Download - National Taiwan University

Integer
transform
Wen - Chih Hong
E-mail: [email protected]
Graduate Institute of Communication Engineering
National Taiwan University, Taipei, Taiwan, ROC
1
Outline
1. Introduction
what is integer
why integer
2. General method
3. Matrix factorization
4. Modified matrix factorization
5. Conclusions
6. Reference
2
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.
1
3
2 2 2 2
2
0
3
Introduction :
Why integer transform

We can use the fix-point multiplication operation
to replace floating-point one to implement it
4
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))
5
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)))
N 1

4.
 B(m, k )  B(n, k )  C
k 0
m
  mn
(*)
6
Introduction :
algorithms
We have three algorithms to approximate noninteger transform



1. General method
2. Matrix factorization
3. Modified matrix factorization
7
General method

1. Forming the prototype matrix

2. Constraints for Orthogonality

3. Constraints for Inequality (Inequality Constraints)

4. Assign the values
(Equality Constraints)
8
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
9
Appendix - Principle of Dyadic
symmetry (2/2)

For a vector of eight elements, there are seven possible dyadic
symmetries.
10
General method :
Generation of the order-8 ICTs(1/8)

Take 8-order discrete cosine transform for
example
for i =0
T8  i, j   1/ 8
 (2 / 8)cos{(i  0.5) j / 8}
for i =1~7
where j=0,1,…,6,7
(1)
11
General method :
Generation of the order-8 ICTs (2/8)

0.35 36
0.4904
0.4619
0.4157
0.3536
0.2778
0.1913
0.0975
0.3536
0 . 41 5 7
0.1913
-0.0975
-0.3536
-0.4904
-0.4619
-0.2778
0.3536
0.2778
-0.1913
-0.4904
-0.3536
0.0975
0.4619
0.4157
0.3536
0 . 0 97 5
-0.4619
-0.2778
0.3536
0.4157
-0.1913
-0.4904
0.3536
- 0 . 09 7 5
-0.4619
0.2778
0.3536
-0.4157
-0.1913
0.4904
0.3536
-0.2778
-0.1913
0.4904
-0.3536
-0.0975
0.4619
-0.4157
0.3536
-0.4157
0.1913
0.0975
-0.3536
0.4904
-0.4619
0.2778
0.3536
-0.4904
0.4619
-0.4157
0.3536
-0.2778
0.1913
-0.0975
Let T=kJ , k is diagonal matrix
12
General method :
Generation of the order-8 ICTs(3/8)

1. Forming the prototype
13
General method :
Generation of the order-8 ICTs(4/8)
2. Constraints for Orthogonality
Sth dyadic symmetry type in basis vector Ji
14
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)
15
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)
16
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)
17
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
18
General method :
Performance(1/2)

In the transform coding of pictures
n

Efficiency:

where
s
i 1
n
n
 s
p 1 q 1
(5)
ii
pq
 s11

[CY ]  E[Y  Y t ]  [T ][Cx ][T ]t  
s
 n1
s1n 


snn 
19
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
20
General method :
Disadvantage of general method
1.
2.
3.
Too much unknowns.
need to satisfy a lot of equations ,C(n,2).
It has no reversibility.
A’≈A , (A^-1)’≈(A^-1),
but (A’)^-1≠ (A^-1)’
21
Matrix Factorization

Reversible integer mapping is essential for lossless
source coding by transformation.

General method can not solve the problem of
reversibility
22
Matrix Factorization :
algorithm(1/9)
Goal :
A  PS N S N 1
S1S0
1. Suppose A , and det(A) ≠ 0
 a11

A
a
 n1
a1n 


ann 
(6)
23
Matrix Factorization :
algorithm(2/9)
2. There must exist a permutation matrix P1
for row interchanges s.t.
(1)
(1)
 p1,1
p1,2
 (1) (1)
p p
P1 A   2,1 2,2

 p (1) p (1)
 N ,1 N ,2
and
(1)
1, N
p
p1,(1)N 

(1)
p2, N 


pN(1), N 
(7)
0
24
Matrix Factorization :
algorithm(3/9)
(1)
(1)
p

s

p
s
3. There must exist a number 1 s.t. 1,1 1 1, N  1
then we get s1  ( p1,1(1)  1) p1,(1)N and a product
 1


P1 AS0,1  P1 A 
I

 s 0 1 
 1

(1)

1
p1,2
 (1)
(1)
p2,1  s1 p2,(1)N p2,2



 (1)
 p  s p (1) p (1)
 N ,1 1 N , N N ,2
p1,(1)N 

p2,(1)N 


pN(1), N 
(8)
25
Matrix Factorization :
algorithm(4/9)
4. Multiplying an elementary Gauss matrix
1

 s p (1)  p (1) 1
1 2, N
2,1
L1P1 AS0,1  

 (1)
(1)
 s1 pN , N  pN ,1
(2)
1 a1,2

(2)
 0 a2,2


 0 a (2)
N ,2

a1,(2)N 

(2)
a2, N 


(2) 
aN , N 
I
L1


 P AS
 1 0,1

1
(9)
26
Matrix Factorization :
algorithm(5/9)
5. Continuing in this way for k=1,2,…,N-1.
Pk defines the row interchanges among the kth
through the Nth rows to guarantee pk( k, N)  0
then we get
LN 1PN 1
L2 P2 L1 AS0,1S0,2
where
aN( N, N1)  ei
1

0

0, N 1 

0

S
( N 1)
a1,2
1
0
a1,( NN1) 

a2,( NN1) 


( N 1) 
aN , N 
 DRU
(10)
27
Matrix Factorization :
algorithm(6/9)
DRU where
DR  diag (1,1, ,1, ei )
1

0
U 

0

( N 1)
a1,2
1
0
a1,( NN1) 

( N 1)
a2, N 


1 
(11)
28
Matrix Factorization :
algorithm(7/9)
6. Multiplying all the SERMs (S0,k ) together
S0,1S0,2
1

S0, N 1 


  s1
I
1
 sN 1


  S 1
0


1
(12)
29
Matrix Factorization :
algorithm(8/9)
7.
LN 1PN 1
L2 P2 L1P1
 LN 1 ( PN 1LN 2 PNT1 )
( PN 1
P2 L1P2T
1 T

L
P
where
L1  LN 1 ( PN 1LN 2 PNT1 )
PT  PN 1
PNT1 )( PN 1
(13)
( PN 1
P2 L1P2T
P2 P1 )
PNT1 )
P2 P1
30
Matrix Factorization :
algorithm
8. we obtain
9. Theorem
(9/9)
L1PT AS01  DRUor
A  PLDRUS0
LU  ( LUS11 ) S1  ( LUS11S21 ) S 2 S1 
( LUS11S21
S N11 ) S N 1
S2 S1  S N S N 1
S2 S1
(14)
i.i.f. det(LU) is integer.
31
Matrix Factorization :
Advantage
if det(A) is integer
then A  PS N S N 1 S1S0
It is easy to derive the inverse of A
32
Modified matrix factorization:
algorithm(1/6)
If A is a NxN reversible transform
1. First, scale A by a constant
  det A
1/ N
such that
G  A
,where
det G :1
(15)
33
Modified matrix factorization:
algorithm(2/6)

2. Do permutation and sign-changing
operations for G:
R  D1PGQD2 P, Q:arbitrary permutation matrix
D1 , D2:diagonal matrix
D1[n, n]  1
D 2 [n, n]  1
(16)
34
Modified matrix factorization:
algorithm(3/6)

3. Do triangular matrix decomposition. First, we find L1such that
has the following form:
H  RL1
where
 h1,1

 h2,1
h
3,1
H  RL1  


 hN 1,1
h
 N ,1
1

0
0
L1  


0
0

h1,2
h1,3
1
0
h3,2
1
h1, N 1 h1, N 

0
0 
0
0 



1
0 
hN , N 1 1 
hN 1,2 hN 1,3
hN ,2
hN ,3
 1,2  1,3
 1, N 1  1, N
1  2,3
 2, N 1
0 1
 3, N 1
0
0
1
0
0
0


 2, N 
 3, N 



 N 1, N 

1

(17)
 r2,1 r2,2

r r
I n  Sn1 ( zˆn  tˆn ) Sn   3,1 3,2

r r
 n ,1 n ,2
I n  [ 1,n  2,n
r2,n1 

r3,n1 


rn ,n1 
 n2,n  n1,n ]T
tˆn  [r2,n r3,n
rn1,n rn ,n ]T
zˆn  [0 0
0 1]T
35
Modified matrix factorization:
algorithm(4/6)
1
T

L
4. 1
1 : Note that
1

0
T1  L11   0



0
T1is also an upper-triangular matrix.
1,2 1,3
1
 2,3
0
1
0
0
1, N 1 1, N 

 2, N 1  2, N 
(18)
3, N 1 3, N 
0
1




36
Modified matrix factorization:
algorithm(5/6)
5. Decompose H into T2and T3 .
H  T3T2
1
h
 2,1
 h3,1
T2  

h
 N 1,1
h
 N ,1
where [ 12
0
0
1
0
h3,2
1
hN 1,2 hN 1,3
hN ,2
hN ,3
 N ]  [h1,1h1,2
0
0
0 
0
0


1
0

hN , N 1 1 
0
 1

0
0
T3  

0

0
h1, N ]  T2,1
 2 3
 N 1  N 
1
0
0
0
1
0
(19)
0
0
1
0
0
0

0
0


0

1 
1  det( R)  1
From (8)(9)(10) we decompose R  T3T2T1,then,
G  PT D1T3T2T1D2QT
37
Modified matrix factorization:
algorithm(6/6)
6. Approximates T1, T2, and T3 by binary valued matrices J 1, J 2 ,
and J 3:
1

0
J1   0



0
t1,2 t1,3
1
t2,3
0
1
0
0
t1,N -1 t1, N 

t2, N 1 t2, N 
t3, N 1 t3, N 



0
1
tm,n  Qb (m,n )
1
c
 2,1
 c3,1
J2  

c
 N 1,1
c
 N ,1
0
0
1
0
c3,2
1
cN 1,2 cN 1,3
cN ,2
cN ,3
cm,n  Qb (hm,n )
0
0
0 
0
0


1
0

cN , N 1 1 
0
 1

0
0
J3  

0

0
s2 s3
1
0
0
1
0
0
0
0
s N 1 s N 

0
0
0
0


1
0

0 1 
sm  Qb (  m )
Qb (a)  2b round (2b a)
38
Modified matrix factorization:
the process of forward transform
Step 1:
x1  D2QT x
N
Step 2: x2[n]  x1[n]  Qr {  tn,m x1[m]} for
m n 1
n  1 ~ N 1
x2 [ N ]  x1[ N ]
39
Modified matrix factorization:
the process of forward transform
n 1
Step 3: x3[n]  x2[n]  Qr { cn,m x2[m]}
m 1
for n  2 ~ N
x3[1]  x2 [1]
Step 4:
N
x4 [1]   x3[1]  Qr { sk x3[k ]}
k 2
x4 [n]  x3[n] when n  1
40
Modified matrix factorization:
the process of forward transform
Step 5:
z  PT D1x4
41
Modified matrix factorization:
Accuracy analysis

Preliminaries
1. Qr [a ]  a   where 2r1    2r1
2.  is a random variable and uniformly distribute
in [ 2 r ,1 2r 1 ]
3. E[]  0 and E[2 ]  4 r 12
42
Modified matrix factorization:
Accuracy analysis
The process in step (2)-(4) can be rewritten as:
z  PT D1{J 3[ J 2 ( J1D2QT x  1 )   2 ]  3}
43
Modified matrix factorization:
Accuracy analysis
If y=Gx=σAx
then the difference between z, y is:
z  y  P DT
1 3T2 1  P DT
1 3 2  P D13 (20)
T
T
T
if b is very large in page.39
44
Modified matrix factorization:
Accuracy analysis

Using (20) to estimate the normalized root mean
square error (NRMSE) and use it to measure the
accuracy:
E[( z  y ) H ( z  y )]
NRMSE 
E[ y H y ]
45
Modified matrix factorization:
Accuracy analysis



Notice (20) ,we find the NRMSE which is dominated
by T2 and T3 .
So we let the entries of T2 and T3 as small as
possible .
That why we multiply P, D, and Q to G.
46
Modified matrix factorization:
Accuracy analysis

Example
47
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
48
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.
49
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”
50