# Patent

## Title
Processes and apparatus for deriving order-16 integer transforms

## Abstract

Apparatus, systems and techniques based on an integer transform for encoding and decoding video or image signals, including apparatus, systems and techniques for deriving an order-16 integer transform from an order-8 integer transform in image and video coding. In some implementations, eight additions and eight subtractions are used to assign the data elements to be transformed to an intermediate matrix; and then two fast algorithms for the computation of the order-8 transform may be applied to the first 8 vectors of the intermediate matrix, and the last 8 vectors of the intermediate matrix, respectively. The derived order-16 integer transform tends to produce small magnitude and high frequency transformed coefficients, and thus achieve high compressibility.

## Background
CROSS-REFERENCES TO RELATED APPLICATION
This application claims the benefit of U.S. provisional patent application Ser. No. 60/983,862, filed on Oct. 30, 2007, entitled “Order-16 Integer Transform from Order-8 Integer Cosine Transform”, which is incorporated by reference in its entirety as part of the specification of this application.


BACKGROUND
This application relates to digital image and video processing.
Nowadays, there are various applications for digital video communication and storage, and corresponding international standards have been and are continuing to be developed. To achieve low bit rate communications, MPEG-2, MPEG-4 Part 2, and H.263 standards divide a picture into 16×16 non-overlapping blocks called macro block and then use 2 dimensional (2D) order-8 discrete cosine transform (DCT) in digital video compression algorithms used in the coding standards. Recently, new video coding standards, such as H.264/AVC, AVS and SMPTE 421M etc, have been proposed that use 2D order-4 or order-8 transforms to provide a better compressibility than the video standards based on the 2D order-8 DCT. In those standards, the 2D order-4 or order-8 transforms are used so as to seek a trade-off between computational efficiency and coding efficiency.
SUMMARY
This application describes examples and implementations of apparatus, systems and techniques based on an integer transform for encoding and decoding video or image signals, including apparatus, systems and techniques for providing an order-16 integer transform from a H.264/AVC order-8 integer transform with a high energy-packing ability and an improved data compression in the field of image and video coding. The described examples and implementations include apparatus, systems and techniques for deriving an order-16 integer transform from an order-8 integer transform in image and video coding. In some implementations, eight additions and eight subtractions are used to assign the data elements to be transformed to an intermediate matrix; and then two fast algorithms for the computation of the order-8 transform may be applied to the first 8 vectors of the intermediate matrix, and the last 8 vectors of the intermediate matrix, respectively. The derived order-16 integer transform tends to produce small magnitude and high frequency transformed coefficients, and thus achieve high compressibility.
In one aspect, a process is provided to derive an order-16 integer transform from an order-8 integer transform in image and video coding, comprising:
retrieving a data matrix X16×16 to be transformed from a video signal stream;
assigning the retrieved data to an intermediate matrix C16×16 by eight additions and eight subtractions; and
deriving the order-16 integer transform by carrying out the order-8 integer transform for the first 8 vectors of the matrix C16×16, and carrying out one more order-8 integer transform for the last 8 vectors of the matrix C16×16.
In another aspect, there is provided an apparatus for deriving an order-16 integer transform from an order-8 integer transform in image and video coding, comprising:
an assignment unit configured to receive a data matrix X16×16 to be transformed, and then assign the received data to an intermediate matrix C16×16 by eight additions and eight subtractions; and
a transform unit configured to derive the order-16 integer transform by carrying out the order-8 transform for the first 8 vectors of the matrix C16×16, and carrying out one more order-8 integer transform for the last 8 vectors of the matrix C16×16, respectively.
In one implementation, the step of assigning the data matrix X16×16 to the intermediate matrix C16×16 may be carried out according to a rule of




  
    C
    m
  
  =
  
    
      [
      
        
          
            
              c
              
                0
                ,
                m
              
            
          
        
        
          
            
              c
              
                1
                ,
                m
              
            
          
        
        
          
            ...
          
        
        
          
            
              c
              
                7
                ,
                m
              
            
          
        
        
          
            
              c
              
                8
                ,
                m
              
            
          
        
        
          
            
              c
              
                9
                ,
                m
              
            
          
        
        
          
            ...
          
        
        
          
            
              c
              
                15
                ,
                m
              
            
          
        
      
      ]
    
    =
    
      [
      
        
          
            
              
                x
                
                  0
                  ,
                  m
                
              
              +
              
                x
                
                  1
                  ,
                  m
                
              
            
          
        
        
          
            
              
                x
                
                  2
                  ,
                  m
                
              
              +
              
                x
                
                  3
                  ,
                  m
                
              
            
          
        
        
          
            ...
          
        
        
          
            
              
                x
                
                  14
                  ,
                  m
                
              
              +
              
                x
                
                  15
                  ,
                  m
                
              
            
          
        
        
          
            
              
                x
                
                  0
                  ,
                  m
                
              
              -
              
                x
                
                  1
                  ,
                  m
                
              
            
          
        
        
          
            
              
                x
                
                  2
                  ,
                  m
                
              
              -
              
                x
                
                  3
                  ,
                  m
                
              
            
          
        
        
          
            ...
          
        
        
          
            
              
                x
                
                  14
                  ,
                  m
                
              
              -
              
                x
                
                  15
                  ,
                  m
                
              
            
          
        
      
      ]
    
  




wherein Cm is the mth vector of the matrix C16×16, ci,m is the (i, m)th element of C16×16, and xi,m is the (i, m)th element of X16×16, 0≦m≦15 and 0≦i≦15.
Alternatively, the step of assigning the data matrix X16×16 to the intermediate matrix C16×16 may be carried out according to a rule of




  
    C
    m
  
  =
  
    
      [
      
        
          
            
              c
              
                0
                ,
                m
              
            
          
        
        
          
            
              c
              
                1
                ,
                m
              
            
          
        
        
          
            ⋮
          
        
        
          
            
              c
              
                7
                ,
                m
              
            
          
        
        
          
            
              c
              
                8
                ,
                m
              
            
          
        
        
          
            
              c
              
                9
                ,
                m
              
            
          
        
        
          
            
              c
              
                10
                ,
                m
              
            
          
        
        
          
            
              c
              
                11
                ,
                m
              
            
          
        
        
          
            
              c
              
                12
                ,
                m
              
            
          
        
        
          
            
              c
              
                13
                ,
                m
              
            
          
        
        
          
            
              c
              
                14
                ,
                m
              
            
          
        
        
          
            
              c
              
                15
                ,
                m
              
            
          
        
      
      ]
    
    =
    
      [
      
        
          
            
              
                x
                
                  0
                  ,
                  m
                
              
              +
              
                x
                
                  1
                  ,
                  m
                
              
            
          
        
        
          
            
              
                x
                
                  2
                  ,
                  m
                
              
              +
              
                x
                
                  3
                  ,
                  m
                
              
            
          
        
        
          
            ⋮
          
        
        
          
            
              
                x
                
                  14
                  ,
                  m
                
              
              +
              
                x
                
                  15
                  ,
                  m
                
              
            
          
        
        
          
            
              
                x
                
                  0
                  ,
                  m
                
              
              -
              
                x
                
                  1
                  ,
                  m
                
              
            
          
        
        
          
            
              
                x
                
                  3
                  ,
                  m
                
              
              -
              
                x
                
                  2
                  ,
                  m
                
              
            
          
        
        
          
            
              
                x
                
                  4
                  ,
                  m
                
              
              -
              
                x
                
                  5
                  ,
                  m
                
              
            
          
        
        
          
            
              
                x
                
                  7
                  ,
                  m
                
              
              -
              
                x
                
                  6
                  ,
                  m
                
              
            
          
        
        
          
            
              
                x
                
                  8
                  ,
                  m
                
              
              -
              
                x
                
                  9
                  ,
                  m
                
              
            
          
        
        
          
            
              
                x
                
                  11
                  ,
                  m
                
              
              -
              
                x
                
                  10
                  ,
                  m
                
              
            
          
        
        
          
            
              
                x
                
                  12
                  ,
                  m
                
              
              -
              
                x
                
                  13
                  ,
                  m
                
              
            
          
        
        
          
            
              
                x
                
                  15
                  ,
                  m
                
              
              -
              
                x
                
                  14
                  ,
                  m
                
              
            
          
        
      
      ]
    
  




wherein Cm is the mth vector of the matrix C16×16, ci,m is the (i, m)th element of C16×16, and xi,m is the (i, m)th element of X16×16, 0≦m≦15 and 0≦i≦15.
In some implementations, only additions and subtraction operations may be needed. Hence, if an order-8 transform is an integer transform, the resultant order-16 integer transform is also an integer transform and thus can be easily implemented with a good precision if the integers are small. In addition, the derived order-16 transform based on what is described in this application can pack high energy to low frequency coefficients, i.e. the transformed coefficient zu,v with small u and v, and leave little energy to high frequency coefficients, i.e. the transformed coefficient zu,v with large u and v in some cases. Hence, it is an effective tool to improve the compression ability of a coder.
The details of the above and other aspects of the described apparatus, systems and techniques are set forth in the accompanying drawings, the description and claims below.



BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 illustrates an example of a flow chart of a process for deriving a forward 2D order-16 integer transform from an order-8 integer transform;
FIG. 2 illustrates an example of a fast algorithm of 1D order-16 integer transform carried out at the step 102 of FIG. 1;
FIG. 3 illustrates an alternative fast algorithm of 1D order-16 integer transform carried out at the step 102 of FIG. 1; and
FIG. 4 illustrates an example of an apparatus of deriving a forward 2D order-16 integer transform from an order-8 integer transform.



DETAILED DESCRIPTION
A exemplary process for deriving an order-16 transform from an order-8 transform is now described.
FIG. 1 illustrates an example of a process 100 for deriving a forward 2D order-16 integer transform from an order-8 transform. The process 100 begins at step 101, where a data matrix X16×16 is retrieved from video or image signal stream. As an example, the data matrix X16×16 is given as below




  
    
      
        
          X
          
            16
            ×
            16
          
        
        =
        
          
            [
            
              
                
                  
                    x
                    
                      0
                      ,
                      0
                    
                  
                
                
                  
                    x
                    
                      0
                      ,
                      1
                    
                  
                
                
                  ...
                
                
                  
                    x
                    
                      0
                      ,
                      15
                    
                  
                
              
              
                
                  
                    x
                    
                      1
                      ,
                      0
                    
                  
                
                
                  
                    x
                    
                      1
                      ,
                      1
                    
                  
                
                
                  
                    
                  
                
                
                  
                    x
                    
                      1
                      ,
                      15
                    
                  
                
              
              
                
                  ⋮
                
                
                  
                    
                  
                
                
                  ⋱
                
                
                  ⋮
                
              
              
                
                  
                    x
                    
                      15
                      ,
                      0
                    
                  
                
                
                  
                    x
                    
                      15
                      ,
                      1
                    
                  
                
                
                  ...
                
                
                  
                    x
                    
                      15
                      ,
                      15
                    
                  
                
              
            
            ]
          
          .
        
      
    
    
      
        (
        1
        )
      
    
  




At step 102, a 1D (one dimension) integer transform is carried out for each of the 16 column vectors. For the purpose of description, let Xm be one vector of the 16 column vectors of the matrix X16×16, which is denoted as

Xm=[x0,m x1,m . . . x15,m]T  (2).

Herein, the resultant of 1D integer transform is denoted as a matrix Z16×16. Let vector Zm be one vector of the matrix Z16×16, vector Zm=E16×16 Xm, i.e.




  
    
      
        
          Z
          m
        
        =
        
          
            [
            
              
                
                  
                    z
                    
                      0
                      ,
                      m
                    
                  
                
              
              
                
                  
                    z
                    
                      1
                      ,
                      m
                    
                  
                
              
              
                
                  ⋮
                
              
              
                
                  
                    z
                    
                      15
                      ,
                      m
                    
                  
                
              
            
            ]
          
          =
          
            
              
                E
                
                  16
                  ×
                  16
                
              
              ⁢
              
                X
                m
              
            
            =
            
              
                E
                
                  16
                  ×
                  16
                
              
              ⁡
              
                [
                
                  
                    
                      
                        x
                        
                          0
                          ,
                          m
                        
                      
                    
                  
                  
                    
                      
                        x
                        
                          1
                          ,
                          m
                        
                      
                    
                  
                  
                    
                      ⋮
                    
                  
                  
                    
                      
                        x
                        
                          15
                          ,
                          m
                        
                      
                    
                  
                
                ]
              
            
          
        
      
    
    
      
        (
        3
        )
      
    
  




wherein E16×16 is an order-16 transform, which is to be explained in detail later.

To improve the computational efficiency, a fast algorithm can be used to derive an order-16 integer transform from a conventional order-8 transform, the fast algorithm in one implementation includes:
a) eight additions and eight subtractions; and
b) computing the order-8 transform E8×8 twice by using a known fast algorithm of the order-8 transform.
Specific examples of the fast algorithm are discussed below.
Example 1
This Example will be discussed in referring to FIG. 2, in which an example of a fast algorithm of 1D order-16 integer transform carried out at step 102 of FIG. 1 is illustrated.
Referring to FIG. 2, the vector Xm is firstly transformed into an intermediate vector Cm by eight additions and eight subtractions as given in equation (4)




  
    
      
        
          C
          m
        
        =
        
          
            [
            
              
                
                  
                    c
                    
                      0
                      ,
                      m
                    
                  
                
              
              
                
                  
                    c
                    
                      1
                      ,
                      m
                    
                  
                
              
              
                
                  ...
                
              
              
                
                  
                    c
                    
                      7
                      ,
                      m
                    
                  
                
              
              
                
                  
                    c
                    
                      8
                      ,
                      m
                    
                  
                
              
              
                
                  
                    c
                    
                      9
                      ,
                      m
                    
                  
                
              
              
                
                  ...
                
              
              
                
                  
                    c
                    
                      15
                      ,
                      m
                    
                  
                
              
            
            ]
          
          =
          
            
              [
              
                
                  
                    
                      
                        x
                        
                          0
                          ,
                          m
                        
                      
                      +
                      
                        x
                        
                          1
                          ,
                          m
                        
                      
                    
                  
                
                
                  
                    
                      
                        x
                        
                          2
                          ,
                          m
                        
                      
                      +
                      
                        x
                        
                          3
                          ,
                          m
                        
                      
                    
                  
                
                
                  
                    ...
                  
                
                
                  
                    
                      
                        x
                        
                          14
                          ,
                          m
                        
                      
                      +
                      
                        x
                        
                          15
                          ,
                          m
                        
                      
                    
                  
                
                
                  
                    
                      
                        x
                        
                          0
                          ,
                          m
                        
                      
                      -
                      
                        x
                        
                          1
                          ,
                          m
                        
                      
                    
                  
                
                
                  
                    
                      
                        x
                        
                          2
                          ,
                          m
                        
                      
                      -
                      
                        x
                        
                          3
                          ,
                          m
                        
                      
                    
                  
                
                
                  
                    ...
                  
                
                
                  
                    
                      
                        x
                        
                          14
                          ,
                          m
                        
                      
                      -
                      
                        x
                        
                          15
                          ,
                          m
                        
                      
                    
                  
                
              
              ]
            
            .
          
        
      
    
    
      
        (
        4
        )
      
    
  




That is, ci,m=x2i,m+x2i+1,m, ci+8,m=x2i,m−x2i+1,m, where 0≦i≦7;
Then, a resultant matrix Z16×16 is calculated by the following equations:

[z0,m z1,m . . . z7,m]T=E8×8[c0,m c1,m . . . c7,m]T  (5.1), and

[z15,m z14,m . . . z8,m]T=E8×8[c8,m c9,m . . . c15,m]T  (5.2).

Zm is of one vector of the matrix Z16×16 (0≦m≦15), and E8×8 is a standard transform used in H.264/AVC, which is showed as follows




  
    
      
        
          E
          
            8
            ×
            8
          
        
        =
        
          
            [
            
              
                
                  8
                
                
                  8
                
                
                  8
                
                
                  8
                
                
                  8
                
                
                  8
                
                
                  8
                
                
                  8
                
              
              
                
                  12
                
                
                  10
                
                
                  6
                
                
                  3
                
                
                  
                    -
                    3
                  
                
                
                  
                    -
                    6
                  
                
                
                  
                    -
                    10
                  
                
                
                  
                    -
                    12
                  
                
              
              
                
                  8
                
                
                  4
                
                
                  
                    -
                    4
                  
                
                
                  
                    -
                    8
                  
                
                
                  
                    -
                    8
                  
                
                
                  
                    -
                    4
                  
                
                
                  4
                
                
                  8
                
              
              
                
                  10
                
                
                  3
                
                
                  12
                
                
                  6
                
                
                  6
                
                
                  12
                
                
                  3
                
                
                  10
                
              
              
                
                  8
                
                
                  8
                
                
                  8
                
                
                  8
                
                
                  8
                
                
                  8
                
                
                  8
                
                
                  8
                
              
              
                
                  6
                
                
                  12
                
                
                  3
                
                
                  10
                
                
                  10
                
                
                  3
                
                
                  12
                
                
                  6
                
              
              
                
                  4
                
                
                  8
                
                
                  8
                
                
                  4
                
                
                  4
                
                
                  8
                
                
                  8
                
                
                  4
                
              
              
                
                  3
                
                
                  6
                
                
                  10
                
                
                  12
                
                
                  12
                
                
                  10
                
                
                  6
                
                
                  3
                
              
            
            ]
          
          .
        
      
    
    
      
        (
        6
        )
      
    
  




Since the order-8 transform is well known to those skilled in the art, the detailed discussion thereof is omitted.
The transformation of the data matrix X16×16 into the matrix Z16×16 using the above order-8 transform can be represented as one order-16 transform by the following equation:

Z16×16=E16×16X16×16  (7)

where,





  
    
      
        
          E
          
            16
            ×
            16
          
        
        =
        
          
            [
            
              
                
                  8
                
                
                  8
                
                
                  8
                
                
                  8
                
                
                  8
                
                
                  8
                
                
                  8
                
                
                  8
                
                
                  8
                
                
                  8
                
                
                  8
                
                
                  8
                
                
                  8
                
                
                  8
                
                
                  8
                
                
                  8
                
              
              
                
                  12
                
                
                  12
                
                
                  10
                
                
                  10
                
                
                  6
                
                
                  6
                
                
                  3
                
                
                  3
                
                
                  
                    -
                    3
                  
                
                
                  
                    -
                    3
                  
                
                
                  
                    -
                    6
                  
                
                
                  
                    -
                    6
                  
                
                
                  
                    -
                    10
                  
                
                
                  
                    -
                    10
                  
                
                
                  
                    -
                    12
                  
                
                
                  
                    -
                    12
                  
                
              
              
                
                  8
                
                
                  8
                
                
                  4
                
                
                  4
                
                
                  
                    -
                    4
                  
                
                
                  
                    -
                    4
                  
                
                
                  
                    -
                    8
                  
                
                
                  
                    -
                    8
                  
                
                
                  
                    -
                    8
                  
                
                
                  
                    -
                    8
                  
                
                
                  
                    -
                    4
                  
                
                
                  
                    -
                    4
                  
                
                
                  4
                
                
                  4
                
                
                  8
                
                
                  8
                
              
              
                
                  10
                
                
                  10
                
                
                  
                    -
                    3
                  
                
                
                  
                    -
                    3
                  
                
                
                  
                    -
                    12
                  
                
                
                  
                    -
                    12
                  
                
                
                  
                    -
                    6
                  
                
                
                  
                    -
                    6
                  
                
                
                  6
                
                
                  6
                
                
                  12
                
                
                  12
                
                
                  3
                
                
                  3
                
                
                  
                    -
                    10
                  
                
                
                  
                    -
                    10
                  
                
              
              
                
                  8
                
                
                  8
                
                
                  
                    -
                    8
                  
                
                
                  
                    -
                    8
                  
                
                
                  
                    -
                    8
                  
                
                
                  
                    -
                    8
                  
                
                
                  8
                
                
                  8
                
                
                  8
                
                
                  8
                
                
                  
                    -
                    8
                  
                
                
                  
                    -
                    8
                  
                
                
                  
                    -
                    8
                  
                
                
                  
                    -
                    8
                  
                
                
                  8
                
                
                  8
                
              
              
                
                  6
                
                
                  6
                
                
                  
                    -
                    12
                  
                
                
                  
                    -
                    12
                  
                
                
                  3
                
                
                  3
                
                
                  10
                
                
                  10
                
                
                  
                    -
                    10
                  
                
                
                  
                    -
                    10
                  
                
                
                  
                    -
                    3
                  
                
                
                  
                    -
                    3
                  
                
                
                  12
                
                
                  12
                
                
                  
                    -
                    6
                  
                
                
                  
                    -
                    6
                  
                
              
              
                
                  4
                
                
                  4
                
                
                  
                    -
                    8
                  
                
                
                  
                    -
                    8
                  
                
                
                  8
                
                
                  8
                
                
                  
                    -
                    4
                  
                
                
                  
                    -
                    4
                  
                
                
                  
                    -
                    4
                  
                
                
                  
                    -
                    4
                  
                
                
                  8
                
                
                  8
                
                
                  
                    -
                    8
                  
                
                
                  
                    -
                    8
                  
                
                
                  4
                
                
                  4
                
              
              
                
                  3
                
                
                  3
                
                
                  
                    -
                    6
                  
                
                
                  
                    -
                    6
                  
                
                
                  10
                
                
                  10
                
                
                  
                    -
                    12
                  
                
                
                  
                    -
                    12
                  
                
                
                  12
                
                
                  12
                
                
                  
                    -
                    10
                  
                
                
                  
                    -
                    10
                  
                
                
                  6
                
                
                  6
                
                
                  
                    -
                    3
                  
                
                
                  
                    -
                    3
                  
                
              
              
                
                  3
                
                
                  
                    -
                    3
                  
                
                
                  
                    -
                    6
                  
                
                
                  6
                
                
                  10
                
                
                  
                    -
                    10
                  
                
                
                  
                    -
                    12
                  
                
                
                  12
                
                
                  12
                
                
                  
                    -
                    12
                  
                
                
                  
                    -
                    10
                  
                
                
                  10
                
                
                  6
                
                
                  
                    -
                    6
                  
                
                
                  
                    -
                    3
                  
                
                
                  3
                
              
              
                
                  4
                
                
                  
                    -
                    4
                  
                
                
                  
                    -
                    8
                  
                
                
                  8
                
                
                  8
                
                
                  
                    -
                    8
                  
                
                
                  
                    -
                    4
                  
                
                
                  4
                
                
                  
                    -
                    4
                  
                
                
                  4
                
                
                  8
                
                
                  
                    -
                    8
                  
                
                
                  
                    -
                    8
                  
                
                
                  8
                
                
                  4
                
                
                  
                    -
                    4
                  
                
              
              
                
                  6
                
                
                  
                    -
                    6
                  
                
                
                  
                    -
                    12
                  
                
                
                  12
                
                
                  3
                
                
                  
                    -
                    3
                  
                
                
                  10
                
                
                  
                    -
                    10
                  
                
                
                  
                    -
                    10
                  
                
                
                  10
                
                
                  
                    -
                    3
                  
                
                
                  3
                
                
                  12
                
                
                  
                    -
                    12
                  
                
                
                  
                    -
                    6
                  
                
                
                  6
                
              
              
                
                  8
                
                
                  
                    -
                    8
                  
                
                
                  
                    -
                    8
                  
                
                
                  8
                
                
                  
                    -
                    8
                  
                
                
                  8
                
                
                  8
                
                
                  
                    -
                    8
                  
                
                
                  8
                
                
                  
                    -
                    8
                  
                
                
                  
                    -
                    8
                  
                
                
                  8
                
                
                  
                    -
                    8
                  
                
                
                  8
                
                
                  8
                
                
                  
                    -
                    8
                  
                
              
              
                
                  10
                
                
                  
                    -
                    10
                  
                
                
                  
                    -
                    3
                  
                
                
                  3
                
                
                  
                    -
                    12
                  
                
                
                  12
                
                
                  
                    -
                    6
                  
                
                
                  6
                
                
                  6
                
                
                  
                    -
                    6
                  
                
                
                  12
                
                
                  
                    -
                    12
                  
                
                
                  3
                
                
                  
                    -
                    3
                  
                
                
                  
                    -
                    10
                  
                
                
                  10
                
              
              
                
                  8
                
                
                  
                    -
                    8
                  
                
                
                  4
                
                
                  
                    -
                    4
                  
                
                
                  
                    -
                    4
                  
                
                
                  4
                
                
                  
                    -
                    8
                  
                
                
                  8
                
                
                  
                    -
                    8
                  
                
                
                  8
                
                
                  
                    -
                    4
                  
                
                
                  4
                
                
                  4
                
                
                  
                    -
                    4
                  
                
                
                  8
                
                
                  
                    -
                    8
                  
                
              
              
                
                  12
                
                
                  
                    -
                    12
                  
                
                
                  10
                
                
                  
                    -
                    10
                  
                
                
                  6
                
                
                  
                    -
                    6
                  
                
                
                  3
                
                
                  
                    -
                    3
                  
                
                
                  
                    -
                    3
                  
                
                
                  3
                
                
                  
                    -
                    6
                  
                
                
                  6
                
                
                  
                    -
                    10
                  
                
                
                  10
                
                
                  
                    -
                    12
                  
                
                
                  12
                
              
              
                
                  8
                
                
                  
                    -
                    8
                  
                
                
                  8
                
                
                  
                    -
                    8
                  
                
                
                  8
                
                
                  
                    -
                    8
                  
                
                
                  8
                
                
                  
                    -
                    8
                  
                
                
                  8
                
                
                  
                    -
                    8
                  
                
                
                  8
                
                
                  
                    -
                    8
                  
                
                
                  8
                
                
                  
                    -
                    8
                  
                
                
                  8
                
                
                  
                    -
                    8
                  
                
              
            
            ]
          
          .
        
      
    
    
      
        (
        8
        )
      
    
  




Example 2
This Example will be discussed in referring to FIG. 3, which illustrates another example of a fast algorithm of 1D order-16 integer transform carried out at step 102 of FIG. 1.
Referring to FIG. 3, the vector Xm is firstly transformed into an intermediate vector Cm by eight additions and eight subtractions as given in equation (9)




  
    
      
        
          C
          m
        
        =
        
          
            [
            
              
                
                  
                    c
                    
                      0
                      ,
                      m
                    
                  
                
              
              
                
                  
                    c
                    
                      1
                      ,
                      m
                    
                  
                
              
              
                
                  ⋮
                
              
              
                
                  
                    c
                    
                      7
                      ,
                      m
                    
                  
                
              
              
                
                  
                    c
                    
                      8
                      ,
                      m
                    
                  
                
              
              
                
                  
                    c
                    
                      9
                      ,
                      m
                    
                  
                
              
              
                
                  
                    c
                    
                      10
                      ,
                      m
                    
                  
                
              
              
                
                  
                    c
                    
                      11
                      ,
                      m
                    
                  
                
              
              
                
                  
                    c
                    
                      12
                      ,
                      m
                    
                  
                
              
              
                
                  
                    c
                    
                      13
                      ,
                      m
                    
                  
                
              
              
                
                  
                    c
                    
                      14
                      ,
                      m
                    
                  
                
              
              
                
                  
                    c
                    
                      15
                      ,
                      m
                    
                  
                
              
            
            ]
          
          =
          
            
              [
              
                
                  
                    
                      
                        x
                        
                          0
                          ,
                          m
                        
                      
                      +
                      
                        x
                        
                          1
                          ,
                          m
                        
                      
                    
                  
                
                
                  
                    
                      
                        x
                        
                          2
                          ,
                          m
                        
                      
                      +
                      
                        x
                        
                          3
                          ,
                          m
                        
                      
                    
                  
                
                
                  
                    ⋮
                  
                
                
                  
                    
                      
                        x
                        
                          14
                          ,
                          m
                        
                      
                      +
                      
                        x
                        
                          15
                          ,
                          m
                        
                      
                    
                  
                
                
                  
                    
                      
                        x
                        
                          0
                          ,
                          m
                        
                      
                      -
                      
                        x
                        
                          1
                          ,
                          m
                        
                      
                    
                  
                
                
                  
                    
                      
                        x
                        
                          3
                          ,
                          m
                        
                      
                      -
                      
                        x
                        
                          2
                          ,
                          m
                        
                      
                    
                  
                
                
                  
                    
                      
                        x
                        
                          4
                          ,
                          m
                        
                      
                      -
                      
                        x
                        
                          5
                          ,
                          m
                        
                      
                    
                  
                
                
                  
                    
                      
                        x
                        
                          7
                          ,
                          m
                        
                      
                      -
                      
                        x
                        
                          6
                          ,
                          m
                        
                      
                    
                  
                
                
                  
                    
                      
                        x
                        
                          8
                          ,
                          m
                        
                      
                      -
                      
                        x
                        
                          9
                          ,
                          m
                        
                      
                    
                  
                
                
                  
                    
                      
                        x
                        
                          11
                          ,
                          m
                        
                      
                      -
                      
                        x
                        
                          10
                          ,
                          m
                        
                      
                    
                  
                
                
                  
                    
                      
                        x
                        
                          12
                          ,
                          m
                        
                      
                      -
                      
                        x
                        
                          13
                          ,
                          m
                        
                      
                    
                  
                
                
                  
                    
                      
                        x
                        
                          15
                          ,
                          m
                        
                      
                      -
                      
                        x
                        
                          14
                          ,
                          m
                        
                      
                    
                  
                
              
              ]
            
            .
          
        
      
    
    
      
        (
        9
        )
      
    
  




That is, ci,m=x2i,m+x2i+1,m, ci+8,m=x2i,m−x2i+1,m, where 0≦i≦7 and i is even, ci+8,m=x2i+1,m−x2i,m, where 0≦i≦7 and i is odd.
Then, a resultant matrix Z16×16 is calculated by using the standard transform E8×8 of Standard H.264/AVC, according to the following equations:

[z0,m z1,m . . . z7,m]T=E8×8[c0,m c1,m . . . c7,m]T  (5.1′), and

[z8,m z9,m . . . z15,m]T=E8×8[c8,m c9,m . . . c15,m]T  (5.2′).

Then, the transformation of the data matrix X16×16 into the matrix Z16×16 using the above order-8 transform E8×8 can be represented as one order-16 transform by the following equation:

Z16×16=E16×16X16×16  (10)

where,




  
    
      
        
          E
          
            16
            ×
            16
          
        
        =
        
          
            [
            
              
                
                  8
                
                
                  8
                
                
                  8
                
                
                  8
                
                
                  8
                
                
                  8
                
                
                  8
                
                
                  8
                
                
                  8
                
                
                  8
                
                
                  8
                
                
                  8
                
                
                  8
                
                
                  8
                
                
                  8
                
                
                  8
                
              
              
                
                  12
                
                
                  12
                
                
                  10
                
                
                  10
                
                
                  6
                
                
                  6
                
                
                  3
                
                
                  3
                
                
                  
                    -
                    3
                  
                
                
                  
                    -
                    3
                  
                
                
                  
                    -
                    6
                  
                
                
                  
                    -
                    6
                  
                
                
                  
                    -
                    10
                  
                
                
                  
                    -
                    10
                  
                
                
                  
                    -
                    12
                  
                
                
                  
                    -
                    12
                  
                
              
              
                
                  8
                
                
                  8
                
                
                  4
                
                
                  4
                
                
                  
                    -
                    4
                  
                
                
                  
                    -
                    4
                  
                
                
                  
                    -
                    8
                  
                
                
                  
                    -
                    8
                  
                
                
                  
                    -
                    8
                  
                
                
                  
                    -
                    8
                  
                
                
                  
                    -
                    4
                  
                
                
                  
                    -
                    4
                  
                
                
                  4
                
                
                  4
                
                
                  8
                
                
                  8
                
              
              
                
                  10
                
                
                  10
                
                
                  
                    -
                    3
                  
                
                
                  
                    -
                    3
                  
                
                
                  
                    -
                    12
                  
                
                
                  
                    -
                    12
                  
                
                
                  
                    -
                    6
                  
                
                
                  
                    -
                    6
                  
                
                
                  6
                
                
                  6
                
                
                  12
                
                
                  12
                
                
                  3
                
                
                  3
                
                
                  
                    -
                    10
                  
                
                
                  
                    -
                    10
                  
                
              
              
                
                  8
                
                
                  8
                
                
                  
                    -
                    8
                  
                
                
                  
                    -
                    8
                  
                
                
                  
                    -
                    8
                  
                
                
                  
                    -
                    8
                  
                
                
                  8
                
                
                  8
                
                
                  8
                
                
                  8
                
                
                  
                    -
                    8
                  
                
                
                  
                    -
                    8
                  
                
                
                  
                    -
                    8
                  
                
                
                  
                    -
                    8
                  
                
                
                  8
                
                
                  8
                
              
              
                
                  6
                
                
                  6
                
                
                  
                    -
                    12
                  
                
                
                  
                    -
                    12
                  
                
                
                  3
                
                
                  3
                
                
                  10
                
                
                  10
                
                
                  
                    -
                    10
                  
                
                
                  
                    -
                    10
                  
                
                
                  
                    -
                    3
                  
                
                
                  
                    -
                    3
                  
                
                
                  12
                
                
                  12
                
                
                  
                    -
                    6
                  
                
                
                  
                    -
                    6
                  
                
              
              
                
                  4
                
                
                  4
                
                
                  
                    -
                    8
                  
                
                
                  
                    -
                    8
                  
                
                
                  8
                
                
                  8
                
                
                  
                    -
                    4
                  
                
                
                  
                    -
                    4
                  
                
                
                  
                    -
                    4
                  
                
                
                  
                    -
                    4
                  
                
                
                  8
                
                
                  8
                
                
                  
                    -
                    8
                  
                
                
                  
                    -
                    8
                  
                
                
                  4
                
                
                  4
                
              
              
                
                  3
                
                
                  3
                
                
                  
                    -
                    6
                  
                
                
                  
                    -
                    6
                  
                
                
                  10
                
                
                  10
                
                
                  
                    -
                    12
                  
                
                
                  
                    -
                    12
                  
                
                
                  12
                
                
                  12
                
                
                  
                    -
                    10
                  
                
                
                  
                    -
                    10
                  
                
                
                  6
                
                
                  6
                
                
                  
                    -
                    3
                  
                
                
                  
                    -
                    3
                  
                
              
              
                
                  8
                
                
                  
                    -
                    8
                  
                
                
                  
                    -
                    8
                  
                
                
                  8
                
                
                  8
                
                
                  
                    -
                    8
                  
                
                
                  
                    -
                    8
                  
                
                
                  8
                
                
                  8
                
                
                  
                    -
                    8
                  
                
                
                  
                    -
                    8
                  
                
                
                  8
                
                
                  8
                
                
                  
                    -
                    8
                  
                
                
                  
                    -
                    8
                  
                
                
                  8
                
              
              
                
                  12
                
                
                  
                    -
                    12
                  
                
                
                  
                    -
                    10
                  
                
                
                  10
                
                
                  6
                
                
                  
                    -
                    6
                  
                
                
                  
                    -
                    3
                  
                
                
                  3
                
                
                  
                    -
                    3
                  
                
                
                  3
                
                
                  6
                
                
                  
                    -
                    6
                  
                
                
                  
                    -
                    10
                  
                
                
                  10
                
                
                  12
                
                
                  
                    -
                    12
                  
                
              
              
                
                  8
                
                
                  
                    -
                    8
                  
                
                
                  
                    -
                    4
                  
                
                
                  4
                
                
                  
                    -
                    4
                  
                
                
                  4
                
                
                  8
                
                
                  
                    -
                    8
                  
                
                
                  
                    -
                    8
                  
                
                
                  8
                
                
                  4
                
                
                  
                    -
                    4
                  
                
                
                  4
                
                
                  
                    -
                    4
                  
                
                
                  
                    -
                    8
                  
                
                
                  8
                
              
              
                
                  10
                
                
                  
                    -
                    10
                  
                
                
                  3
                
                
                  
                    -
                    3
                  
                
                
                  
                    -
                    12
                  
                
                
                  12
                
                
                  6
                
                
                  
                    -
                    6
                  
                
                
                  6
                
                
                  
                    -
                    6
                  
                
                
                  
                    -
                    12
                  
                
                
                  12
                
                
                  3
                
                
                  
                    -
                    3
                  
                
                
                  10
                
                
                  
                    -
                    10
                  
                
              
              
                
                  8
                
                
                  
                    -
                    8
                  
                
                
                  8
                
                
                  
                    -
                    8
                  
                
                
                  
                    -
                    8
                  
                
                
                  8
                
                
                  
                    -
                    8
                  
                
                
                  8
                
                
                  8
                
                
                  
                    -
                    8
                  
                
                
                  8
                
                
                  
                    -
                    8
                  
                
                
                  
                    -
                    8
                  
                
                
                  8
                
                
                  
                    -
                    8
                  
                
                
                  8
                
              
              
                
                  6
                
                
                  
                    -
                    6
                  
                
                
                  12
                
                
                  
                    -
                    12
                  
                
                
                  3
                
                
                  
                    -
                    3
                  
                
                
                  
                    -
                    10
                  
                
                
                  10
                
                
                  
                    -
                    10
                  
                
                
                  10
                
                
                  3
                
                
                  
                    -
                    3
                  
                
                
                  12
                
                
                  
                    -
                    12
                  
                
                
                  6
                
                
                  
                    -
                    6
                  
                
              
              
                
                  4
                
                
                  
                    -
                    4
                  
                
                
                  8
                
                
                  
                    -
                    8
                  
                
                
                  8
                
                
                  
                    -
                    8
                  
                
                
                  4
                
                
                  
                    -
                    4
                  
                
                
                  
                    -
                    4
                  
                
                
                  4
                
                
                  
                    -
                    8
                  
                
                
                  8
                
                
                  
                    -
                    8
                  
                
                
                  8
                
                
                  
                    -
                    4
                  
                
                
                  4
                
              
              
                
                  3
                
                
                  
                    -
                    3
                  
                
                
                  6
                
                
                  
                    -
                    6
                  
                
                
                  10
                
                
                  
                    -
                    10
                  
                
                
                  12
                
                
                  
                    -
                    12
                  
                
                
                  12
                
                
                  
                    -
                    12
                  
                
                
                  6
                
                
                  
                    -
                    6
                  
                
                
                  6
                
                
                  
                    -
                    6
                  
                
                
                  3
                
                
                  
                    -
                    3
                  
                
              
            
            ]
          
          .
        
      
    
    
      
        (
        11
        )
      
    
  




A 2D fast algorithm for the order-16 integer transform can be achieved by performing the 1D integer transform in vertical for each column in X16×16 to form matrix Z16×16 of elements zi,j and then in horizontal for each row in Z16×16 to form matrix Z16×16′ of elements zi,j′ as given in equation (12)

Z16×16′=E16×16X16×16E16×16T.  (12).

Specifically, referring to FIG. 1 again, at step 102, for each of 16 column vectors of the matrix X16×16, the above mentioned 1D integer transform is carried out to achieve an intermediate matrix Z16×16, and then at step 103, the above mentioned 1D integer transform is carried out again for each of the 16 row vectors of the intermediate matrix Z16×16 so as to achieve the resultant matrix Z16×16′.
In FIG. 1, step 102 is performed before step 103. Alternatively, one may choose to perform step 103 before step 102. In other words, one may perform the 1D transform in horizontal for each row in X16×16 first. Such step is represented by the operation X16×16ET16×16 in (12). It is then followed by the 1D transform in vertical, which is represented by the multiplication of E16×16 in (12). The same Z16×16′ can be obtained in either way.
As is well known in the art, a transform is good if it can pack more energy into low frequency coefficients (i.e. z′u,v with small u, v) and leave little energy to high frequency coefficients (i.e. z′u,v with large u, v). In general, a transform with its low frequency basis vectors (i.e. the first few row of E16×16) resembling a slowly changing vector xi of the matrix X16×16 is good. As is shown in equation (8), the first three basis vectors of transform E16×16 are as follows

E1=[12 12 10 10 6 6 3 3 −3 −3 −6 −6 −10 −10 −12 −12]

E2=[8 8 4 4 −4 −4 −8 −8 −8 −8 −4 −4 4 4 8 8]

E3=[10 10 −3 −3 −12 −12 −6 −6 6 6 12 12 3 3 −10 −10].

In most cases, most of the energy of X16×16 is packed into low frequency coefficients related to the first several basis vectors of transform E16×16 like E1, E2, and E3 etc. The remaining small amount of energy of X16×16 is represented by other coefficients which become zero after quantization. As a result, higher compression rates can be achieved.
Then, at step 104, the matrix Z16×16′ is normalized by a scaling matrix K16×16 to form a matrix Y16×16, as given in equation (10)

yi,j=z′i,j·K16×16(i,j)  (10)

where K16×16 (i,j) is the (i,j)th element of K16×16.

The scaling matrix K16×16 is generated to ensure that all the row vectors of the transform E16×16 have a magnitude of 1. It should be appreciated that the scaling step 104 is not necessary and thus can be removed in some implementations.
The scaling matrix K16×16 is derived from a scaling matrix K8×8, which is listed as follows




  
    
      
        
          K
          
            8
            ×
            8
          
        
        =
        
          
            [
            
              
                
                  
                    a
                    2
                  
                
                
                  ab
                
                
                  ac
                
                
                  ab
                
                
                  
                    a
                    2
                  
                
                
                  ab
                
                
                  ac
                
                
                  ab
                
              
              
                
                  ab
                
                
                  
                    b
                    2
                  
                
                
                  bc
                
                
                  
                    b
                    2
                  
                
                
                  ab
                
                
                  
                    b
                    2
                  
                
                
                  bc
                
                
                  
                    b
                    2
                  
                
              
              
                
                  ac
                
                
                  bc
                
                
                  
                    c
                    2
                  
                
                
                  bc
                
                
                  ac
                
                
                  bc
                
                
                  
                    c
                    2
                  
                
                
                  bc
                
              
              
                
                  ab
                
                
                  
                    b
                    2
                  
                
                
                  bc
                
                
                  
                    b
                    2
                  
                
                
                  ab
                
                
                  
                    b
                    2
                  
                
                
                  bc
                
                
                  
                    b
                    2
                  
                
              
              
                
                  
                    a
                    2
                  
                
                
                  ab
                
                
                  ac
                
                
                  ab
                
                
                  
                    a
                    2
                  
                
                
                  ab
                
                
                  ac
                
                
                  ab
                
              
              
                
                  ab
                
                
                  
                    b
                    2
                  
                
                
                  bc
                
                
                  
                    b
                    2
                  
                
                
                  ab
                
                
                  
                    b
                    2
                  
                
                
                  bc
                
                
                  
                    b
                    2
                  
                
              
              
                
                  ac
                
                
                  bc
                
                
                  
                    c
                    2
                  
                
                
                  bc
                
                
                  ac
                
                
                  bc
                
                
                  
                    c
                    2
                  
                
                
                  bc
                
              
              
                
                  ab
                
                
                  
                    b
                    2
                  
                
                
                  bc
                
                
                  
                    b
                    2
                  
                
                
                  ab
                
                
                  
                    b
                    2
                  
                
                
                  bc
                
                
                  
                    b
                    2
                  
                
              
            
            ]
          
          ⁢
          
            
          
          ⁢
          where
          ⁢
          
            
          
          ⁢
          
            {
            
              
                
                  
                    
                      a
                      =
                      
                        1
                        
                          512
                        
                      
                    
                  
                
                
                  
                    
                      b
                      =
                      
                        1
                        
                          578
                        
                      
                    
                  
                
                
                  
                    
                      c
                      =
                      
                        1
                        
                          320
                        
                      
                    
                  
                
              
              .
            
          
        
      
    
    
      
        (
        11
        )
      
    
  




And then, where the transform E16×16 is given as equation (8), the scaling matrix K16×16 is represented as




Where the transform E16×16 is given as equation (11), the scaling matrix K16×16 is represented as




From the above, the 2D integer transform of X into Y is discussed. It should be appreciated that an inverse 2D integer transform of Y into X may be easily implemented based on the above mentioned 2D transform of X into Y, and be represented as




  
    
      
        
          
            
              X
              =
              
              ⁢
              
                
                  
                    E
                    
                      16
                      ×
                      16
                    
                    T
                  
                  ⁡
                  
                    (
                    
                      Y
                      ⊗
                      
                        K
                        
                          16
                          ×
                          16
                        
                      
                    
                    )
                  
                
                ⁢
                
                  E
                  
                    16
                    ×
                    16
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    E
                    
                      16
                      ×
                      16
                    
                    T
                  
                  ⁡
                  
                    (
                    
                      
                        [
                        
                          
                            
                              
                                y
                                
                                  0
                                  ,
                                  0
                                
                              
                            
                            
                              
                                y
                                
                                  0
                                  ,
                                  1
                                
                              
                            
                            
                              ...
                            
                            
                              
                                y
                                
                                  0
                                  ,
                                  15
                                
                              
                            
                          
                          
                            
                              
                                y
                                
                                  1
                                  ,
                                  0
                                
                              
                            
                            
                              
                                y
                                
                                  1
                                  ,
                                  1
                                
                              
                            
                            
                              
                                
                              
                            
                            
                              
                                y
                                
                                  1
                                  ,
                                  15
                                
                              
                            
                          
                          
                            
                              ⋮
                            
                            
                              
                                
                              
                            
                            
                              ⋱
                            
                            
                              ⋮
                            
                          
                          
                            
                              
                                y
                                
                                  15
                                  ,
                                  0
                                
                              
                            
                            
                              
                                y
                                
                                  15
                                  ,
                                  1
                                
                              
                            
                            
                              ...
                            
                            
                              
                                y
                                
                                  15
                                  ,
                                  15
                                
                              
                            
                          
                        
                        ]
                      
                      ⊗
                      
                        K
                        
                          16
                          ×
                          16
                        
                      
                    
                    )
                  
                
                ⁢
                
                  
                    E
                    
                      16
                      ×
                      16
                    
                  
                  .
                
              
            
          
        
      
    
    
      
        (
        14
        )
      
    
  




Since the inverse 2D integer transform of Y into X can be computed by a similar way as stated above, the detailed description thereof is then omitted.
Then, the process 100 goes to step 105, where a quantization is carried out to convert transform coefficients yi,j of the matrix Y into quantized transform coefficients YQ(i,j), which require less bits for representation. The amount of bit reduction is controlled by a quantization parameter denoted as QP.
In H.264/AVC, a quantization factor A is generated from QP by

A(QP)=2(QP 4)/6 for QP=[0,1,2, . . . ,51].

Note that A(QP) can also be written as follows:

A(QP)=2k·B(QP%6)where B(QP)≈0.625(2QP/6)and k=floor(QP/6)  (15).

The quantized forward transform coefficients are calculated as:




  
    
      
        
          
            
              
                
                  Y
                  Q
                
                ⁡
                
                  (
                  
                    i
                    ,
                    j
                  
                  )
                
              
              =
              
              ⁢
              
                round
                ⁢
                
                  
                
                ⁢
                
                  (
                  
                    
                      
                        (
                        
                          
                            E
                            
                              16
                              ×
                              16
                            
                          
                          ⁢
                          
                            XE
                            
                              16
                              ×
                              16
                            
                            T
                          
                        
                        )
                      
                      ·
                      
                        
                          K
                          
                            16
                            ×
                            16
                          
                        
                        ⁡
                        
                          (
                          
                            i
                            ,
                            j
                          
                          )
                        
                      
                    
                    
                      A
                      ⁡
                      
                        (
                        
                          Q
                          ⁢
                          
                            
                          
                          ⁢
                          P
                        
                        )
                      
                    
                  
                  )
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                round
                (
                
                  
                    
                      
                        (
                        
                          
                            E
                            
                              16
                              ×
                              16
                            
                          
                          ⁢
                          
                            XE
                            
                              16
                              ×
                              16
                            
                            T
                          
                        
                        )
                      
                      ·
                      
                        
                          K
                          
                            16
                            ×
                            16
                          
                        
                        ⁡
                        
                          (
                          
                            i
                            ,
                            j
                          
                          )
                        
                      
                    
                    
                      B
                      ⁡
                      
                        (
                        
                          QP
                          ⁢
                          
                            
                          
                          ⁢
                          %6
                        
                        )
                      
                    
                  
                  ·
                  
                    2
                    
                      -
                      k
                    
                  
                
                )
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  (
                  
                    
                      
                        (
                        
                          
                            E
                            
                              16
                              ×
                              16
                            
                          
                          ⁢
                          
                            XE
                            
                              16
                              ×
                              16
                            
                            T
                          
                        
                        )
                      
                      ·
                      
                        Q
                        ⁡
                        
                          (
                          
                            
                              QP
                              ⁢
                              
                                
                              
                              ⁢
                              %6
                            
                            ,
                            i
                            ,
                            j
                          
                          )
                        
                      
                    
                    +
                    f
                  
                  )
                
                ⁢
                
                  /
                
                ⁢
                
                  2
                  
                    k
                    +
                    n
                  
                
              
            
          
        
      
    
    
      
        (
        16
        )
      
    
  




where f is the rounding offset and ‘/’ denotes division with truncation, and the fixed-point quantization matrix Q(QP) is built up by the scaling matrix K16×16 and the quantization factor A(QP). Q(QP) is equal to

Q(QP%6,i,j)=round(2n·K16×16(i,j)/B(QP%6))  (17)

where n is an integer to control the precision of Q(QP), and the rescaled coefficients YR are computed by equation (18):

YR(i,j)=round(2m·B(QP%6)·YQ(i,j)·K16×16(i,j)).  (18)

where m is the number of fractional bits used for fixed-point rescaling to reduce the rounding error. Notice that YR can be also calculated as

YR(i,j)=R(QP%6,i,j)·YQ(i,j)  (19).

The resealing matrix R(QP%6) is given by:

R(QP%6,i,j)=round(2m·K16×16(i,j)·B(QP%6))  (20).

Finally, the reconstructed values XR given the rescaled transform coefficients YR are equal to:

XR=(E16×16TYRE16×16)//2m  (21).

Here ‘//’ we denote division with rounding. Only values of Q and R for QP%6=0, 1, . . . , 5 have to be pre-calculated and stored. Moreover elements of R or Q for a given QP%6 have only 6 different values.
FIG. 4 illustrates exemplarily an embodiment of an apparatus 1000 for the implementation of the process as shown in FIG. 1. As is showed therein, the apparatus 1000 comprises an assignment unit 10, a transforming unit 20, a scaling unit 30 and a quantization unit 40.
The assignment unit 10 is configured to retrieve a data matrix X16×16 to be transformed, and then to assign of the retrieved data to an intermediate matrix C16×16 under the rule set forth in equation (4) or equation (9).
The transform unit 20 carries out two order-8 transforms for the matrix C16×16 by using the conventional E8×8, which is showed in equation (6), so as to output the matrix Z16×16. E8×8 may be embedded in the transform unit 20. Alternatively, E8×8 may be stored in any other separately memory (not shown). In this case, the transform unit 20 may be capable of retrieving E8×8 from the memory.
The scaling unit 30 receives the transformed data, and then uses a scaling matrix K16×16 to make a Z16×16 to be normalized into a matrix Y16×16. Herein, K16×16 embedded in the scaling unit 30. Alternatively, K16×16 may be stored in any other separately memory (not shown). In this case, the scaling unit 30 may be capable of retrieving K16×16 from the memory.
The quantization unit 40 receives the normalized data Y16×16, and then convert the data Y16×16 into quantized transform coefficients YQ(ij), which requires less bits for representation.
While the assignment unit 10, the transforming unit 20, the scaling unit 30 and the quantization unit 40 are shown in FIG. 4 and described herein as four separate units. Other implementations can also be used. It should be understood that the units 10, 20, 30 and 40 may be integrated into one chip or arranged into multiple units separately, and may be implemented either by software or hardware, or the combination of software and hardware.
The disclosed and other embodiments and the functional operations described in this specification can be implemented in digital electronic circuitry, or in computer software, firmware, or hardware, including the structures disclosed in this specification and their structural equivalents, or in combinations of one or more of them. The disclosed and other embodiments can be implemented as one or more computer program products, i.e., one or more modules of computer program instructions encoded on a computer readable medium for execution by, or to control the operation of, data processing apparatus. The computer readable medium can be a machine-readable storage device, a machine-readable storage substrate, a memory device, a composition of matter effecting a machine-readable propagated signal, or a combination of one or more them. The term “data processing apparatus” encompasses all apparatus, devices, and machines for processing data, including by way of example a programmable processor, a computer, or multiple processors or computers. The apparatus can include, in addition to hardware, code that creates an execution environment for the computer program in question, e.g., code that constitutes processor firmware, a protocol stack, a database management system, an operating system, or a combination of one or more of them. A propagated signal is an artificially generated signal, e.g., a machine-generated electrical, optical, or electromagnetic signal, that is generated to encode information for transmission to suitable receiver apparatus.
A computer program (also known as a program, software, software application, script, or code) can be written in any form of programming language, including compiled or interpreted languages, and it can be deployed in any form, including as a stand alone program or as a module, component, subroutine, or other unit suitable for use in a computing environment. A computer program does not necessarily correspond to a file in a file system. A program can be stored in a portion of a file that holds other programs or data (e.g., one or more scripts stored in a markup language document), in a single file dedicated to the program in question, or in multiple coordinated files (e.g., files that store one or more modules, sub programs, or portions of code). A computer program can be deployed to be executed on one computer or on multiple computers that are located at one site or distributed across multiple sites and interconnected by a communication network.
The processes and logic flows described in this specification can be performed by one or more programmable processors executing one or more computer programs to perform functions by operating on input data and generating output. The processes and logic flows can also be performed by, and apparatus can also be implemented as, special purpose logic circuitry, e.g., an FPGA (field programmable gate array) or an ASIC (application specific integrated circuit).
Processors suitable for the execution of a computer program include, by way of example, both general and special purpose microprocessors, and any one or more processors of any kind of digital computer. Generally, a processor will receive instructions and data from a read only memory or a random access memory or both. The essential elements of a computer are a processor for performing instructions and one or more memory devices for storing instructions and data. Generally, a computer will also include, or be operatively coupled to receive data from or transfer data to, or both, one or more mass storage devices for storing data, e.g., magnetic, magneto optical disks, or optical disks. However, a computer need not have such devices. Computer readable media suitable for storing computer program instructions and data include all forms of non volatile memory, media and memory devices, including by way of example semiconductor memory devices, e.g., EPROM, EEPROM, and flash memory devices; magnetic disks, e.g., internal hard disks or removable disks; magneto optical disks; and CD ROM and DVD-ROM disks. The processor and the memory can be supplemented by, or incorporated in, special purpose logic circuitry.
The disclosed embodiments can be implemented in a computing system that includes a back end component, e.g., as a data server, or that includes a middleware component, e.g., an application server, or that includes a front end component, e.g., a client computer having a graphical user interface or a Web browser through which a user can interact with an implementation of what is disclosed here, or any combination of one or more such back end, middleware, or front end components. The components of the system can be interconnected by any form or medium of digital data communication, e.g., a communication network. Examples of communication networks include a local area network (“LAN”) and a wide area network (“WAN”), e.g., the Internet.
A computer system for implementing the disclosed embodiments can include client computers (clients) and server computers (servers). A client and a server are generally remote from each other and typically interact through a communication network. The relationship of client and server can arise by virtue of computer programs running on the respective computers and having a client-server relationship to each other.
While this specification contains many specifics, these should not be construed as limitations on the scope of any invention or of what may be claimed, but rather as descriptions of features specific to particular embodiments. Certain features that are described in this specification in the context of separate embodiments can also be implemented in combination in a single embodiment. Conversely, various features that are described in the context of a single embodiment can also be implemented in multiple embodiments separately or in any suitable subcombination. Moreover, although features may be described above as acting in certain combinations and even initially claimed as such, one or more features from a claimed combination can in some cases be excised from the combination, and the claimed combination may be directed to a subcombination or variation of a subcombination.
Similarly, operations are depicted in the drawings in a particular order, and such operations should be performed in the particular order shown or in sequential order, and that all illustrated operations be performed, to achieve desirable results. In certain circumstances, multitasking and parallel processing may be advantageous. Moreover, the separation of various system components in the embodiments described above should not be understood as requiring such separation in all embodiments, and it should be understood that the described program components and systems can generally be integrated together in a single software product or packaged into multiple software products.
Thus, particular embodiments have been described. Other embodiments are within the scope of the following claims.

## Claims

1. A process implemented using instructions encoded on computer readable storage medium for deriving an order-16 integer transform using a plurality of order-8 integer transforms in image and video coding, the process comprising:
retrieving a data matrix X16×16 to be transformed;
determining Cm, the mth vector of an intermediate matrix C16×16, based on the retrieved matrix X16×16, by carrying out eight additions and eight subtractions in accordance with a rule of





  
    C
    m
  
  =
  
    
      [
      
        
          
            
              c
              
                0
                ,
                m
              
            
          
        
        
          
            
              c
              
                1
                ,
                m
              
            
          
        
        
          
            ...
          
        
        
          
            
              c
              
                7
                ,
                m
              
            
          
        
        
          
            
              c
              
                8
                ,
                m
              
            
          
        
        
          
            
              c
              
                9
                ,
                m
              
            
          
        
        
          
            ...
          
        
        
          
            
              c
              
                15
                ,
                m
              
            
          
        
      
      ]
    
    =
    
      [
      
        
          
            
              
                x
                
                  0
                  ,
                  m
                
              
              +
              
                x
                
                  1
                  ,
                  m
                
              
            
          
        
        
          
            
              
                x
                
                  2
                  ,
                  m
                
              
              +
              
                x
                
                  3
                  ,
                  m
                
              
            
          
        
        
          
            ...
          
        
        
          
            
              
                x
                
                  14
                  ,
                  m
                
              
              +
              
                x
                
                  15
                  ,
                  m
                
              
            
          
        
        
          
            
              
                x
                
                  0
                  ,
                  m
                
              
              -
              
                x
                
                  1
                  ,
                  m
                
              
            
          
        
        
          
            
              
                x
                
                  2
                  ,
                  m
                
              
              -
              
                x
                
                  3
                  ,
                  m
                
              
            
          
        
        
          
            ...
          
        
        
          
            
              
                x
                
                  14
                  ,
                  m
                
              
              -
              
                x
                
                  15
                  ,
                  m
                
              
            
          
        
      
      ]
    
  



wherein ci,m is the (i, m)th element of C16×16, and xi,m is the (i, m)th element of X16×16, 0≦m≦15 and 0≦i≦15;
deriving the order-16 integer transform by carrying out the order-8 integer transform for the first 8 vectors of the matrix C16×16, and carrying out one more order-8 integer transform for the last 8 vectors of the matrix C16×16; and
normalizing the order-16 integer transform by using a scaling matrix K16×16, where,





and where




  
    a
    =
    
      1
      
        512
      
    
  
  ,
  
    b
    =
    
      
        
          1
          
            578
          
        
        ⁢
        
          
        
        ⁢
        and
        ⁢
        
          
        
        ⁢
        c
      
      =
      
        
          1
          
            320
          
        
        .

2. The process according to claim 1, wherein the deriving is carried out to form a resultant matrix Z16×16 by a rule of


[z0,m z1,m . . . z7,m]T=E8×8[c0,m c1,m . . . c7,m]T 



[z15,m z14,m . . . z8,m]T=E8×8[c8,m c9,m . . . c15,m]T 

wherein zn,m is of the (n,m)th element of the matrix Z16×16, and E8×8 is the standard transform used in H.264/AVC.

3. The process according to claim 1, further comprising:
of quantizing the normalized transform.

4. A process implemented using instructions encoded on computer readable storage medium for deriving an order-16 integer transform using a plurality of order-8 integer transforms in image and video coding, comprising:
retrieving a data matrix X16×16 to be transformed;
determining Cm, the mth vector of an intermediate matrix C16×16, based on the retrieved matrix X16×16, by carrying out eight additions and eight subtractions in accordance with a rule of





  
    C
    m
  
  =
  
    
      [
      
        
          
            
              c
              
                0
                ,
                m
              
            
          
        
        
          
            
              c
              
                1
                ,
                m
              
            
          
        
        
          
            ⋮
          
        
        
          
            
              c
              
                7
                ,
                m
              
            
          
        
        
          
            
              c
              
                8
                ,
                m
              
            
          
        
        
          
            
              c
              
                9
                ,
                m
              
            
          
        
        
          
            
              c
              
                10
                ,
                m
              
            
          
        
        
          
            
              c
              
                11
                ,
                m
              
            
          
        
        
          
            
              c
              
                12
                ,
                m
              
            
          
        
        
          
            
              c
              
                13
                ,
                m
              
            
          
        
        
          
            
              c
              
                14
                ,
                m
              
            
          
        
        
          
            
              c
              
                15
                ,
                m
              
            
          
        
      
      ]
    
    =
    
      [
      
        
          
            
              
                x
                
                  0
                  ,
                  m
                
              
              +
              
                x
                
                  1
                  ,
                  m
                
              
            
          
        
        
          
            
              
                x
                
                  2
                  ,
                  m
                
              
              +
              
                x
                
                  3
                  ,
                  m
                
              
            
          
        
        
          
            ⋮
          
        
        
          
            
              
                x
                
                  14
                  ,
                  m
                
              
              +
              
                x
                
                  15
                  ,
                  m
                
              
            
          
        
        
          
            
              
                x
                
                  0
                  ,
                  m
                
              
              -
              
                x
                
                  1
                  ,
                  m
                
              
            
          
        
        
          
            
              
                x
                
                  3
                  ,
                  m
                
              
              -
              
                x
                
                  2
                  ,
                  m
                
              
            
          
        
        
          
            
              
                x
                
                  4
                  ,
                  m
                
              
              -
              
                x
                
                  5
                  ,
                  m
                
              
            
          
        
        
          
            
              
                x
                
                  7
                  ,
                  m
                
              
              -
              
                x
                
                  6
                  ,
                  m
                
              
            
          
        
        
          
            
              
                x
                
                  8
                  ,
                  m
                
              
              -
              
                x
                
                  9
                  ,
                  m
                
              
            
          
        
        
          
            
              
                x
                
                  11
                  ,
                  m
                
              
              -
              
                x
                
                  10
                  ,
                  m
                
              
            
          
        
        
          
            
              
                x
                
                  12
                  ,
                  m
                
              
              -
              
                x
                
                  13
                  ,
                  m
                
              
            
          
        
        
          
            
              
                x
                
                  15
                  ,
                  m
                
              
              -
              
                x
                
                  14
                  ,
                  m
                
              
            
          
        
      
      ]
    
  



wherein ci,m is the (i, m)th element of C16×16, and xi,m is the (i, m)th element of X16×16, 0≦m≦15 and 0≦i≦15;
deriving the order-16 integer transform by carrying out the order-8 integer transform for the first 8 vectors of the matrix C16×16, and carrying out one more order-8 integer transform for the last 8 vectors of the matrix C16×16; and
normalizing the order-16 integer transform by using a scaling matrix K16×16, where,





and where




  
    a
    =
    
      1
      
        512
      
    
  
  ,
  
    b
    =
    
      
        
          1
          
            578
          
        
        ⁢
        
          
        
        ⁢
        and
        ⁢
        
          
        
        ⁢
        c
      
      =
      
        
          1
          
            320
          
        
        .

5. The process according to claim 4, wherein the deriving is carried out to form a resultant matrix Z16×16 by a rule of


[z0,m z1,m . . . z7,m]T=E8×8[c0,m c1,m . . . c7,m]T 



[z8,m z9,m . . . z15,m]T=E8×8[c8,m c9,m . . . c15,m]T 

wherein zn,m is of the (n,m)th element of the matrix Z16×16, and E8×8 is the standard transform used in H.264/AVC.

6. The process according to claim 4, further comprising:
quantizing the normalized transform.

7. A process implemented using instructions encoded on computer readable storage medium for transforming an order-16 transform in image and video coding, the process comprising:
receiving a data matrix X16×16 to be transformed; and
transforming the received data matrix X16×16 with an order-16 transform matrix E16×16 to generate an output Z16×16 by a rule of Z16×16=E16×16X16×16,
wherein the transform matrix E16×16 is represented as





  
    E
    
      116
      ×
      16
    
  
  =
  
    
      [
      
        
          
            8
          
          
            8
          
          
            8
          
          
            8
          
          
            8
          
          
            8
          
          
            8
          
          
            8
          
          
            8
          
          
            8
          
          
            8
          
          
            8
          
          
            8
          
          
            8
          
          
            8
          
          
            8
          
        
        
          
            12
          
          
            12
          
          
            10
          
          
            10
          
          
            6
          
          
            6
          
          
            3
          
          
            3
          
          
            
              -
              3
            
          
          
            
              -
              3
            
          
          
            
              -
              6
            
          
          
            
              -
              6
            
          
          
            
              -
              10
            
          
          
            
              -
              10
            
          
          
            
              -
              12
            
          
          
            
              -
              12
            
          
        
        
          
            8
          
          
            8
          
          
            4
          
          
            4
          
          
            
              -
              4
            
          
          
            
              -
              4
            
          
          
            
              -
              8
            
          
          
            
              -
              8
            
          
          
            
              -
              8
            
          
          
            
              -
              8
            
          
          
            
              -
              4
            
          
          
            
              -
              4
            
          
          
            4
          
          
            4
          
          
            8
          
          
            8
          
        
        
          
            10
          
          
            10
          
          
            
              -
              3
            
          
          
            
              -
              3
            
          
          
            
              -
              12
            
          
          
            
              -
              12
            
          
          
            
              -
              6
            
          
          
            
              -
              6
            
          
          
            6
          
          
            6
          
          
            12
          
          
            12
          
          
            3
          
          
            3
          
          
            
              -
              10
            
          
          
            
              -
              10
            
          
        
        
          
            8
          
          
            8
          
          
            
              -
              8
            
          
          
            
              -
              8
            
          
          
            
              -
              8
            
          
          
            
              -
              8
            
          
          
            8
          
          
            8
          
          
            8
          
          
            8
          
          
            
              -
              8
            
          
          
            
              -
              8
            
          
          
            
              -
              8
            
          
          
            
              -
              8
            
          
          
            8
          
          
            8
          
        
        
          
            6
          
          
            6
          
          
            
              -
              12
            
          
          
            
              -
              12
            
          
          
            3
          
          
            3
          
          
            10
          
          
            10
          
          
            
              -
              10
            
          
          
            
              -
              10
            
          
          
            
              -
              3
            
          
          
            
              -
              3
            
          
          
            12
          
          
            12
          
          
            
              -
              6
            
          
          
            
              -
              6
            
          
        
        
          
            4
          
          
            4
          
          
            
              -
              8
            
          
          
            
              -
              8
            
          
          
            8
          
          
            8
          
          
            
              -
              4
            
          
          
            
              -
              4
            
          
          
            
              -
              4
            
          
          
            
              -
              4
            
          
          
            8
          
          
            8
          
          
            
              -
              8
            
          
          
            
              -
              8
            
          
          
            4
          
          
            4
          
        
        
          
            3
          
          
            3
          
          
            
              -
              6
            
          
          
            
              -
              6
            
          
          
            10
          
          
            10
          
          
            
              -
              12
            
          
          
            
              -
              12
            
          
          
            12
          
          
            12
          
          
            
              -
              10
            
          
          
            
              -
              10
            
          
          
            6
          
          
            6
          
          
            
              -
              3
            
          
          
            
              -
              3
            
          
        
        
          
            3
          
          
            
              -
              3
            
          
          
            
              -
              6
            
          
          
            6
          
          
            10
          
          
            
              -
              10
            
          
          
            
              -
              12
            
          
          
            12
          
          
            12
          
          
            
              -
              12
            
          
          
            
              -
              10
            
          
          
            10
          
          
            6
          
          
            
              -
              6
            
          
          
            
              -
              3
            
          
          
            3
          
        
        
          
            4
          
          
            
              -
              4
            
          
          
            
              -
              8
            
          
          
            8
          
          
            8
          
          
            
              -
              8
            
          
          
            
              -
              4
            
          
          
            4
          
          
            
              -
              4
            
          
          
            4
          
          
            8
          
          
            
              -
              8
            
          
          
            
              -
              8
            
          
          
            8
          
          
            4
          
          
            
              -
              4
            
          
        
        
          
            6
          
          
            
              -
              6
            
          
          
            
              -
              12
            
          
          
            12
          
          
            3
          
          
            
              -
              3
            
          
          
            10
          
          
            
              -
              10
            
          
          
            
              -
              10
            
          
          
            10
          
          
            
              -
              3
            
          
          
            3
          
          
            12
          
          
            
              -
              12
            
          
          
            
              -
              6
            
          
          
            6
          
        
        
          
            8
          
          
            
              -
              8
            
          
          
            
              -
              8
            
          
          
            8
          
          
            
              -
              8
            
          
          
            8
          
          
            8
          
          
            
              -
              8
            
          
          
            8
          
          
            
              -
              8
            
          
          
            
              -
              8
            
          
          
            8
          
          
            
              -
              8
            
          
          
            8
          
          
            8
          
          
            
              -
              8
            
          
        
        
          
            10
          
          
            
              -
              10
            
          
          
            
              -
              3
            
          
          
            3
          
          
            
              -
              12
            
          
          
            12
          
          
            
              -
              6
            
          
          
            6
          
          
            6
          
          
            
              -
              6
            
          
          
            12
          
          
            
              -
              12
            
          
          
            3
          
          
            
              -
              3
            
          
          
            
              -
              10
            
          
          
            10
          
        
        
          
            8
          
          
            
              -
              8
            
          
          
            4
          
          
            
              -
              4
            
          
          
            
              -
              4
            
          
          
            4
          
          
            
              -
              8
            
          
          
            8
          
          
            
              -
              8
            
          
          
            8
          
          
            
              -
              4
            
          
          
            4
          
          
            4
          
          
            
              -
              4
            
          
          
            8
          
          
            
              -
              8
            
          
        
        
          
            12
          
          
            
              -
              12
            
          
          
            10
          
          
            
              -
              10
            
          
          
            6
          
          
            
              -
              6
            
          
          
            3
          
          
            
              -
              3
            
          
          
            
              -
              3
            
          
          
            3
          
          
            
              -
              6
            
          
          
            6
          
          
            
              -
              10
            
          
          
            10
          
          
            
              -
              12
            
          
          
            12
          
        
        
          
            8
          
          
            
              -
              8
            
          
          
            8
          
          
            
              -
              8
            
          
          
            8
          
          
            
              -
              8
            
          
          
            8
          
          
            
              -
              8
            
          
          
            8
          
          
            
              -
              8
            
          
          
            8
          
          
            
              -
              8
            
          
          
            8
          
          
            
              -
              8
            
          
          
            8
          
          
            
              -
              8
            
          
        
      
      ]
    
    .

8. The process according to claim 7, further comprising:
normalizing the order-16 transform matrix E16×16 by using a scaling matrix K16×16, where,





and where




  
    a
    =
    
      1
      
        512
      
    
  
  ,
  
    b
    =
    
      
        
          1
          
            578
          
        
        ⁢
        
          
        
        ⁢
        and
        ⁢
        
          
        
        ⁢
        c
      
      =
      
        
          1
          
            320
          
        
        .

9. The process according to claim 8, further comprising:
quantizing the normalized transform.

10. A process implemented using instructions encoded on computer readable storage medium for deriving an order-16 transform in image and video coding, the process comprising:
receiving a data matrix X16×16 to be transformed; and
transforming the received data matrix X16×16 with an order-16 transform matrix E16×16 to generate an output Z16×16 by a rule of Z16×16=E16×16X16×16,
wherein the transform matrix E16×16 is represented as





  
    E
    
      16
      ×
      16
    
  
  =
  
    
      [
      
        
          
            8
          
          
            8
          
          
            8
          
          
            8
          
          
            8
          
          
            8
          
          
            8
          
          
            8
          
          
            8
          
          
            8
          
          
            8
          
          
            8
          
          
            8
          
          
            8
          
          
            8
          
          
            8
          
        
        
          
            10
          
          
            10
          
          
            9
          
          
            9
          
          
            6
          
          
            6
          
          
            2
          
          
            2
          
          
            
              -
              2
            
          
          
            
              -
              2
            
          
          
            
              -
              6
            
          
          
            
              -
              6
            
          
          
            
              -
              9
            
          
          
            
              -
              9
            
          
          
            
              -
              10
            
          
          
            
              -
              10
            
          
        
        
          
            10
          
          
            10
          
          
            4
          
          
            4
          
          
            
              -
              4
            
          
          
            
              -
              4
            
          
          
            
              -
              10
            
          
          
            
              -
              10
            
          
          
            
              -
              10
            
          
          
            
              -
              10
            
          
          
            
              -
              4
            
          
          
            
              -
              4
            
          
          
            4
          
          
            4
          
          
            10
          
          
            10
          
        
        
          
            9
          
          
            9
          
          
            
              -
              2
            
          
          
            
              -
              2
            
          
          
            
              -
              10
            
          
          
            
              -
              10
            
          
          
            
              -
              6
            
          
          
            
              -
              6
            
          
          
            6
          
          
            6
          
          
            10
          
          
            10
          
          
            2
          
          
            2
          
          
            
              -
              9
            
          
          
            
              -
              9
            
          
        
        
          
            8
          
          
            8
          
          
            
              -
              8
            
          
          
            
              -
              8
            
          
          
            
              -
              8
            
          
          
            
              -
              8
            
          
          
            8
          
          
            8
          
          
            8
          
          
            8
          
          
            
              -
              8
            
          
          
            
              -
              8
            
          
          
            
              -
              8
            
          
          
            
              -
              8
            
          
          
            8
          
          
            8
          
        
        
          
            6
          
          
            6
          
          
            
              -
              10
            
          
          
            
              -
              10
            
          
          
            2
          
          
            2
          
          
            9
          
          
            9
          
          
            
              -
              9
            
          
          
            
              -
              9
            
          
          
            
              -
              2
            
          
          
            
              -
              2
            
          
          
            10
          
          
            10
          
          
            
              -
              6
            
          
          
            
              -
              6
            
          
        
        
          
            4
          
          
            4
          
          
            
              -
              10
            
          
          
            
              -
              10
            
          
          
            10
          
          
            10
          
          
            
              -
              4
            
          
          
            
              -
              4
            
          
          
            
              -
              4
            
          
          
            
              -
              4
            
          
          
            10
          
          
            10
          
          
            
              -
              10
            
          
          
            
              -
              10
            
          
          
            4
          
          
            4
          
        
        
          
            2
          
          
            2
          
          
            
              -
              6
            
          
          
            
              -
              6
            
          
          
            9
          
          
            9
          
          
            
              -
              10
            
          
          
            
              -
              10
            
          
          
            10
          
          
            10
          
          
            
              -
              9
            
          
          
            
              -
              9
            
          
          
            6
          
          
            6
          
          
            
              -
              2
            
          
          
            
              -
              2
            
          
        
        
          
            8
          
          
            
              -
              8
            
          
          
            
              -
              8
            
          
          
            8
          
          
            8
          
          
            
              -
              8
            
          
          
            
              -
              8
            
          
          
            8
          
          
            8
          
          
            
              -
              8
            
          
          
            
              -
              8
            
          
          
            8
          
          
            8
          
          
            
              -
              8
            
          
          
            
              -
              8
            
          
          
            8
          
        
        
          
            10
          
          
            
              -
              10
            
          
          
            
              -
              9
            
          
          
            9
          
          
            6
          
          
            
              -
              6
            
          
          
            
              -
              2
            
          
          
            2
          
          
            
              -
              2
            
          
          
            2
          
          
            6
          
          
            
              -
              6
            
          
          
            
              -
              9
            
          
          
            9
          
          
            10
          
          
            
              -
              10
            
          
        
        
          
            10
          
          
            
              -
              10
            
          
          
            
              -
              4
            
          
          
            4
          
          
            
              -
              4
            
          
          
            4
          
          
            10
          
          
            
              -
              10
            
          
          
            
              -
              10
            
          
          
            10
          
          
            4
          
          
            
              -
              4
            
          
          
            4
          
          
            
              -
              4
            
          
          
            
              -
              10
            
          
          
            10
          
        
        
          
            9
          
          
            
              -
              9
            
          
          
            2
          
          
            
              -
              2
            
          
          
            
              -
              10
            
          
          
            10
          
          
            6
          
          
            
              -
              6
            
          
          
            6
          
          
            
              -
              6
            
          
          
            
              -
              10
            
          
          
            10
          
          
            2
          
          
            
              -
              2
            
          
          
            9
          
          
            
              -
              9
            
          
        
        
          
            8
          
          
            
              -
              8
            
          
          
            8
          
          
            
              -
              8
            
          
          
            
              -
              8
            
          
          
            8
          
          
            
              -
              8
            
          
          
            8
          
          
            8
          
          
            
              -
              8
            
          
          
            8
          
          
            
              -
              8
            
          
          
            
              -
              8
            
          
          
            8
          
          
            
              -
              8
            
          
          
            8
          
        
        
          
            6
          
          
            
              -
              6
            
          
          
            10
          
          
            
              -
              10
            
          
          
            2
          
          
            
              -
              2
            
          
          
            
              -
              9
            
          
          
            9
          
          
            
              -
              9
            
          
          
            9
          
          
            2
          
          
            
              -
              2
            
          
          
            10
          
          
            
              -
              10
            
          
          
            6
          
          
            
              -
              6
            
          
        
        
          
            4
          
          
            
              -
              4
            
          
          
            10
          
          
            
              -
              10
            
          
          
            10
          
          
            
              -
              10
            
          
          
            4
          
          
            
              -
              4
            
          
          
            
              -
              4
            
          
          
            4
          
          
            
              -
              10
            
          
          
            10
          
          
            
              -
              10
            
          
          
            10
          
          
            
              -
              4
            
          
          
            4
          
        
        
          
            2
          
          
            
              -
              2
            
          
          
            6
          
          
            
              -
              6
            
          
          
            9
          
          
            
              -
              9
            
          
          
            10
          
          
            
              -
              10
            
          
          
            10
          
          
            
              -
              10
            
          
          
            9
          
          
            
              -
              9
            
          
          
            6
          
          
            
              -
              6
            
          
          
            2
          
          
            
              -
              2
            
          
        
      
      ]
    
    .

11. The process according to claim 10, further comprising:
quantizing the normalized transform.

12. The process of claim 10 further comprising:
normalizing the order-16 integer transform by using a scaling matrix K16×16, where,





and where




  
    a
    =
    
      1
      
        512
      
    
  
  ,
  
    b
    =
    
      
        
          1
          
            578
          
        
        ⁢
        
          
        
        ⁢
        and
        ⁢
        
          
        
        ⁢
        c
      
      =
      
        
          1
          
            320
          
        
        .

13. The process according to claim 12, further comprising:
quantizing the normalized transform.

14. An apparatus for deriving an order-16 integer transform using a plurality of order-8 integer transforms in image and video coding, the apparatus comprising:
an assignment unit including digital electronic circuitry configured to
retrieve a data matrix X16×16 to be transformed, and then
determine Cm, the mth vector of an intermediate matrix C16×16, based on the retrieved matrix X16×16, by carrying out eight additions and eight subtractions in accordance with a rule of






  
    C
    m
  
  =
  
    
      [
      
        
          
            
              c
              
                0
                ,
                m
              
            
          
        
        
          
            
              c
              
                1
                ,
                m
              
            
          
        
        
          
            ...
          
        
        
          
            
              c
              
                7
                ,
                m
              
            
          
        
        
          
            
              c
              
                8
                ,
                m
              
            
          
        
        
          
            
              c
              
                9
                ,
                m
              
            
          
        
        
          
            ...
          
        
        
          
            
              c
              
                15
                ,
                m
              
            
          
        
      
      ]
    
    =
    
      [
      
        
          
            
              
                x
                
                  0
                  ,
                  m
                
              
              +
              
                x
                
                  1
                  ,
                  m
                
              
            
          
        
        
          
            
              
                x
                
                  2
                  ,
                  m
                
              
              +
              
                x
                
                  3
                  ,
                  m
                
              
            
          
        
        
          
            ...
          
        
        
          
            
              
                x
                
                  14
                  ,
                  m
                
              
              +
              
                x
                
                  15
                  ,
                  m
                
              
            
          
        
        
          
            
              
                x
                
                  0
                  ,
                  m
                
              
              -
              
                x
                
                  1
                  ,
                  m
                
              
            
          
        
        
          
            
              
                x
                
                  2
                  ,
                  m
                
              
              -
              
                x
                
                  3
                  ,
                  m
                
              
            
          
        
        
          
            ...
          
        
        
          
            
              
                x
                
                  14
                  ,
                  m
                
              
              -
              
                x
                
                  15
                  ,
                  m
                
              
            
          
        
      
      ]
    
  




wherein ci,m is the (i, m)th element of C16×16, and xi,m is the (i, m)th element of X16×16, 0≦m≦15 and 0≦i≦15;

a transform unit including digital electronic circuitry configured to derive the order-16 integer transform by carrying out the order-8 transform for the first 8 vectors of the matrix C16×16, and carrying out one more order-8 transform for the last 8 vectors of the matrix C16×16; and
a scaling unit including digital electronic circuitry configured to normalize the order-16 integer transform by using a scaling matrix K16×16, where,





and where




  
    a
    =
    
      1
      
        512
      
    
  
  ,
  
    b
    =
    
      
        
          1
          
            578
          
        
        ⁢
        
          
        
        ⁢
        and
        ⁢
        
          
        
        ⁢
        c
      
      =
      
        
          1
          
            320
          
        
        .

15. The apparatus according to claim 14, wherein the digital electronic circuitry of the transform unit is configured to form a resultant matrix Z16×16 by a rule of


[z0,m z1,m . . . z7,m]T=E8×8[c0,m c1,m . . . c7,m]T 



[z15,m z14,m . . . z8,m]T=E8×8[c8,m c9,m . . . c15,m]T 

wherein zn,m is of the (n, m)th element of the matrix Z16×16, and E8×8 is the standard transform used in H.264/AVC.

16. The apparatus according to claim 14, further comprising:
a quantization unit including digital electronic circuitry configured to quantize the normalized transform.

17. An apparatus for deriving an order-16 integer transform using a plurality of order-8 integer transforms in image and video coding, the apparatus comprising:
an assignment unit including digital electronic circuitry configured to
retrieve a data matrix X16×16 to be transformed, and then
determine Cm, the mth vector of an intermediate matrix C16×16, based on the retrieved matrix X16×16, by carrying out eight additions and eight subtractions in accordance with a rule of






  
    C
    m
  
  =
  
    
      [
      
        
          
            
              c
              
                0
                ,
                m
              
            
          
        
        
          
            
              c
              
                1
                ,
                m
              
            
          
        
        
          
            ⋮
          
        
        
          
            
              c
              
                7
                ,
                m
              
            
          
        
        
          
            
              c
              
                8
                ,
                m
              
            
          
        
        
          
            
              c
              
                9
                ,
                m
              
            
          
        
        
          
            
              c
              
                10
                ,
                m
              
            
          
        
        
          
            
              c
              
                11
                ,
                m
              
            
          
        
        
          
            
              c
              
                12
                ,
                m
              
            
          
        
        
          
            
              c
              
                13
                ,
                m
              
            
          
        
        
          
            
              c
              
                14
                ,
                m
              
            
          
        
        
          
            
              c
              
                15
                ,
                m
              
            
          
        
      
      ]
    
    =
    
      [
      
        
          
            
              
                x
                
                  0
                  ,
                  m
                
              
              +
              
                x
                
                  1
                  ,
                  m
                
              
            
          
        
        
          
            
              
                x
                
                  2
                  ,
                  m
                
              
              +
              
                x
                
                  3
                  ,
                  m
                
              
            
          
        
        
          
            ⋮
          
        
        
          
            
              
                x
                
                  14
                  ,
                  m
                
              
              +
              
                x
                
                  15
                  ,
                  m
                
              
            
          
        
        
          
            
              
                x
                
                  0
                  ,
                  m
                
              
              +
              
                x
                
                  1
                  ,
                  m
                
              
            
          
        
        
          
            
              
                x
                
                  3
                  ,
                  m
                
              
              +
              
                x
                
                  2
                  ,
                  m
                
              
            
          
        
        
          
            
              
                x
                
                  4
                  ,
                  m
                
              
              +
              
                x
                
                  5
                  ,
                  m
                
              
            
          
        
        
          
            
              
                x
                
                  7
                  ,
                  m
                
              
              +
              
                x
                
                  6
                  ,
                  m
                
              
            
          
        
        
          
            
              
                x
                
                  8
                  ,
                  m
                
              
              +
              
                x
                
                  9
                  ,
                  m
                
              
            
          
        
        
          
            
              
                x
                
                  11
                  ,
                  m
                
              
              +
              
                x
                
                  10
                  ,
                  m
                
              
            
          
        
        
          
            
              
                x
                
                  12
                  ,
                  m
                
              
              +
              
                x
                
                  13
                  ,
                  m
                
              
            
          
        
        
          
            
              
                x
                
                  15
                  ,
                  m
                
              
              +
              
                x
                
                  14
                  ,
                  m
                
              
            
          
        
      
      ]
    
  




wherein ci,m is the (i, m)th element of C16×16, and xi,m is the (i, m)th element of X16×16, 0≦m≦15 and 0≦i≦15;

a transform unit including digital electronic circuitry configured to derive the order-16 integer transform by carrying out the order-8 transform for the first 8 vectors of the matrix C16×16, and carrying out one more order-8 transform for the last 8 vectors of the matrix C16×16; and
a scaling unit including digital electronic circuitry configured to normalize the order-16 integer transform by using a scaling matrix K16×16, where,





and where




  
    a
    =
    
      1
      
        512
      
    
  
  ,
  
    b
    =
    
      
        
          1
          
            578
          
        
        ⁢
        
          
        
        ⁢
        and
        ⁢
        
          
        
        ⁢
        c
      
      =
      
        
          1
          
            320
          
        
        .

18. The apparatus according to claim 17, wherein the digital electronic circuitry of the transform unit is configured to form a resultant matrix Z16×16 by a rule of


[z0,m z1,m . . . z7,m]T=E8×8[c0,m c1,m . . . c7,m]T 



[z8,m z9,m . . . z15,m]t=E8×8[c8,m C9,m . . . c15,m]T 

wherein zn,m is of the (n,m)th element of the matrix Z16×16, and E8×8 is the standard transform used in H.264/AVC.

19. The apparatus according to claim 17, further comprising:
a quantization unit including digital electronic circuitry configured to quantize the normalized transform.

20. An apparatus for processing an order-16 transform in image and video coding, the apparatus comprising:
a receiving unit including digital electronic circuitry configured to receive a data matrix X16×16 to be transformed; and
a transform unit including digital electronic circuitry configured to transform the received data matrix X16×16 with an order-16 transform matrix E16×16 to generate an output Z16×16 by a rule of Z16×16==E16×16X16×16,
wherein the transform matrix E16×16 is represented as





  
    E
    
      116
      ×
      16
    
  
  =
  
    
      [
      
        
          
            8
          
          
            8
          
          
            8
          
          
            8
          
          
            8
          
          
            8
          
          
            8
          
          
            8
          
          
            8
          
          
            8
          
          
            8
          
          
            8
          
          
            8
          
          
            8
          
          
            8
          
          
            8
          
        
        
          
            12
          
          
            12
          
          
            10
          
          
            10
          
          
            6
          
          
            6
          
          
            3
          
          
            3
          
          
            
              -
              3
            
          
          
            
              -
              3
            
          
          
            
              -
              6
            
          
          
            
              -
              6
            
          
          
            
              -
              10
            
          
          
            
              -
              10
            
          
          
            
              -
              12
            
          
          
            
              -
              12
            
          
        
        
          
            8
          
          
            8
          
          
            4
          
          
            4
          
          
            
              -
              4
            
          
          
            
              -
              4
            
          
          
            
              -
              8
            
          
          
            
              -
              8
            
          
          
            
              -
              8
            
          
          
            
              -
              8
            
          
          
            
              -
              4
            
          
          
            
              -
              4
            
          
          
            4
          
          
            4
          
          
            8
          
          
            8
          
        
        
          
            10
          
          
            10
          
          
            
              -
              3
            
          
          
            
              -
              3
            
          
          
            
              -
              12
            
          
          
            
              -
              12
            
          
          
            
              -
              6
            
          
          
            
              -
              6
            
          
          
            6
          
          
            6
          
          
            12
          
          
            12
          
          
            3
          
          
            3
          
          
            
              -
              10
            
          
          
            
              -
              10
            
          
        
        
          
            8
          
          
            8
          
          
            
              -
              8
            
          
          
            
              -
              8
            
          
          
            
              -
              8
            
          
          
            
              -
              8
            
          
          
            8
          
          
            8
          
          
            8
          
          
            8
          
          
            
              -
              8
            
          
          
            
              -
              8
            
          
          
            
              -
              8
            
          
          
            
              -
              8
            
          
          
            8
          
          
            8
          
        
        
          
            6
          
          
            6
          
          
            
              -
              12
            
          
          
            
              -
              12
            
          
          
            3
          
          
            3
          
          
            10
          
          
            10
          
          
            
              -
              10
            
          
          
            
              -
              10
            
          
          
            
              -
              3
            
          
          
            
              -
              3
            
          
          
            12
          
          
            12
          
          
            
              -
              6
            
          
          
            
              -
              6
            
          
        
        
          
            4
          
          
            4
          
          
            
              -
              8
            
          
          
            
              -
              8
            
          
          
            8
          
          
            8
          
          
            
              -
              4
            
          
          
            
              -
              4
            
          
          
            
              -
              4
            
          
          
            
              -
              4
            
          
          
            8
          
          
            8
          
          
            
              -
              8
            
          
          
            
              -
              8
            
          
          
            4
          
          
            4
          
        
        
          
            3
          
          
            3
          
          
            
              -
              6
            
          
          
            
              -
              6
            
          
          
            10
          
          
            10
          
          
            
              -
              12
            
          
          
            
              -
              12
            
          
          
            12
          
          
            12
          
          
            
              -
              10
            
          
          
            
              -
              10
            
          
          
            6
          
          
            6
          
          
            
              -
              3
            
          
          
            
              -
              3
            
          
        
        
          
            3
          
          
            
              -
              3
            
          
          
            
              -
              6
            
          
          
            6
          
          
            10
          
          
            
              -
              10
            
          
          
            
              -
              12
            
          
          
            12
          
          
            12
          
          
            
              -
              12
            
          
          
            
              -
              10
            
          
          
            10
          
          
            6
          
          
            
              -
              6
            
          
          
            
              -
              3
            
          
          
            3
          
        
        
          
            4
          
          
            
              -
              4
            
          
          
            
              -
              8
            
          
          
            8
          
          
            8
          
          
            
              -
              8
            
          
          
            
              -
              4
            
          
          
            4
          
          
            
              -
              4
            
          
          
            4
          
          
            8
          
          
            
              -
              8
            
          
          
            
              -
              8
            
          
          
            8
          
          
            4
          
          
            
              -
              4
            
          
        
        
          
            6
          
          
            
              -
              6
            
          
          
            
              -
              12
            
          
          
            12
          
          
            3
          
          
            
              -
              3
            
          
          
            10
          
          
            
              -
              10
            
          
          
            
              -
              10
            
          
          
            10
          
          
            
              -
              3
            
          
          
            3
          
          
            12
          
          
            
              -
              12
            
          
          
            
              -
              6
            
          
          
            6
          
        
        
          
            8
          
          
            
              -
              8
            
          
          
            
              -
              8
            
          
          
            8
          
          
            
              -
              8
            
          
          
            8
          
          
            8
          
          
            
              -
              8
            
          
          
            8
          
          
            
              -
              8
            
          
          
            
              -
              8
            
          
          
            8
          
          
            
              -
              8
            
          
          
            8
          
          
            8
          
          
            
              -
              8
            
          
        
        
          
            10
          
          
            
              -
              10
            
          
          
            
              -
              3
            
          
          
            3
          
          
            
              -
              12
            
          
          
            12
          
          
            
              -
              6
            
          
          
            6
          
          
            6
          
          
            
              -
              6
            
          
          
            12
          
          
            
              -
              12
            
          
          
            3
          
          
            
              -
              3
            
          
          
            
              -
              10
            
          
          
            10
          
        
        
          
            8
          
          
            
              -
              8
            
          
          
            4
          
          
            
              -
              4
            
          
          
            
              -
              4
            
          
          
            4
          
          
            
              -
              8
            
          
          
            8
          
          
            
              -
              8
            
          
          
            8
          
          
            
              -
              4
            
          
          
            4
          
          
            4
          
          
            
              -
              4
            
          
          
            8
          
          
            
              -
              8
            
          
        
        
          
            12
          
          
            
              -
              12
            
          
          
            10
          
          
            
              -
              10
            
          
          
            6
          
          
            
              -
              6
            
          
          
            3
          
          
            
              -
              3
            
          
          
            
              -
              3
            
          
          
            3
          
          
            
              -
              6
            
          
          
            6
          
          
            
              -
              10
            
          
          
            10
          
          
            
              -
              12
            
          
          
            12
          
        
        
          
            8
          
          
            
              -
              8
            
          
          
            8
          
          
            
              -
              8
            
          
          
            8
          
          
            
              -
              8
            
          
          
            8
          
          
            
              -
              8
            
          
          
            8
          
          
            
              -
              8
            
          
          
            8
          
          
            
              -
              8
            
          
          
            8
          
          
            
              -
              8
            
          
          
            8
          
          
            
              -
              8
            
          
        
      
      ]
    
    .

21. The apparatus according to claim 20, further comprising a normalizing unit including digital electronic circuitry configured to normalize the transform matrix E16×16 by using a scaling matrix K16×16, where,




and where




  
    a
    =
    
      1
      
        512
      
    
  
  ,
  
    b
    =
    
      
        
          1
          
            578
          
        
        ⁢
        
          
        
        ⁢
        and
        ⁢
        
          
        
        ⁢
        c
      
      =
      
        
          1
          
            320
          
        
        .

22. The apparatus according to claim 21, further comprising a quantizing unit including digital electronic circuitry configured to quantize the normalized transform.

23. An apparatus for processing an order-16 transform in image and video coding, the apparatus comprising:
a receiving unit including digital electronic circuitry configured to receive a data matrix X16×16 to be transformed; and
a transform unit including digital electronic circuitry configured to transform the received data matrix X16×16 with an order-16 transform matrix E16×16 to generate an output Z16×16 by a rule of Z16×16=E16×16X16×16,
wherein the transform matrix E16×16 is represented as





  
    E
    
      16
      ×
      16
    
  
  =
  
    
      [
      
        
          
            8
          
          
            8
          
          
            8
          
          
            8
          
          
            8
          
          
            8
          
          
            8
          
          
            8
          
          
            8
          
          
            8
          
          
            8
          
          
            8
          
          
            8
          
          
            8
          
          
            8
          
          
            8
          
        
        
          
            10
          
          
            10
          
          
            9
          
          
            9
          
          
            6
          
          
            6
          
          
            2
          
          
            2
          
          
            
              -
              2
            
          
          
            
              -
              2
            
          
          
            
              -
              6
            
          
          
            
              -
              6
            
          
          
            
              -
              9
            
          
          
            
              -
              9
            
          
          
            
              -
              10
            
          
          
            
              -
              10
            
          
        
        
          
            10
          
          
            10
          
          
            4
          
          
            4
          
          
            
              -
              4
            
          
          
            
              -
              4
            
          
          
            
              -
              10
            
          
          
            
              -
              10
            
          
          
            
              -
              10
            
          
          
            
              -
              10
            
          
          
            
              -
              4
            
          
          
            
              -
              4
            
          
          
            4
          
          
            4
          
          
            10
          
          
            10
          
        
        
          
            9
          
          
            9
          
          
            
              -
              2
            
          
          
            
              -
              2
            
          
          
            
              -
              10
            
          
          
            
              -
              10
            
          
          
            
              -
              6
            
          
          
            
              -
              6
            
          
          
            6
          
          
            6
          
          
            10
          
          
            10
          
          
            2
          
          
            2
          
          
            
              -
              9
            
          
          
            
              -
              9
            
          
        
        
          
            8
          
          
            8
          
          
            
              -
              8
            
          
          
            
              -
              8
            
          
          
            
              -
              8
            
          
          
            
              -
              8
            
          
          
            8
          
          
            8
          
          
            8
          
          
            8
          
          
            
              -
              8
            
          
          
            
              -
              8
            
          
          
            
              -
              8
            
          
          
            
              -
              8
            
          
          
            8
          
          
            8
          
        
        
          
            6
          
          
            6
          
          
            
              -
              10
            
          
          
            
              -
              10
            
          
          
            2
          
          
            2
          
          
            9
          
          
            9
          
          
            
              -
              9
            
          
          
            
              -
              9
            
          
          
            
              -
              2
            
          
          
            
              -
              2
            
          
          
            10
          
          
            10
          
          
            
              -
              6
            
          
          
            
              -
              6
            
          
        
        
          
            4
          
          
            4
          
          
            
              -
              10
            
          
          
            
              -
              10
            
          
          
            10
          
          
            10
          
          
            
              -
              4
            
          
          
            
              -
              4
            
          
          
            
              -
              4
            
          
          
            
              -
              4
            
          
          
            10
          
          
            10
          
          
            
              -
              10
            
          
          
            
              -
              10
            
          
          
            4
          
          
            4
          
        
        
          
            2
          
          
            2
          
          
            
              -
              6
            
          
          
            
              -
              6
            
          
          
            9
          
          
            9
          
          
            
              -
              10
            
          
          
            
              -
              10
            
          
          
            10
          
          
            10
          
          
            
              -
              9
            
          
          
            
              -
              9
            
          
          
            6
          
          
            6
          
          
            
              -
              2
            
          
          
            
              -
              2
            
          
        
        
          
            8
          
          
            
              -
              8
            
          
          
            
              -
              8
            
          
          
            8
          
          
            8
          
          
            
              -
              8
            
          
          
            
              -
              8
            
          
          
            8
          
          
            8
          
          
            
              -
              8
            
          
          
            
              -
              8
            
          
          
            8
          
          
            8
          
          
            
              -
              8
            
          
          
            
              -
              8
            
          
          
            8
          
        
        
          
            10
          
          
            
              -
              10
            
          
          
            
              -
              9
            
          
          
            9
          
          
            6
          
          
            
              -
              6
            
          
          
            
              -
              2
            
          
          
            2
          
          
            
              -
              2
            
          
          
            2
          
          
            6
          
          
            
              -
              6
            
          
          
            
              -
              9
            
          
          
            9
          
          
            10
          
          
            
              -
              10
            
          
        
        
          
            10
          
          
            
              -
              10
            
          
          
            
              -
              4
            
          
          
            4
          
          
            
              -
              4
            
          
          
            4
          
          
            10
          
          
            
              -
              10
            
          
          
            
              -
              10
            
          
          
            10
          
          
            4
          
          
            
              -
              4
            
          
          
            4
          
          
            
              -
              4
            
          
          
            
              -
              10
            
          
          
            10
          
        
        
          
            9
          
          
            
              -
              9
            
          
          
            2
          
          
            
              -
              2
            
          
          
            
              -
              10
            
          
          
            10
          
          
            6
          
          
            
              -
              6
            
          
          
            6
          
          
            
              -
              6
            
          
          
            
              -
              10
            
          
          
            10
          
          
            2
          
          
            
              -
              2
            
          
          
            9
          
          
            
              -
              9
            
          
        
        
          
            8
          
          
            
              -
              8
            
          
          
            8
          
          
            
              -
              8
            
          
          
            
              -
              8
            
          
          
            8
          
          
            
              -
              8
            
          
          
            8
          
          
            8
          
          
            
              -
              8
            
          
          
            8
          
          
            
              -
              8
            
          
          
            
              -
              8
            
          
          
            8
          
          
            
              -
              8
            
          
          
            8
          
        
        
          
            6
          
          
            
              -
              6
            
          
          
            10
          
          
            
              -
              10
            
          
          
            2
          
          
            
              -
              2
            
          
          
            
              -
              9
            
          
          
            9
          
          
            
              -
              9
            
          
          
            9
          
          
            2
          
          
            
              -
              2
            
          
          
            10
          
          
            
              -
              10
            
          
          
            6
          
          
            
              -
              6
            
          
        
        
          
            4
          
          
            
              -
              4
            
          
          
            10
          
          
            
              -
              10
            
          
          
            10
          
          
            
              -
              10
            
          
          
            4
          
          
            
              -
              4
            
          
          
            
              -
              4
            
          
          
            4
          
          
            
              -
              10
            
          
          
            10
          
          
            
              -
              10
            
          
          
            10
          
          
            
              -
              4
            
          
          
            4
          
        
        
          
            2
          
          
            
              -
              2
            
          
          
            6
          
          
            
              -
              6
            
          
          
            9
          
          
            
              -
              9
            
          
          
            10
          
          
            
              -
              10
            
          
          
            10
          
          
            
              -
              10
            
          
          
            9
          
          
            
              -
              9
            
          
          
            6
          
          
            
              -
              6
            
          
          
            2
          
          
            
              -
              2
            
          
        
      
      ]
    
    .

24. The apparatus according to claim 23, further comprising a normalizing unit including digital electronic circuitry configured to normalize the transform E16×16 by using a scaling matrix K16×16, where,




and where




  
    a
    =
    
      1
      
        512
      
    
  
  ,
  
    b
    =
    
      
        
          1
          
            578
          
        
        ⁢
        
          
        
        ⁢
        and
        ⁢
        
          
        
        ⁢
        c
      
      =
      
        
          1
          
            320
          
        
        .

25. The apparatus according to claim 23, further comprising a quantizing unit including digital electronic circuitry configured to quantize the normalized transform.