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Integer Integer transformtransformInteger Integer transformtransformWen - Chih HongWen - Chih Hong

E-mail: pippo.cm93@nctu.edu.tw E-mail: pippo.cm93@nctu.edu.tw

Graduate Institute of Communication Engineering Graduate Institute of Communication Engineering

National Taiwan University, Taipei, Taiwan, ROCNational Taiwan University, Taipei, Taiwan, ROC

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Outline 1. Introduction

what is integer

why integer

2. General method

3. Matrix factorization

4. Modified matrix factorization

5. Conclusions

6. Reference

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Introduce : what is integer transform

1. all entries are integer

i.e. -5,-3,0,2,…

2.all entries are sum of power of 2

i.e.

2 0 1 32 2 2 2

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Introduction : Why integer transform

We can use the fix-point multiplication operation to replace floating-point one to implement it

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Introduction : The constraints of approximated integer transform

1. if A(m,n1)=τA(m,n2) , τ=1,-1,j,-j then B(m,n1)=τB(m,n2)

2. if Re(A(m,n1)) ≥ Re(A(m,n2)) Im(A(m,n1)) ≥ Im(A(m,n2)) then Re(B(m,n1)) ≥ Re(B(m,n2)) Im(B(m,n1)) ≥ Im(B(m,n2))

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Introduction : The constraints of Approximated integer transform

3. if sgn(Re(A(m,n1)))=sgn((Re(A(m,n1)))

then sgn(Re(B(m,n1)))=sgn((Re(B(m,n1)))

4. (*) 1

0

( , ) ( , )N

m mnk

B m k B n k C

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Introduction : algorithms

We have three algorithms to approximate non-

integer transform 1. General method 2. Matrix factorization 3. Modified matrix factorization

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General method 1. Forming the prototype matrix

2. Constraints for Orthogonality (Equality Constraints) 3. Constraints for Inequality (Inequality Constraints) 4. Assign the values

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Appendix - Principle of Dyadic symmetry(1/2) Definition: a vector [a0,a1,…,am-1] m=2^n

we say it has the ith dyadic symmetry

i.i.f. aj=s ． a i j

where i,j in the range [0,m-1]

s= 1 when symmetry is even

s=-1 when symmetry is odd

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Appendix - Principle of Dyadic symmetry (2/2) For a vector of eight elements, there are seven possible dyadic

symmetries.

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General method : Generation of the order-8 ICTs(1/8)

Take 8-order discrete cosine transform for example

for i =0

for i =1~7

where j=0,1,…,6,7 (1)

8T , 1/ 8i j (2 / 8)cos{( 0.5) /8}i j

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General method : Generation of the order-8 ICTs (2/8)

0.3536 0.3536 0.3536 0.3536 0.3536 0.3536 0.3536 0.3536 0.4904 0.4157 0.2778 0.0975 -0.0975 -0.2778 -0.4157 -0.4904 0.4619 0.1913 -0.1913 -0.4619 -0.4619 -0.1913 0.1913 0.4619 0.4157 -0.0975 -0.4904 -0.2778 0.2778 0.4904 0.0975 -0.4157 0.3536 -0.3536 -0.3536 0.3536 0.3536 -0.3536 -0.3536 0.3536 0.2778 -0.4904 0.0975 0.4157 -0.4157 -0.0975 0.4904 -0.2778 0.1913 -0.4619 0.4619 -0.1913 -0.1913 0.4619 -0.4619 0.1913 0.0975 -0.2778 0.4157 -0.4904 0.4904 -0.4157 0.2778 -0.0975

Let T=kJ , k is diagonal matrix

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General method : Generation of the order-8 ICTs(3/8)

1. Forming the prototype

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General method : Generation of the order-8 ICTs(4/8)

2. Constraints for Orthogonality Sth dyadic symmetry type in basis vector Ji

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General method : Generation of the order-8 ICTs(5/8)

2. Constraints for Orthogonality

(i) Totally C(8,2)=28 equations must be satisfied

(ii) Using principle of dyadic symmetry*, we can reduce 28 to 1 equation:

a ． b=a ． c + b ． d + c ． d (2)

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General method : Generation of the order-8 ICTs (6/8)

3. Constraints for Inequality

the equation in page.8 imply

a ≥ b ≥ c ≥ d ,and e ≥ f (3)

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General method : Generation of the order-8 ICTs(7/8)

4. Assign the values :

use computer to find all possible values

(of course the values must be integer)

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General method : Generation of the order-8 ICTs(8/8)

Some example of (a, b, c, d, e, f): (3,2,1,1,3,1) ,(5,3,2,1,3,1),… infinity set

solutions we need to define a tool to recognize

which one is better

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General method : Performance(1/2)

In the transform coding of pictures

Efficiency: (5)

where

1

1 1

n

iii

n n

pqp q

s

s

11 1

1

[ ] [ ] [ ][ ][ ]n

t tY x

n nn

s s

C E Y Y T C T

s s

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General method : Performance(2/2)

The twelve order-8 ICTs that have the highest transform efficiencies for p equal 0.9 and a less than or equal to 255

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General method : Disadvantage of general method

1. Too much unknowns.

2. need to satisfy a lot of equations ,C(n,2).

3. It has no reversibility.

A’≈A , (A^-1)’≈(A^-1),

but (A’)^-1≠ (A^-1)’

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Matrix Factorization Reversible integer mapping is essential for lossless

source coding by transformation.

General method can not solve the problem of reversibility

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Matrix Factorization :algorithm(1/9)

Goal :

1. Suppose A , and det(A) ≠ 0

(6)11 1

1

n

n nn

a a

A

a a

1 1 0N NA PS S S S

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Matrix Factorization :algorithm(2/9) 2. There must exist a permutation matrix

for row interchanges s.t.

(7)

and

1P

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

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

1

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

N

N

N N N N

p p p

p p pPA

p p p

(1)1, 0Np

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Matrix Factorization :algorithm(3/9) 3. There must exist a number s.t.

then we get and a product

(8)

1s(1) (1)1,1 1 1, 1Np s p

(1) (1)1 1,1 1,( 1) Ns p p

1 0,1 1

1

1

0 1

PAS PA I

s

(1) (1)1,2 1,

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

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

1

N

N N

N N N N N N

p

p s p p p

p s p p p

p

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Matrix Factorization :algorithm(4/9) 4. Multiplying an elementary Gauss matrix

(9)

1L

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

1 1 0,1 1 0,1

(1) (1)1 , ,1

1

1

1

N

N N N

s p pL PAS PAS

I

s p p

(2) (2)1,2 1,

(2) (2)2,2 2,

(2) (2),2 ,

1

0

0

N

N

N N N

a a

a a

a a

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Matrix Factorization :algorithm(5/9) 5. Continuing in this way for k=1,2,…,N-1.

defines the row interchanges among the through the rows to guarantee

then we get

where

(10)

kP thkthN ( )

, 0kk Np

1 1 2 2 1 0,1 0,2 0, 1N N NL P L P L AS S S

( 1) ( 1)1,2 1,

( 1)2,

( 1),

1

0 1

0 0

N NN

NN

NN N

a a

a

a

RD U( 1)

,N iN Na e

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Matrix Factorization :algorithm(6/9) where

(11)

RD U (1,1, ,1, )iRD diag e

( 1) ( 1)1,2 1,

( 1)2,

1

0 1

0 0 1

N NN

NN

a a

aU

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Matrix Factorization :algorithm(7/9) 6. Multiplying all the SERMs ( ) together

(12)

0,kS

0,1 0,2 0, 1NS S S 10

1 1

1

1

1N

IS

s s

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Matrix Factorization :algorithm(8/9)7.

(13)

where

1 1 2 2 1 1N NL P L P L P

1 1 2 1( )TN N N NL P L P

1 2 1 2 1 1 2 1( )( )T TN N NP P L P P P P P

1 TL P1

1 1 2 1 1 2 1 2 1( ) ( )T T TN N N N N NL L P L P P P L P P

1 2 1T

NP P P P

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Matrix Factorization :algorithm (9/9)8. we obtain or

9. Theorem

(14)

i.i.f. det(LU) is integer.

1 10

TRL P AS D U

0RA PLD US

1 1 11 1 1 2 2 1( ) ( )LU LUS S LUS S S S

1 1 11 2 1 1 2 1 1 2 1( )N N N NLUS S S S S S S S S S

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Matrix Factorization :Advantage if det(A) is integer

then

It is easy to derive the inverse of A 1 1 0N NA PS S S S

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Modified matrix factorization:algorithm(1/6)

If A is a NxN reversible transform 1. First, scale A by a constant , where

such that :

(15)

1/

detN

A det 1G

G A

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Modified matrix factorization:algorithm(2/6)

2. Do permutation and sign-changing operations for :

, :arbitrary permutation matrix

, :diagonal matrix

(16)

G

1 2R D PGQD P Q

1D 2D

1D [ , ] 1n n 2D [ , ] 1n n

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Modified matrix factorization:algorithm(3/6) 3. Do triangular matrix decomposition. First, we find such that

has the following form:

(17)

where

1L1H RL

1,1 1,2 1,3 1, 1 1,

2,1

3,1 3,21

1,1 1,2 1,3

1 0 0 0

1 0 0

N N

N N N

h h h h h

h

h hH RL

h h h

,1 ,2 ,3 , 1

1 0

1N N N N Nh h h h

1,2 1,3 1, 1 1,

2,3 2, 1 2,

3, 1 3,1

1,

1

0 1

0 0 1

0 0 0 1

0 0 0 0 1

N N

N N

N N

N N

L

1 ˆˆ( )n n n nI S z t

2,1 2,2 2, 1

3,1 3,2 3, 1

,1 ,2 , 1

n

nn

n n n n

r r r

r r rS

r r r

1, 2, 2, 1,[ ]Tn n n n n n nI

2, 3, 1, ,ˆ [ ]Tn n n n n n nt r r r r

ˆ [0 0 0 1]Tnz

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Modified matrix factorization:algorithm(4/6)

4. : Note that is also an upper-triangular matrix.

(18)

11 1T L

1T

1,2 1,3 1, 1 1,

2,3 2, 1 2,

11 1 3, 1 3,

1

0 1

0 0 1

0 0 0 0 1

N N

N N

N NT L

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Modified matrix factorization:algorithm(5/6)

5. Decompose into and .

(19)

where ,

From (8)(9)(10) we decompose ,then,

H 2T 3T

3 2H TT2,1

3,1 3,2

2

1,1 1,2 1,3

1 0 0 0 0

1 0 0 0

1 0 0

N N N

h

h hT

h h h

,1 ,2 ,3 , 1

1 0

1N N N N Nh h h h

1 2 3 1

3

0 1 0 0 0

0 0 1 0 0

0 0 0 1 0

0 0 0 0 1

N N

T

11 2 1,1 1,2 1, 2[ ] [ ]N Nh h h T 1 det( ) 1R

3 2 1R TT T

1 3 2 1 2T TG P DTT TD Q

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Modified matrix factorization:algorithm(6/6)

6. Approximates , , and by binary valued matrices , , and :

1T 2T 3T 1J 2J

3J

1,2 1,3 1, -1 1,

2,3 2, 1 2,

1 3, 1 3,

1

0 1

0 0 1

0 0 0 0 1

N N

N N

N N

t t t t

t t t

J t t

2,1

3,1 3,2

2

1,1 1,2 1,3

1 0 0 0 0

1 0 0 0

1 0 0

N N N

c

c cJ

c c c

,1 ,2 ,3 , 1

1 0

1N N N N Nc c c c

2 3 1

3

1

0 1 0 0 0

0 0 1 0 0

0 0 0 1 0

0 0 0 0 1

N Ns s s s

J

, ,( )m n b m nt Q , ,( )m n b m nc Q h ( )m b ms Q

( ) 2 (2 )b bbQ a round a

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Modified matrix factorization:the process of forward transform

Step 1:

Step 2: for

1 2Tx D Q x

2 1 , 11

[ ] [ ] { [ ]}N

r n mm n

x n x n Q t x m

1 ~ 1n N

2 1[ ] [ ]x N x N

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Modified matrix factorization:the process of forward transform

Step 3: for

Step 4:

when

1

3 2 , 21

[ ] [ ] { [ ]}n

r n mm

x n x n Q c x m

2 ~n N

3 2[1] [1]x x

4 3 32

[1] [1] { [ ]}N

r kk

x x Q s x k

4 3[ ] [ ]x n x n 1n

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Modified matrix factorization:the process of forward transform

Step 5: 1 4Tz P D x

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Modified matrix factorization:Accuracy analysis

Preliminaries

1. where

2. is a random variable and uniformly distribute in [ , ]

3. and

[ ]rQ a a 1 12 2r r

12 r 12 r

[ ] 0E 2[ ] 4 12rE

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Modified matrix factorization:Accuracy analysis

The process in step (2)-(4) can be rewritten as:

1 3 2 1 2 1 2 3{ [ ( ) ] }T Tz P D J J J D Q x

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Modified matrix factorization:Accuracy analysis

If y=Gx=σAx

then the difference between z, y is:

(20)

if b is very large in page.39

1 3 2 1 1 3 2 1 3T T Tz y P DT T P DT P D

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Modified matrix factorization:Accuracy analysis

Using (20) to estimate the normalized root mean square error (NRMSE) and use it to measure the accuracy:

[( ) ( )]

[ ]

H

H

E z y z yNRMSE

E y y

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Modified matrix factorization: Accuracy analysis

Notice (20) ,we find the NRMSE which is dominated by and .

So we let the entries of T2 and T3 as small as possible .

That why we multiply P, D, and Q to G.

2T 3T

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Modified matrix factorization:Accuracy analysis

Example

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Modified matrix factorization:Advantages

Compare to matrix factorization :

1. simpler and faster way to derive the integer transform

2. Easier to design 3. Higher accuracy

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Conclusions

If a transform A ,which det(A) is an integer factor, we can convert it into integer transform.

Integer transform is easy to implement, but is less accuracy than non-integer transform. It is a trade-off.

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Reference 1. W.-K. Cham, PhD “Development of integer cosine transforms by the

principle of dyadic symmetry” 2. W. K. Cham and Y. T. Chan” An Order-16 Integer Cosine Transform” 3. Pengwei Hao and Qingyun Shi “Matrix Factorizations for Reversible

Integer Mapping” 4. Soo-Chang Pei and Jian-Jiun Ding” The Integer Transforms

Analogous to Discrete Trigonometric Transforms” 5. Soo-Chang Pei and Jian-Jiun Ding” Improved Reversible Integer

Transform”