

n

NnNn

N

Nn

Nn

3. Retrieval of x(n1,n2) from í(n1,n2) by using the following formula:

for:


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