WO2013127366A1 - Process for coding data in image or video and apparatus thereof - Google Patents

Process for coding data in image or video and apparatus thereof Download PDF

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WO2013127366A1
WO2013127366A1 PCT/CN2013/072084 CN2013072084W WO2013127366A1 WO 2013127366 A1 WO2013127366 A1 WO 2013127366A1 CN 2013072084 W CN2013072084 W CN 2013072084W WO 2013127366 A1 WO2013127366 A1 WO 2013127366A1
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transform
order
matrix
coding
integer
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Chi Keung Fong
Wai Kuen Cham
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Chinese University of Hong Kong CUHK
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Chinese University of Hong Kong CUHK
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    • G—PHYSICS
    • G06—COMPUTING OR CALCULATING; COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
    • G06F17/10—Complex mathematical operations
    • G06F17/14—Fourier, Walsh or analogous domain transformations, e.g. Laplace, Hilbert, Karhunen-Loeve, transforms
    • G06F17/147—Discrete orthonormal transforms, e.g. discrete cosine transform, discrete sine transform, and variations therefrom, e.g. modified discrete cosine transform, integer transforms approximating the discrete cosine transform
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04N—PICTORIAL COMMUNICATION, e.g. TELEVISION
    • H04N19/00—Methods or arrangements for coding, decoding, compressing or decompressing digital video signals
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04N—PICTORIAL COMMUNICATION, e.g. TELEVISION
    • H04N19/00—Methods or arrangements for coding, decoding, compressing or decompressing digital video signals
    • H04N19/10—Methods or arrangements for coding, decoding, compressing or decompressing digital video signals using adaptive coding
    • H04N19/102—Methods or arrangements for coding, decoding, compressing or decompressing digital video signals using adaptive coding characterised by the element, parameter or selection affected or controlled by the adaptive coding
    • H04N19/12—Selection from among a plurality of transforms or standards, e.g. selection between discrete cosine transform [DCT] and sub-band transform or selection between H.263 and H.264
    • H04N19/122—Selection of transform size, e.g. 8x8 or 2x4x8 DCT; Selection of sub-band transforms of varying structure or type
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04N—PICTORIAL COMMUNICATION, e.g. TELEVISION
    • H04N19/00—Methods or arrangements for coding, decoding, compressing or decompressing digital video signals
    • H04N19/10—Methods or arrangements for coding, decoding, compressing or decompressing digital video signals using adaptive coding
    • H04N19/169—Methods or arrangements for coding, decoding, compressing or decompressing digital video signals using adaptive coding characterised by the coding unit, i.e. the structural portion or semantic portion of the video signal being the object or the subject of the adaptive coding
    • H04N19/17—Methods or arrangements for coding, decoding, compressing or decompressing digital video signals using adaptive coding characterised by the coding unit, i.e. the structural portion or semantic portion of the video signal being the object or the subject of the adaptive coding the unit being an image region, e.g. an object
    • H04N19/176—Methods or arrangements for coding, decoding, compressing or decompressing digital video signals using adaptive coding characterised by the coding unit, i.e. the structural portion or semantic portion of the video signal being the object or the subject of the adaptive coding the unit being an image region, e.g. an object the region being a block, e.g. a macroblock

Definitions

  • the present application relates to an integer transform of encoding and decoding for image or video signals, specifically, to a process for coding data in image or video and an apparatus thereof.
  • the present application also relates to a process and a system for deriving an order-2N integer transform from an order-N integer transform in image or video coding.
  • MPEG-2, MPEG-4 Part 2, and H.263 standards divide a picture into 16 x 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.
  • Newer video coding standards such as H.264/AVC, AVS and SMPTE 421M etc, have adopted to use 2D order-4 or order-8 transforms to provide a better compressibility than the video standards based on the 2D order-8 DCT.
  • the 2D order-4 or order-8 transforms are used so as to seek a trade-off between computational efficiency and coding efficiency.
  • a process for coding data in image or video comprising: retrieving a data matrix X 2j v,x2jvto be transformed;
  • ⁇ 2N X N represents a resultant matrix through the coding
  • an apparatus for coding data in image or video comprising:
  • a retrieving unit for retrieving a data matrix be transformed
  • ⁇ 2N X N represents a resultant matrix through the coding
  • P 2JV is a permutation matrix
  • ⁇ 2 ⁇ is a diagonal matrix with integer elements to make every basis norms in E 2J V similar
  • J J V is a mirrored version of IN
  • JV is a recursive matrix from an integer matrix
  • a system for deriving an order-2N integer transform from an order-N integer transform in image and video coding comprising: means for retrieving an order-N transform EN, and
  • P 2JV is a permutation matrix
  • M 2J V is a diagonal matrix with integer elements to make every basis norms in E 2J V similar
  • J J V is a mirrored version of l N .
  • JV is a recursive matrix from an integer matrix K K [0009]
  • FIG. 1 is a block diagram of an apparatus for coding data in image and video according to one embodiment of the present application.
  • FIG. 2 illustrates exemplarily a method for coding data in image and video according to one embodiment of the present application.
  • the apparatus 1000 includes a retrieving unit 10, a generating unit 20 and a coder 30.
  • the retrieving unit 10 is utilized to retrieve an order-N transform E N , which may be stored in a memory (not shown) in the apparatus 1000 or any storage means external to the apparatus 1000 so long as the apparatus 1000 may obtain it if necessary.
  • the order-N transform may be input into the retrieving unit 10, for example, through a key board or the like.
  • E N may be an order-N integer transform in image or video coding in the prior art.
  • E N may be an order-8 transform E VC adopted in H.264/AVC, or an order-8 transform E VS adopted in AVS. It should be understood that those skilled in the art could select other Ejvif required based on the disclosures of the application.
  • the generating unit 20 unitizes the retrieved order-N transform E N from the unit 10 to generate an order-2N transform E 2N by computing the transform E N , which will be detailed later.
  • the coder 30 has two inputs, E 2 N from the generating unit 20 and a data matrix ⁇ 2Nx N to be transformed.
  • the coder 30 operates to code the data matrix X 2 N* N into a data matrix Z 2 NX2N by a rule of
  • the coder 30 may carry out a transform through two cascading ID transforms to obtain Z ⁇ N ⁇ y a rule of
  • ⁇ 2 ⁇ is the k-th column vector in X 2 N* N
  • ⁇ 2NX N is a 2Nx2N matrix built byi7 2Jv t for k e ⁇ o,l,...,2N- l ⁇
  • V 2N k is the k-t row vector in V 2 NX N for k e ⁇ o,l,...,2N- l ⁇ .
  • [ 00193 E 2N can be constructed by the generating unit 20 by
  • P 2JV is a permutation matrix
  • M 2J V is a diagonal matrix with integer elements to make every basis norms in E 2J V similar
  • I J V is an order-N identity matrix
  • J J V is a mirrored version of IN
  • JV is a recursive matrix from an integer matrix K K ,
  • RAT may be constructed by a rule of
  • AT may be constructed by the generating unit 20 or may be retrieved from the memory internal or external to the apparatus 1000.
  • a ⁇ ,k are selected by a rule of
  • Table 1 , Table 2A-B, Table 3A-D and Table 4A-H A list of ciN,k in co-prime pairs are listed in Table 1 , Table 2A-B, Table 3A-D and Table 4A-H for different N.
  • Table 1 , Table 2A-B, Table 3A-D and Table 4A-H may be stored in a memory (not shown) in the apparatus 1000 or any storage means external to the apparatus 1000 so long as the apparatus 1000 may obtain it when necessary.
  • (ON C , ciN,2N-k) can be selected as a pair in any column in these tables. Larger value of a N ⁇ usually means that the resultant E 2 N has a higher coding performance. But this also means that the transform process requires more computation and the resultant transform coefficients have larger dynamic ranges.
  • E N may be an order-N integer transform in image and video coding in the prior art.
  • E N may be an order-8 transform EA VC adopted in H.264/AVC, or an order-8 transform EA VS adopted in AVS. It should be also understood that those skilled in the art could select other Ejvif required based on the disclosures of the application.
  • an order-N transform E N is retrieved and then E 2JV ZS generated from the retrieved Ejv by performing a transform according to the above mentioned equations 3) ⁇ 5).
  • the retrieved data matrix is coded according to the above mentioned equation 1).
  • the coding may further comprise a 2D transform carried out by two cascading ID transforms according to a rule as set in equation 2).
  • the process 2000 may be implemented by one or more processor of a computer using instructions encoded in computer readable storage medium.
  • the steps S 101-103 may be respectively implemented by the retrieving unit 10, the generating unit 20 and the coder 30 as discussed in the above.
  • each of units 10-30 may include electronic circuits to carry out the corresponding functions or utilize the programs stored therein to carry out the corresponding functions.
  • 2 j i and 2j 3 are taken as 5 and 2 respectively.
  • E 4 is built as
  • E 8 is built as:
  • M 8o R 4 is built as:
  • E 2N is built as:
  • E 2 N is orthogonal (up to scale) for any real b NJ .
  • b N are integers that satisfy:
  • B is a matrix consisting of
  • 5TM n is a minimal value of the sum of b 2 N and b 2 N,2N
  • 5TM x is a maximal value of the sum of b 2 N and b 2
  • ?TM n is a minimal ratio between bN and
  • ?TM x is a maximal ratio between bN and bN,2N
  • #8,3 j #8,5, #8,7, #8,9, #8,11, #8,1 3 , #8,1 5 ) (10, 10, 9, 6, 5, 5, 3, 1)
  • #8,3 j #8,5, #8,7, #8,9, #8,11 , #8,1 3 , #8,1 5 ) (10, 10, 9, 6, 5, 5, 3, 1)

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Description

PROCESS FOR CODING DATA IN IMAGE OR VIDEO AND APPARATUS
THEREOF
CROSS REFERENCE OF RELATED APPLICATION
[0001 J The application claims the benefit of the US application No. 61/605,478, filed on March 1, 2013, which is incorporated by reference in this entirety as part of this application.
TECHNICAL FIELD
[0002] The present application relates to an integer transform of encoding and decoding for image or video signals, specifically, to a process for coding data in image or video and an apparatus thereof. The present application also relates to a process and a system for deriving an order-2N integer transform from an order-N integer transform in image or video coding.
BACKGROUND
[0003] 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 x 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. Newer video coding standards, such as H.264/AVC, AVS and SMPTE 421M etc, have adopted to 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.
[0004] Recently, in the development of standard HEVC, 16x 16, 32x32 and higher order transforms has been proposed. A paper issued by Siwei Ma and C.-C Jay Kuo proposed a super-macroblock coding scheme with a high coding gain, see the paper "High-definition Video Coding with Super-macroblocks", SPIE-IS &T/V0I. 650816 (1-12), 2007. In this paper, Siwei Ma proposed an order- 16 transform, which can be decomposed and implemented with existing order-8 transform modules. US Patent Application Publication No. 2007/0223590 filed by Siwei Ma discloses an apparatus for processing a 2N><2N integer transform by means of a unit that generates an NxN transform. Both the Paper and the US patent publication deal with reduction of computational complexity of the image and video coding, failure to address the issue of improvement of compression rates.
SUMMARY
[ 00051 According to one aspect of the invention, there is provided a process for coding data in image or video, comprising: retrieving a data matrix X2jv,x2jvto be transformed; and
coding the retrieved data matrix by a rule of
Figure imgf000003_0001
wherein ∑2NX N represents a resultant matrix through the coding, E2JV is an order-2N integer transform and N= 2" where n is a positive integer no less than 2.
[ 0006J According to the other aspect of the invention, there is provided an apparatus for coding data in image or video, comprising:
a retrieving unit for retrieving a data matrix
Figure imgf000003_0002
be transformed; and
a coder for coding the retrieved data matrix by a rule of
Figure imgf000003_0003
wherein ∑2NX N represents a resultant matrix through the coding, E2JV is an order-2N integer transform and N= 2" where n is a positive integer no less than 2.
[ 0007] According to the other aspect of the invention, there is provided a process for deriving an order-2N integer transform from an order-N integer transform in image or video coding, comprising: retrieving an order-N transform EN, and
generating E2JV in response to the retrieved EN by performing a transform of
Έ2Ν = V2NM2N \ - " ' \\ 'I~ N" J \ N" I or
L N
Figure imgf000004_0001
where
P2JV is a permutation matrix,
Μ2Λί is a diagonal matrix with integer elements to make every basis norms in E2JV similar,
IN is an order-N identity matrix,
JJV is a mirrored version of IN, and
JV is a recursive matrix from an integer matrix
[ 0008] According to the other aspect of the invention, there is provided a system for deriving an order-2N integer transform from an order-N integer transform in image and video coding, comprising: means for retrieving an order-N transform EN, and
means for generating E2JV in response the retrieved EN by performing a transform of
Έ2Μ = V2MM2M \ - " \\ ~" \" I or
A "A
Figure imgf000004_0002
where
P2JV is a permutation matrix,
M2JV is a diagonal matrix with integer elements to make every basis norms in E2JV similar,
IN is an order-N identity matrix,
JJV is a mirrored version of lN, and
JV is a recursive matrix from an integer matrix KK [0009] The detailed technology and preferred embodiments implemented for the subject invention are described in the following paragraphs accompanying the appended drawings for those skilled in this field to well appreciate the features of the claimed invention.
BRIEF DESCRIPTION OF THE DRAWINGS
[0010 J FIG. 1 is a block diagram of an apparatus for coding data in image and video according to one embodiment of the present application.
[0011 J FIG. 2 illustrates exemplarily a method for coding data in image and video according to one embodiment of the present application.
DETAILED DESCRIPTION
[0012] Embodiments of the present application will be described with reference to the accompanying drawings, but these drawings are presented only for the illustrative purpose and in no respect, are intended to limit the present invention anymore.
[0013] Referring now to FIG. 1, an apparatus 1000 for coding data in image and video according to one embodiment of the present application is illustrated. As shown, the apparatus 1000 includes a retrieving unit 10, a generating unit 20 and a coder 30.
[0014] The retrieving unit 10 is utilized to retrieve an order-N transform EN, which may be stored in a memory (not shown) in the apparatus 1000 or any storage means external to the apparatus 1000 so long as the apparatus 1000 may obtain it if necessary. According to one embodiment, the order-N transform may be input into the retrieving unit 10, for example, through a key board or the like. According to one embodiment, EN may be an order-N integer transform in image or video coding in the prior art. For example, EN may be an order-8 transform E VC adopted in H.264/AVC, or an order-8 transform E VS adopted in AVS. It should be understood that those skilled in the art could select other Ejvif required based on the disclosures of the application.
[ 0015] The generating unit 20 unitizes the retrieved order-N transform EN from the unit 10 to generate an order-2N transform E2N by computing the transform EN, which will be detailed later.
[ 0016] The coder 30 has two inputs, E2N from the generating unit 20 and a data matrix ^2Nx N to be transformed. The coder 30 operates to code the data matrix X2N* N into a data matrix Z2NX2N by a rule of
[ 0017] According to one embodiment of the application, the coder 30 may carry out a transform through two cascading ID transforms to obtain Z^^N ^y a rule of
U -X
ΰ
Figure imgf000006_0001
where
Χ2Μ is the k-th column vector in X2N* N
^2NX N is a 2Nx2N matrix built byi72Jv t for k e {o,l,...,2N- l}
V2N k is the k-t row vector in V2NX N for k e {o,l,...,2N- l} .
[ 0018] Hereinafter, a process for generating an order-2N transform E2N from an order-N transform EN according to the embodiment of the application will be discussed.
[ 00193 E2N can be constructed by the generating unit 20 by
E, P M 0* ~
3)
.0* - J N _ wherein Ejvis an order-N transform,
P2JV is a permutation matrix,
M2JV is a diagonal matrix with integer elements to make every basis norms in E2JV similar,
IJV is an order-N identity matrix,
JJV is a mirrored version of IN, and
JV is a recursive matrix from an integer matrix KK ,
[ 0020 ] In this embodiment, RAT may be constructed by a rule of
Figure imgf000007_0001
wherein <¾v, i, <¾v,3, . . MN,2N-I are positive integers.
[ 0021 According to one embodiment, AT may be constructed by the generating unit 20 or may be retrieved from the memory internal or external to the apparatus 1000.
[ 0022 ] To ensure a good coding performance, a^,k are selected by a rule of
Figure imgf000007_0002
[ 0023 ] A list of ciN,k in co-prime pairs are listed in Table 1 , Table 2A-B, Table 3A-D and Table 4A-H for different N. In one embodiment, Table 1 , Table 2A-B, Table 3A-D and Table 4A-H may be stored in a memory (not shown) in the apparatus 1000 or any storage means external to the apparatus 1000 so long as the apparatus 1000 may obtain it when necessary. (ON C, ciN,2N-k) can be selected as a pair in any column in these tables. Larger value of aN^ usually means that the resultant E2N has a higher coding performance. But this also means that the transform process requires more computation and the resultant transform coefficients have larger dynamic ranges.
Table 1
Figure imgf000008_0001
Table 2A
Figure imgf000008_0002
Table 2B
N = 4
«4,3 1 3 4 5 5 7 6 7 8 9
«4,5 1 2 3 3 4 4 5 5 5 7
Table 3 A
N = 8
«8,1 9 10 1 1 12 17 19 21 23 25 26
«8,15 1 1 1 1 2 2 2 2 3 3
Figure imgf000008_0003
Table 3C
Figure imgf000008_0004
#8,11 1 2 3 4 5 5 5 6 6 7 7 7
Figure imgf000009_0001
Table 4A
N = 16
«16,1 17 18 19 20 21 22 23 24 33 35 37 39 41 43 45 47
#16,31 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2
JV = 16
#16,3 6 7 8 11 13 15 17 19 20 22 23 23 25 27 29
#16,29 1 1 1 2 2 2 3 3 3 3 3 4 4 4 4
Figure imgf000009_0002
Table 4D
Figure imgf000009_0003
4E
Figure imgf000009_0004
Table 4F
N = = 16
#16,11 2 3 5 7 7 9 11 10 11 12 13 11 13 15 13 14 16
#16,21 1 2 3 4 5 5 6 7 7 7 7 8 8 8 9 9 9 N = 16
#16,13 3 4 5 6 7 8 7 8 9 10 11 9 11 10 11
#16,19 2 3 4 5 5 5 6 7 7 7 7 8 8 9 9
4H
Figure imgf000010_0001
[ 0024 ] The proposed transform E2JV can be implemented with additions, subtractions and multiplications with small integers only.
Hereinafter, a process 2000 for coding data in image or video according to one embodiment of the present application will be discussed.
[ 0026] At step S101, a data matrix Χ2 Χ2Ν ΪΟ be transformed is retrieved. As mentioned in the above, according to one embodiment, EN may be an order-N integer transform in image and video coding in the prior art. For example, EN may be an order-8 transform EAVC adopted in H.264/AVC, or an order-8 transform EAVS adopted in AVS. It should be also understood that those skilled in the art could select other Ejvif required based on the disclosures of the application.
[ 0027] At step SI 02, an order-N transform EN is retrieved and then E2JV ZS generated from the retrieved Ejv by performing a transform according to the above mentioned equations 3)~5). And then at step SI 03, the retrieved data matrix is coded according to the above mentioned equation 1). As discussed in the above, the coding may further comprise a 2D transform carried out by two cascading ID transforms according to a rule as set in equation 2).
[ 0028 ] According to one embodiment, the process 2000 may be implemented by one or more processor of a computer using instructions encoded in computer readable storage medium. Alternatively, the steps S 101-103 may be respectively implemented by the retrieving unit 10, the generating unit 20 and the coder 30 as discussed in the above. It shall be also understood that each of units 10-30 may include electronic circuits to carry out the corresponding functions or utilize the programs stored therein to carry out the corresponding functions.
[0029] Hereinafter, some examples according to the embodiment of the application will be described. These examples are present only for the illustrative purpose and in no respect, are intended to limit the present invention anymore.
Example 1 : Order-4 Transform
[ 0030 ] Herein, starting from N = 2, E2 and 2 are built by rules
l l
; and
l -l
-αΊ
2ji and 2j 3 are taken as 5 and 2 respectively.
[ 00313 As a result, E4 is built as
~E2 02 I2 J2
E4 =P4M4
Figure imgf000011_0001
4 4 4 4
5 -2 -2 5
4 -4 -4 4
2 -5 5 -2
Example 2: Order-8 Transform
[0032] Take 4ji, 4j3, «4,5 and 4j7 as 5, 3, 2 and 1, respectively, R4 is built as R,
02 R2
a47 ■1 1
27 23 15 5
16 -4 -19 -11
11 -19 4 16
5 -15 23 -27
[ 00333 And then E8 is built as
I4 J4
E8 =P8M8 X o4 "
A R4 I4 _J4
22 22 22 22 22 22 22 22
27 23 15 5 -5 -15 -23 -27
25 10 -10 -25 -25 -10 10 25
16 -4 -19 -11 11 19 4 -16
22 -22 -22 22 22 -22 -22 22
11 -19 4 16 -16 -4 19 -11
10 -25 25 -10 -10 25 -25 10
5 -15 23 -27 27 -23 15 -5
wherein
Figure imgf000012_0001
Example 3: Order- 16 Transform
[0034] Take 8ji, 8j3, 8j5, 8j 8jg, 8jn, 8ji3 and 8jis as 10, 17, 15, 10, 8, 8, 5 and 1 respectively. Accordingly, E8 will be built as
Figure imgf000013_0001
256 256 256 256 256 256 256 256 256 ■ 256 256
275 265 245 215 173 127 77 23 -23 ■ • -265 -275
324 276 180 60 -16 -180 -276 -324 -324 ■ 276 324
327 217 27 -163 -303 -343 -267 -107 107 ■ • -217 -327
320 128 -128 -320 -320 -128 128 320 320 ■ 128 320
293 37 -253 -317 -92 212 328 152 -152 ■ • -37 -293
288 -72 -342 -198 198 342 72 -288 -288 ■ • -72 288
266 -166 -334 34 350 110 -310 -230 230 ■ 166 -266
256 -256 -256 256 256 -256 -256 256 256 ■ • -256 256
230 -310 -110 350 -34 -334 166 266 -266 ■ 310 -230
198 -342 72 288 -288 -72 342 -198 -198 ■ • -342 198
152 -328 212 92 -317 253 37 -293 293 ■ 328 -152
128 -320 320 -128 -128 320 -320 128 128 ■ • -320 128
107 -267 343 -303 163 27 -217 327 -327 ■ 267 -107
60 -180 276 -324 324 -276 180 -60 -60 ■ • -180 60
23 -77 127 -173 215 -245 265 -275 275 ■ 77 -23
Example 4 : Order-32 Transform
[0035] Take ι6,ι, i6,3, ... ai6,3i as 20, 13, 16, 14, 13, 13, 12, 15, 14, 9, 8, 6, 5, 4, 2 and 1 respectively. Accordingly, E8 is built
Figure imgf000013_0002
4096 4096 4096 4096 4096 4096 4096 4096 4096 ■ • 4096
5523 5477 5377 5223 5027 4773 4473 4127 3675 ■ • 5523
5500 5300 4900 4300 3460 2540 1540 460 -460 ■ • 5500
4465 4037 3355 2287 1037 -335 -1513 -2725 -3613 ■ • 4465
5616 4784 3120 1040 -1040 -3120 -4784 -5616 -5616 ■ • 5616
5296 4080 1904 -720 -3200 -4896 -5440 -4704 -2740 ■ • 5296
4195 -3505 -4865 2555 5335 -1525 5065 415 5647 ■ • 4195
4096 -4096 -4096 4096 4096 -4096 -4096 4096 4096 ■ • 4096
1712 -4272 5488 -4848 2608 432 -3472 5232 -5232 ■ • 1712
1260 -3604 5100 -5396 4404 -2380 -204 2740 -4704 ■ • 1260
1040 -3120 4784 -5616 5616 -4784 3120 -1040 -1040 ■ • 1040
737 -2045 3037 -3905 4405 -4513 4265 -3613 2725 ■ • 737
460 -1540 2540 -3460 4300 -4900 5300 -5500 5500 ■ • 460
185 -735 1275 -1805 2295 -2785 3245 -3675 4127 ■ • 185
[0036J Alternatively, we may merge the diagonal matrix M into the integer transform kernel.
Example 5: Another Order-4 Transform
[0037] E4 is built as
Figure imgf000014_0001
Example 6: Another Order-8 Transform
[0038] E8 is built as:
Figure imgf000014_0002
[0039] M8oR4 is built as:
Figure imgf000014_0003
[0040] Recursively, E2N is built as:
EN A '
0 A "A_
[ 0041 ] M(2N)oRN is built as :
Figure imgf000014_0004
E2N is orthogonal (up to scale) for any real bNJ.
[ 0042 The values of
Figure imgf000015_0001
should be chosen: (1) to maximize the transform coding gain, and (2) to minimize the differences among the basis vector norms of the transform itself.
[ 00431 According to the above requirements (1) and (2), bN are integers that satisfy:
N > 2 an(J
where B is a matrix consisting of
Figure imgf000015_0002
5™n is a minimal value of the sum of b2N and b2N,2N
5™x is a maximal value of the sum of b2N and b2
?™n is a minimal ratio between bN and
Figure imgf000015_0003
?™x is a maximal ratio between bN and bN,2N
[ 0044 According to the above claim, when N=2:
Figure imgf000015_0004
16 101 104 5.0000 10.000
32 397 425 2.3750 20.000
64 1557 1717 2.3750 41.000
128 6228 6868 2.3750 82.000
256 24905 27521 2.3593 165.00
512 99620 110097 2.3593 331.00
1024 398480 440397 2.3593 663.00
2048 1593841 1761605 2.3592 1327.0
4096 6375346 7046425 2.3591 2654.0
Figure imgf000016_0001
6] when N=4 and (a2, a2 = (5, 2)
Figure imgf000016_0002
Figure imgf000016_0003
256 4297 4745 1.0000 68.000
512 17177 18980 1.0000 137.00
1024 68708 75925 1.0000 275.00
2048 274801 303725 1.0000 551.00
4096 1099201 1214901 1.0000 1102.0
[0047] when N=4 and (a2, a2 = (7, 3)
Figure imgf000017_0001
Figure imgf000017_0002
[0048] when N=4 and (a2, a2 = (12, 5)
Figure imgf000017_0003
4096 188621 208469 2.2857 456.00
Figure imgf000018_0001
91 when 7V=8 and ( 2,ι, 2j3) = (2, 1) and ( 4,ι, 4,3, 4,5, a4j) = (5, 3, 2, 1)
Figure imgf000018_0002
Figure imgf000018_0003
Figure imgf000018_0004
1024 30658 33876 1.0000 184.00
2048 122609 135505 1.0000 368.00
4096 490417 542026 1.0000 736.00
Figure imgf000019_0001
01 when N=8 and ( 2,ι, 2,3) = (5, 2) and ( 4,ι, 4,3, 4,5, a4j) = (5, 3, 2, 1)
Figure imgf000019_0002
Figure imgf000019_0003
Figure imgf000019_0004
min nmax min
B D max
#8,7 #8,7
256 338 362 1.1111 13.000
512 1322 1460 1.0000 27.000
1024 5288 5840 1.0000 54.000
2048 21146 23362 1.0000 108.00
4096 84562 93448 1.0000 216.00
when N=8 and ( 2,ι, 2,3) = (7, 3) and ( 4,ι, 4,3, 4,5, a4j) = (5, 3, 2, 1)
Figure imgf000020_0001
Figure imgf000020_0002
Figure imgf000020_0003
Figure imgf000020_0004
[0052 J when N=8 and ( 2,ι, 2,3) = (12, 5) and ( 4,ι, 4,3, 4,5, a4j)
Figure imgf000021_0001
Figure imgf000021_0002
Figure imgf000021_0003
Figure imgf000021_0004
[00531 when N=16 and ( 2,ι, 2,3) = (2, 1) and ( 4,ι, 4,3, 4,5, 04,7) = (5, 3, 2, 1) and ( 8,ι,
<¾,3, «8,5, «8,7, «8,9, «8,ii j «8,i3, <¾is) - (10, 10, 9, 6, 5, 5, 3, 1)
Figure imgf000021_0005
4096 4500 4969 1.31451 213.00 r>max r>min r> max
B #16,5 #16,5 ^16,5 ^16,5
2048 1157 1277 1.0000 35.000
4096 4628 5113 1.0000 71.000 r>max r>min r> max
B #16,7 #16,7 -«16,7 -«16,7
2048 1009 1109 1.0000 33.000
4096 4021 4442 1.0000 66.000
Figure imgf000022_0001
Figure imgf000022_0002
Figure imgf000022_0003
[0054] when N=16 and ( 2,ι, a2,3) = (5, 2) and ( 4,ι, 4,3, #4,5, 04,7) = (5, 3, 2, 1) and ( 8,ι,
#8,3 j #8,5, #8,7, #8,9, #8,11, #8,13, #8,15) = (10, 10, 9, 6, 5, 5, 3, 1)
Figure imgf000022_0004
2048 194 212 1.2222 14.000
4096 776 857 1.0327 88.000
B
2048 200 218 1.0000 7.0000
4096 800 881 1.0000 29.000
Figure imgf000023_0001
Figure imgf000023_0002
Figure imgf000023_0003
Figure imgf000023_0004
[0055] when N=16 and ( 2,ι, a2,3) = (7, 3) and ( 4,ι, 4,3, #4,5, 04,7) = (5, 3, 2, 1) and ( 8,ι,
#8,3 j #8,5, #8,7, #8,9, #8,11 , #8,13, #8,15) = (10, 10, 9, 6, 5, 5, 3, 1)
Figure imgf000023_0005
B
2048 97 106 1.0476 31.000
4096 388 425 1.0232 62.000
B
2048 100 109 1.3333 10.000
4096 400 436 1.0714 20.000
Figure imgf000024_0001
Figure imgf000024_0002
Figure imgf000024_0003
Figure imgf000024_0004
[0056] when N=16 and ( 2,ι, «2,3) = (2, 1) and (α4;1, «4,3, «4,5, «4,7)
«8,3, «8,5, «8,7, «8,9, «8,11 , «8,13, «8,15) = (10, 13, 13, 6, 5, 5, 3, 1)
Figure imgf000024_0005
16384 38992 42912 1.0000 207.00
32768 155968 171648 1.0000 414.00
Figure imgf000025_0001
Figure imgf000025_0002
Figure imgf000025_0003
Figure imgf000025_0004
Figure imgf000025_0005
16384 36000 39792 1.0000 199.00
32768 14400 159168 1.0000 398.00
Figure imgf000026_0001
Figure imgf000026_0002
[0057] when 7Y=16 and ( 2,ι, 2j3) = (5, 2) and ( 4,ι, 4,3, 4,5, 04,7) = (5, 3, 2, 1) and ( 8,ι,
<¾,3, β8,5, β8,7, β8,9, <¾n, β8,ι3, β8,ι5)— (10, 13, 13, 6, 5, 5, 3, 1)
Figure imgf000026_0003
Figure imgf000026_0004
Figure imgf000026_0005
4096 388 429 1.0000 20.000
8192 1552 1716 1.0000 41.000
16384 6208 6864 1.0000 82.000
32768 24832 27456 1.0000 165.00
Figure imgf000027_0001
Figure imgf000027_0002
Figure imgf000027_0003
Figure imgf000027_0004
Figure imgf000027_0005
4096 425 457 1.0000 4.0000
8192 1700 1828 1.0000 8.2000
16384 6800 7312 1.0000 9.1111
32768 27200 29248 1.0000 9.1111
[0058] when N=16 and ( 2,ι, a2j3) = (7, 3) and ( 4,ι, 4,3, 4,5, 04,7) = (5, 3, 2, 1) and ( 8,ι,
#8,3 #8,5, #8,7, #8,9, «8,11, #8,13, «8,15) _ (10, 13, 13, 6, 5, 5, 3, 1)
Figure imgf000028_0001
Figure imgf000028_0002
Figure imgf000028_0003
Figure imgf000028_0004
B
2048 88 90 1.6000 3.0000
4096 169 184 1.1111 13.000
8192 676 736 1.0000 27.000
16384 2704 2944 1.0000 54.000
32768 10816 11776 1.0000 108.00
Figure imgf000029_0001
Figure imgf000029_0002
Figure imgf000029_0003
[00591 Some suggested value of bN pairs are listed below but does not limit to these.
When N= 2:
Figure imgf000029_0004
Figure imgf000030_0001
WhenN=4, (<¾,i, 2,3)
Figure imgf000030_0002
Figure imgf000031_0001
WhenN=4, ( 2,ι, 2,3)
Figure imgf000031_0002
128 (23, 5), (23, 4), (24, 4), (23, 6), (24, 3)
256 (47, 9), (46, 10), (47, 8), (46, 11), (47, 7)
512 (93, 19), (93, 20), (94, 15), (94, 14), (94, 13)
1024 (187, 35), (186, 39), (187, 34), (186, 40), (185, 44)
2048 (373, 74), (373, 75), (374, 69), (374, 68), (372, 79)
4096 (746, 148), (747, 144), (746, 149), (747, 143), (745, 153)
32 (5, 3), (6, 1)
64 (10, 6), (9, 8), (11, 5), (11, 4), (12, 1)
128 (20, 13), (19, 14), (21, 11), (22, 9), (23, 6)
256 (40, 26), (39, 27), (41, 24), (42, 22), (37, 30)
{b4,3 , b4,5 )
512 (79, 53), (82, 48), (76, 57), (73, 61), (72, 62)
1024 (161, 101), (163, 98), (155, 110), (153, 113), (167, 91)
2048 (319, 207), (317, 210), (321, 204), (315, 213), (313, 216)
4096 (640, 411), (626, 432), (650, 395), (653, 390), (619, 442)
When N=4, (α2,ι, a2,3) = (12, 5):
B Suggested Values
64 (7, 1)
128 (14, 2)
256 (27, 5), (28, 5), (27, 6), (27, 4), (28, 4)
(¾4,1 A,v ) 512 (55, 8), (54, 13), (55, 9), (55, 10), (55, 11)
1024 (108, 27), (110, 17), (110, 18), (109, 23), (109, 22)
2048 (216, 55), (216, 54), (220, 34), (217, 51), (220, 35)
4096 (437, 87), (438, 82), (436, 92), (439, 76), (435, 97)
{b4,3 , b4,5 ) 64 (5, 5), (7, 1)
128 (12, 7), (11, 9), (10, 10), (13, 5), (14, 2)
256 (23, 16), (24, 14), (22, 17), (25, 12), (21, 18) 512 (47, 30), (48, 28), (45, 33), (44, 34), (51, 22)
1024 (92, 63), (95, 58), (89, 67), (98, 53), (99, 51)
2048 (187, 121), (189, 118), (183, 127), (181, 130), (192, 113)
4096 (378, 236), (381, 231), (364, 257), (384, 226), (359, 264)
When N=8, ( 2,ι, a2,3) = (2, 1), ( 4,ι, a4,3, <¾,5, a4j) = (5, 3, 2, 1):
B Suggested Values
128 (16, 2), (16, 1)
256 (32, 3), (32, 4), (32,2), (31, 5), (32, 1)
512 (63, 6), (63, 7), (63, 5), (63, 8), (63, 9)
1024 (126, 14), (126, 15), (127, 9), (126, 16), (127, 8)
2048 (253, 24), (253, 25), (253, 23), (253, 22), (253, 21)
4096 (506, 46), (506, 45), (506, 55), (505, 56), (507, 34)
128 (21, 8), (22, 5), (22, 4), (22, 3), (20, 10)
256 (43, 13), (42, 16), (44, 10), (44, 9), (44, 8)
512 (86, 26), (85, 25), (87, 22), (84, 32), (88, 18)
1024 (172, 52), (171, 55), (173, 48), (170, 58), (169, 61)
2048 (343, 107), (345, 100), (342, 110), (346, 97), (341, 113)
4096 (688, 207), (685, 217), (684, 220), (691, 197), (683, 223)
128 (20, 10), (19, 12), (21, 8), (22, 5), (22, 4)
256 (39, 22), (41, 18), (38, 24), (42, 16), (36, 27)
512 (85, 29), (86, 26), (71, 55), (88, 18), (65, 62)
1024 (156, 89), (165, 71), (170, 58), (171, 55), (145, 106)
2048 (317, 169), (325, 153), (326, 151), (330, 142), (303, 193)
4096 (634, 338), (651, 304), (628, 349), (656, 293), (620, 363)
128 (13, 9), (14, 7), (15, 5), (16, 2), (16, 1)
256 (24, 21), (26, 18), (23, 22), (27, 17), (28, 15) 512 (53, 35), (56, 30), (57, 28), (58, 26), (60, 21)
1024 (101 , 77), (97, 82), (104, 73), (106, 70), (1 12, 60)
2048 (202, 154), (205, 150), (194, 164), (210, 143), (212, 140)
4096 (397, 317), (393, 332), (410, 300), (388, 328), (372, 346)
When N=8, (<¾,i, <¾3) = (5, 2), ( 4,ι, 4j3j 4j5, a4j) = (5, 3, 2, 1):
B Suggested Values
256 (13, 2), (13, 1)
512 (26, 3), (26, 4), (26, 2), (26, 1), (27, 1)
(fc8,l , ) 1024 (52, 7), (53, 3), (52, 8), (53, 2), (52, 9)
2048 (105, 10), (105, 1 1), (105, 9), (105, 12), (105, 8)
4096 (210, 21), (210, 20), (210, 19), (209, 28), (209, 29)
256 (18, 5), (18, 6), (18, 4), (17, 7), (17, 8)
512 (36, 10), (36, 9), (35, 13), (34, 15), (37, 5)
(^8,3 ' ^8,13 ) 1024 (72, 20), (71 , 23), (72, 19), (70, 26), (73, 15)
2048 (143, 42), (142, 46), (144, 39), (141, 49), (145, 35)
4096 (286, 85), (285, 88), (288, 78), (289, 74), (290, 70)
256 (17, 8), (16, 10), (15, 1 1), (18, 5), (14, 12)
512 (32, 19), (34, 15), (35, 13), (30, 22), (36, 10)
1024 (67, 33), (63, 40), (61 , 43), (70, 26), (71 , 23)
2048 (129, 75), (137, 59), (124, 83), (139, 54), (119, 90)
4096 (264, 139), (265, 137), (266, 135), (167, 133), (259, 148)
256 (1 1 , 7), (12, 5), (13, 3), (13, 2), (13, 1)
512 (21 , 16), (20, 17), (23, 13), (24, 1 1), (25, 8)
(¾8,7 , b 9 ) 1024 (41 , 33), (42, 32), (44, 29), (46, 26), (47, 24)
2048 (83, 65), (79, 70), (86, 61), (78, 71), (77, 72)
4096 (163, 134), (170, 125), (156, 142), (180, 1 10), (183, 105) WhenN=8, ( 2,ι, <¾3) = (7, 3), ( 4,ι, a4,3,a4,5, a4j)
Figure imgf000035_0001
1024 (22, 2), (22, 3), (22, 1), (21, 5), (21, 6)
2048 (43, 6), (43, 7), (44, 1), (43, 8), (43, 9)
4096 (87, 9), (87, 8), (87, 10), (87, 7), (87, 6)
512 (15, 4), (15, 3), (14, 6), (14, 7), (13, 8)
1024 (30, 8), (30, 7), (29, 11), (28, 13), (27, 15)
(^8,3 ' ^8,13 )
2048 (59, 18), (60, 15), (58, 21), (57, 24), (61, 10)
4096 (118, 37), (119, 33), (117, 40), (120, 30), (115, 45)
512 (14, 7), (13, 8), (14, 6), (12, 10), (15, 4)
1024 (27, 15), (28, 13), (29, 11), (25, 18), (30, 8)
Figure imgf000036_0001
2048 (55, 28), (54, 30), (56, 26), (53, 32), (57, 24)
4096 (109, 58), (108, 60), (107, 62), (112, 52), (105, 65)
512 (9, 6), (10, 5), (10, 4), (11, 2), (11, 1)
1024 (16, 15), (17, 14), (18, 12), (19, 11), (20, 9)
(^8,7 ' ^8,9 )
2048 (35, 26), (36, 25), (32, 30), (31, 31), (37, 23)
4096 (68, 55), (67, 56), (71, 51), (73, 48), (75, 45)
When N=16, ( 2,ι , a2j3) = (2, 1), ( 4,i, a ,3, a j5, «4 ) = (5, 3, 2, 1), ( 8ji, 8j3, 8>5, «8,7, «8,9, «s,i i, «8,i3, «8,15) = (10, 10, 9, 6, 5, 5, 3, 1):
B Suggested Values
2048 (25, 1), (25, 2), (25, 3), (25, 4), (25, 5)
(^16,1 ' ^16,31 )
4096 (50, 6), (50, 7), (50, 8), (50, 9), (49, 12)
2048 (34, 5), (34, 6), (33, 10), (32, 13), (31, 15)
(^16,3 ' ^16,29 )
4096 (68, 10), (68, 11), (67, 15), (67, 16), (66, 19)
2048 (33, 10), (34, 6), (34, 5), (32, 13), (31, 15)
(^16,5 ' ^16,27 )
4096 (68, 16), (68, 15), (67, 19), (67, 20), (69, 11)
2048 (31, 10), (29, 15), (32, 6), (32, 5), (27, 18)
(^16,7 ' ^16,25 )
4096 (62, 20), (60, 25), (63, 16), (64, 12), (64, 11) 2048 (29, 15), (31, 10), (27, 18), (32, 6), (25, 21)
(^16,9 ' ^16,23 )
4096 (60, 25), (56, 33), (63, 16), (52, 39), (47, 45)
2048 (30, 18), (31, 16), (28, 21), (32, 14), (27, 22)
(^16,11 ' ^16,21 )
4096 (61, 34), (62, 32), (63, 30), (64, 28), (55, 43)
2048 (28, 20), (30, 17), (31, 15), (32, 13), (33, 10)
(^16,13■> ^16,19 )
4096 (56, 40), (58, 37), (53, 44), (52, 45), (61, 32)
2048 (18, 18), (21, 14), (23, 10), (24, 8), (25, 4)
(^16,15■> ^16,17 )
4096 (39, 32), (40, 31), (42, 28), (44, 25), (46, 21)
When N=16, ( 2,ι, 2j3) = (5, 2), ( 4,i, 4j3j 4j5, a4j) = (5, 3, 2, 1), ( 8,i, 8j3, 8j5, a8j, <¾,9, <%,i i, «8,i3, «8,15) = (10, 10, 9, 6, 5, 5, 3, 1):
B Suggested Values
2048 (10, 3)
(^16,1■> ^16,31 )
4096 (21, 1), (21, 2), (21, 3), (21, 4), (20, 5)
2048 (14, 4), (14, 2), (14, 3), (13, 6), (12, 8)
(^16,3■> ^16,29 )
4096 (28, 5), (28, 6), (27, 9), (26, 12), (23, 17)
2048 (14, 4), (14, 3), (13, 6), (13, 7), (12, 8)
(^16,5 ' ^16,27 )
4096 (28, 7), (28, 8), (29, 2), (29, 1), (26, 13)
2048 (13, 4), (13, 3), (12, 6), (11, 8), (10, 9)
(^16,7 ' ^16,25 )
4096 (25, 10), (26, 7), (23, 14), (27, 2), (27, 1)
2048 (12, 6), (13, 4), (11, 8), (13, 3), (10, 9)
(^16,9 ' ^16,23 )
4096 (25, 10), (23, 14), (26, 7), (21, 17), (20, 18)
2048 (12, 8), (13, 7), (13, 6), (11, 9), (14, 4)
(^16,11 ' ^16,21 )
4096 (24, 16), (26, 13), (22, 19), (21, 20), (28, 8)
2048 (11, 9), (12, 8), (10, 10), (13, 6), (14, 3)
(^16,13■> ^16,19 )
4096 (23, 17) , (22, 18), (25, 14), (26, 12), (27, 9)
(^16,15■> ^16,17 ) 2048 (8, 7), (9, 5), (10, 3)
Figure imgf000038_0001
When N=16, (<¾,i, <¾3) = (7, 3), ( 4,i, 4j3j 4j5, a4 ) = (5, 3, 2, 1), ( 8,i, 8j3, 8j5, a8j, <¾,9, <%,ii,
«8,i3, «8,15) = (10, 10, 9, 6, 5, 5, 3, 1):
Figure imgf000038_0002
Example 6: Order-8 Transform
[0060] Taking 6U = 128, (<¾i, 2j3) = (5, 2), (b2,i, /¾3) = (166, 72) and (b4,i, b4,3, b4,s b4 ) = (33, 29, 17, 6), An order-8 transform is built: 128 128 0 0
p 128 -128 0 0 J
0 0 72 166 J
Figure imgf000039_0001
X ! o4 " I4 J4 "
_o4 * J4
"128 128 128 128 128 128 128 128
177 153 96 36 -36 -96 153 -177
166 72 -72 - 166 -166 -72 72 166
143 -27 - 179 - 111 111 179 27 -143
128 - -128 - 128 128 128 128 - 128 128
111 - -179 27 143 -143 -27 179 -111
72 - -166 166 -72 -72 166 - 166 72
36 -96 153 - 177 177 153 96 - 36
[0061 J While we have hereinbefore described the embodiments of this application, it is understood that our basic constructions can be altered to provide other embodiments which utilize the processes and compositions of this application. Consequently, it will be appreciated that the scope of this application is to be defined by the claims appended hereto rather than by the specific embodiments which have been presented hereinbefore by way of examples.

Claims

What is claimed is:
1. A process for coding data in image or video, comprising:
retrieving a data matrix X2NX2N^ be coded; and
coding the retrieved data matrix by a rule of
7 — V . V T
wherein Z2NX2N represents a matrix for the coded data, E2N is an order-2N integer transform and N = 2" where n is a positive integer no less than 2.
2. The process according to claim 1 , further comprising:
generating E2N from an order-N transform EN .
3. The process according to claim 2, wherein the generating further comprises:
generating E2N from an order-N transform EN y performing a transform of
Figure imgf000040_0001
where
F2N is a permutation matrix,
Μ2Λί is a diagonal matrix with integer elements to make every basis norms in E2JV similar,
IN is an order-N identity matrix,
Jv is a mirrored version of IN, and
RN is a recursive matrix from an integer matrix ¾
4. The process according to claim 3, wherein M2NRN is constructed by a rule of
Figure imgf000040_0002
wherein, the process further comprises:
selecting values of bN such that a transform coding gain for the process is maximized, and differences among a basis vector norms of the transform itself is to minimized.
5. The process according to claim 4, wherein,
Figure imgf000041_0001
are integers that satisfy: antJ / (^,. N > 2 amJ " '
where 5 is a matrix consisting of
Figure imgf000041_0002
5™n is a minimal value of the sum of b2N and b2N,2N-i,
B™™ is a maximal value of the sum of b2N and b2N,2N-
R™™ is a minimal ratio between bN and bN> 2N-i,
R™** is a maximal ratio between and bN,2N-i.
6. The process accordin to claim 2 wherein is constructed b a rule of
Figure imgf000041_0003
1 1
1 1
0. 1 1
0. 1 - 1
- 1 1
- 1 1 where <¾ν,ι, <¾v,3, . . . ΟΛΓ,2ΛΓ-Ι are positive integers.
7. The process according to claim 6, wherein aN„ k, and N^N-k are prime pairs for k = \,3,..., N - \ .
. The process according to claim 7, wherein
Figure imgf000042_0001
9. The process according to claim 1 , wherein the coding further comprises a 2D transform carried out by two cascading ID transforms according to a rule of
U -X
v
- \z T ■ z
where
X2N K is a £-th column vector in X2N* N,
is a 2Nx2N matrix built byU2N k for A: e {o,l,...,2N- l} , and
r2JVjt is a &-th row vector in
Figure imgf000042_0002
for A: e {o,l,...,2N - 1} .
10. The process according to claim 2, wherein EN is one selected from a group consisting of an order-8 transform, an order- 16 transform and an order-32 transform t adopted in H.264/AVC.
1 1. The process according to claim 2, wherein EN is an order-8 transform, an order- 16 transform and an order-32 transform adopted in AVS.
12. The process according to claim 1 , wherein the process is implemented by one or more processor executing instructions encoded in computer readable storage medium.
13. The process according to claim 1 , wherein the retrieving and coding are respectively implemented by a retrieving unit and a coding unit in a coding system, each of units including electronic circuits.
14. An apparatus for coding data in image or video, comprising:
a retrieving unit for retrieving a data matrix ¾vx # to be coded; and
a coder for coding the retrieved data matrix by a rule of
7 — V . V T
wherein ∑2NX2N represents a matrix for the coded data, i¾v is an order-2N integer transform and N = 2" where n is a positive integer no less than 2.
15. The apparatus according to claim 14, wherein the retrieving unit is configured to further retrieve an order-N transform EN; and the system further comprises:
a generating unit for generating E2N from the retrieved EN .
16. The apparatus according to claim 15, wherein the generating unit generates E performing a transform of
Ε, P M 0* " 'I N J or
EN J
M2 NORN _ — -J
where
Ρ2Λ, is a permutation matrix,
M2N is a diagonal matrix with integer elements to make every basis norms in i¾v similar,
IN is an order-N identity matrix,
Jv is a mirrored version of IN, and
RN is a recursive matrix from an integer matrix K .
17. The apparatus according to claim 16, wherein M2NRN is constructed by a rule of
Figure imgf000044_0001
where, the values of bN are chosen such that a transform coding gain for the process is maximized, and the differences among a basis vector norms of the transform itself is to minimized.
18. The apparatus according to claim 17, wherein,
Figure imgf000044_0002
are integers that satisfy:
Figure imgf000044_0003
where B is a matrix consisting of bN ,
5™'n is a minimal value of the sum of b2 N i and b2 N 2N-i, is a maximal value of the sum of b2N and ?™n is a minimal ratio between bN and
R™** is a maximal ratio between bN and _W-/.
19. The apparatus according to claim 14, wherein the generating unit is configured to generate RN by a rule of
Figure imgf000045_0001
1 1
1 1
0. 1 1
0. 1 - 1
-1 1
- 1 1 where α^,ι, <¾v,3, ...aN^N-i are positive integers.
20. The apparatus according to claim 19, wherein the generating unit is further configured to select aN^, and aN^N-k from prime pairs for k = 1,3,..., N -1 .
21. The a aratus accordin to claim 20, wherein
Figure imgf000045_0002
22. The apparatus according to claim 14, wherein the coder is further configured to carry out a 2D transform through two cascading ID transforms to obtain Z^^by a rules of
U -X
v
7 - E VT
- \zT ■ z
where
Χ2Ν is a k-t column vector in X2JV^2#
is a 2Nx2N matrix built byU2N k for k e {o,l,...,2N- l}
ν2Ν is a k-t row vector in V2NX N for k e {o,l,...,2N - 1} .
23. The apparatus according to claim 15, wherein EN is an order-8 transform, order-16 transform and an order-32 transform adopted in H.264/AVC.
24. The apparatus according to claim 12, wherein EN is an order-8 transform, order-16 transform and an order-32 transform adopted in AVS.
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