1 Integer transform Wen - Chih Hong Graduate Institute of Communication Engineering National Taiwan University, Taipei, Taiwan, ROC

2 Outline 1. Introduction what is integer why integer 2. General method 3. Matrix factorization 4. Modified matrix factorization 5. Conclusions 6. Reference

3 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.

4 Introduction : Why integer transform We can use the fix-point multiplication operation to replace floating-point one to implement it

5 Introduction : The constraints of approximated integer transform 1. if A(m,n 1 )=τA(m,n 2 ), τ=1,-1,j,-j then B(m,n 1 )=τB(m,n 2 ) 2. if Re(A(m,n 1 )) ≥ Re(A(m,n 2 )) Im(A(m,n 1 )) ≥ Im(A(m,n 2 )) then Re(B(m,n 1 )) ≥ Re(B(m,n 2 )) Im(B(m,n 1 )) ≥ Im(B(m,n 2 ))

6 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. (*)

7 Introduction : algorithms We have three algorithms to approximate non- integer transform 1. General method 2. Matrix factorization 3. Modified matrix factorization

8 General method 1. Forming the prototype matrix 2. Constraints for Orthogonality (Equality Constraints) 3. Constraints for Inequality (Inequality Constraints) 4. Assign the values

9 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. a j =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

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

11 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)

12 General method : Generation of the order-8 ICTs (2/8) Let T=kJ, k is diagonal matrix

13 General method : Generation of the order-8 ICTs(3/8) 1. Forming the prototype

14 General method : Generation of the order-8 ICTs(4/8) 2. Constraints for Orthogonality Sth dyadic symmetry type in basis vector Ji

15 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)

16 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)

17 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)

18 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

19 General method : Performance(1/2) In the transform coding of pictures Efficiency: (5) where

20 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

21 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)’

22 Matrix Factorization Reversible integer mapping is essential for lossless source coding by transformation. General method can not solve the problem of reversibility

23 Matrix Factorization : algorithm(1/9) Goal : 1. Suppose A, and det(A) ≠ 0 (6)

24 Matrix Factorization : algorithm(2/9) 2. There must exist a permutation matrix for row interchanges s.t. (7) and

25 Matrix Factorization : algorithm(3/9) 3. There must exist a number s.t. then we get and a product (8)

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

27 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)

28 Matrix Factorization : algorithm(6/9) where (11)

29 Matrix Factorization : algorithm(7/9) 6. Multiplying all the SERMs ( ) together (12)

30 Matrix Factorization : algorithm (8/9) 7. (13) where

31 Matrix Factorization : algorithm (9/9) 8. we obtain or 9. Theorem (14) i.i.f. det(LU) is integer.

32 Matrix Factorization : Advantage if det(A) is integer then It is easy to derive the inverse of A

33 Modified matrix factorization: algorithm(1/6) If A is a NxN reversible transform 1. First, scale A by a constant, where such that : (15)

34 Modified matrix factorization: algorithm(2/6) 2. Do permutation and sign-changing operations for :, : arbitrary permutation matrix, : diagonal matrix (16)

35 Modified matrix factorization: algorithm(3/6) 3. Do triangular matrix decomposition. First, we find such that has the following form: (17) where

36 Modified matrix factorization: algorithm(4/6) 4. : Note that is also an upper-triangular matrix. (18)

37 Modified matrix factorization: algorithm(5/6) 5. Decompose into and. (19) where, From (8)(9)(10) we decompose,then,

38 Modified matrix factorization: algorithm(6/6) 6. Approximates,, and by binary valued matrices,, and :

39 Modified matrix factorization: the process of forward transform Step 1 : Step 2: for

40 Modified matrix factorization: the process of forward transform Step 3: for Step 4: when

41 Modified matrix factorization: the process of forward transform Step 5:

42 Modified matrix factorization: Accuracy analysis Preliminaries 1. where 2. is a random variable and uniformly distribute in [, ] 3. and

43 Modified matrix factorization: Accuracy analysis The process in step (2)-(4) can be rewritten as:

44 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

45 Modified matrix factorization: Accuracy analysis Using (20) to estimate the normalized root mean square error (NRMSE) and use it to measure the accuracy:

46 Modified matrix factorization: Accuracy analysis Notice (20),we find the NRMSE which is dominated by and. So we let the entries of T 2 and T 3 as small as possible. That why we multiply P, D, and Q to G.

47 Modified matrix factorization: Accuracy analysis Example

48 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

49 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.

50 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”