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    Integer Cosine Transform (ICT)Chia-Hao Tsai ()

    E-mail: [email protected]

    Graduate Institute of Communication Engineering

    National Taiwan University, Taipei, Taiwan, ROC

    Abstract

    The discrete cosine transform (DCT) uses the concept of the transform coding that

    has best performance in image compression and filtering for image. The block sizes of

    most appropriate are 8 or 16 for the transform coding of the image data. Therefore,

    implementation of the order-8 and -16 DCTs has fast computing time and

    cost-effectiveness for realization of a transform coding. However, the components of

    the basis vectors of the DCT exist irrational numbers then cannot be reduced to

    integers by simple scaling. Therefore, it is hard to implement and using floating point

    arithmetic is complex and expensive, so integer cosine transforms (ICTs) are proposed

    to implement the DCT by using simple integer arithmetic. On the basis of theory of

    dyadic symmetry, transforms the order-8 and order-16 cosine transform into a family

    of integer cosine transforms (ICTs). If the better performance is demanded, the large

    magnitudes can be used for the integer transforms. If the fast computation speed and

    the low implementation cost are desired, the integer transforms of small transformcomponent magnitudes are chosen. An engineer can freely choose to tradeoff the

    performance and speed for the ICTs in designing the transform coding.

    Finally, we also show the other integer sinusoidal transforms. Each member of the

    sinusoidal family is the optimal Karhunen-Loeve transform (KLT) of a particular

    Markov process. The cosine and sine transforms are widely used for image coding.

    Equation Chapter 1 Section 1

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    1.Implementation of integer cosine transform by dyadicsymmetry

    1.1 IntroductionThe advantage of the transform coding is high compression ratio for the image

    compression. A transform coding is included two parts. The first part is that using

    orthogonal transform converts highly correlated image data into weakly correlated

    coefficients. The second part is that doing adaptive quantization on coefficients

    reduces the bit transmission rate. The discrete cosine transform (DCT) uses the

    concept of the transform coding that has best performance in image compression and

    filtering for image. The block sizes of most appropriate are 8 or 16 for the transform

    coding of the image data. For that reason, implementation of the order-8 and -16

    DCTs has fast computing time and cost-effectiveness for realization of a transform

    coding.

    However, the components of the basis vectors of the DCT exist irrational numbers

    then cannot be reduced to integers by simple scaling. Therefore, it is hard to

    implement and using floating point arithmetic is complex and expensive.

    Approximating the real magnitudes of the DCT components can eliminate floating

    point arithmetic by M-bit integers then the DCT can be computed using integer

    multiplications and additions. If the order-8 requires 7 bits for representation thecomponents, the 2-D order-8 requires 24-bit multiplication and 25-bit addition

    operations. It is difficult to implement and brings lots of delay in computation, so the

    other researchers have proposed new transform such as the Walsh transform [1], [2],

    slant transform [3] and the high-correlation transform (HCT) [4]. The proposed

    transforms have an advantage of shorter bit lengths, but compares with the DCT then

    their performance are all dissatisfactory. In addition, using the C-matrix transform

    (CMT) to approximate the order-8 DCT was found in Jones et al. [5] with small

    performance degradation. The CMT also can be extended to order-16 [6] and order-32

    [7]. The C-matrix is derived by trial and error.

    In the paper [8], the author proposed a new transform that is simple to implement

    using simple integer arithmetic. On the basis of theory of dyadic symmetry,

    transforms the order-8 cosine transform into a family of integer cosine transforms

    (ICTs). We will introduce the method how use the ICT to achieve the DCT as below.

    1.2 Definition of dyadic symmetryConsider a vector of 2

    m

    elements 0 1 12, , ,m

    a a a , it is said that has ith dyadic

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    Consider the order-8 DCT, we can get its kernel from eq.(1.3):

    0 0 1 1 2 2 3 3 4 4 5 6 6 7 75, , , , , ,,t

    T kk J k J k J k J k J J k J k J

    (1.3)

    where iJ means ith basis vector and ik denotes a scaling constant then makes

    1iik J . Let , J i j be the jth element of iJ . From eq.(1.1), we can know

    8 8 8 8 8 8 8 8(1,0) (1,7) (3,2) (3,5) (5,1) (5,6) (7,3) (7,4)T T T T T T T T then

    we can represent the magnitude of (1,0), (1,7), (3,2), (3,5), (5,1), (5,6), (7,3) J J J J J J J

    , and (7,4)J by a single variable a. As shown Table 1.2, all eight vectors can be

    expressed as variables b, c, d, e, and f in the same manner.

    0 1 1 1 1 1 1 1 1

    12

    3

    4 1 1 1 1 1 1 1 1

    5

    6

    7

    a b c d d c b ae f f e e f f e

    b d a c c a d b

    c a d b b d a c

    f e e f f e e f

    d c b a a b c d

    i iJ

    Table 1.2 The 8 scaled basis vectors in J .

    Step 2: Find the conditions which iJ and jJ are orthogonal.

    0

    1

    2

    3

    4

    5

    6

    7

    E E E E E E E

    O

    O E

    O

    O O E E O O E

    O

    O E

    O

    i Dyadic symmetry S in iJ

    Table 1.3 Sth dyadic symmetry type in each basis vector iJ .

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    We find the condition which the ith basis vector iJ and the jth basis vector jJ

    are orthogonal for all i,j. Seeing Table 1.2 and Table 1.3, we can know that 0J has

    even 7th dyadic symmetry and 1J has odd 7th dyadic symmetry. As we mentioned

    the orthogonal property in section 1.2, so 0J and 1J are always orthogonal.

    Because 0J has even 3rd dyadic symmetry and 2J has odd 3rd dyadic symmetry

    then 0J and 2J are always orthogonal. Table 1.4 lists the variable a, b, c, and d

    mustbe satisfied the condition to make the kernel T be orthogonal:

    .a b a c b d c d

    (1.4)

    *3 *2 *3 *2 *3 *2 *3 0

    *3 *1 *3 *1 *3 *4 1

    *3 *2 *3 *4 *3 2

    *3 *4 *3 *1 3

    *3 *2 *3 4

    *3 *1 5

    *3 6

    1 2 3 4 5 6 7 ji

    *1: if .a b a c b d c d *2: must the orthogonal due to 3rd dyadic symmetry

    *3: must the orthogonal due to 7th dyadic symmetry*4: must the orthogonal due to dot product equals zero

    Table 1.4 The conditions which the ith basis vector and thejth basis vector are

    orthogonal.

    From eq.(1.4) we know, the four variables a, b, c, and d have infinitely many

    solutions then it implies that the infinitely many new orthogonal transforms are

    generated from the DCT.

    Step 3: Set up boundary conditions and generate the new orthogonal transforms.

    From eq.(1.1) and Table 1.2, the relationship of the magnitude of the variables a, b,

    c, d, e, andfis for the DCT:

    and .a b c d e f

    (1.5)

    To let the new orthogonal transforms be like the DCT, so the eq.(1.5) must be satisfied.

    For eliminating truncation error, the eq.(1.6) also must be satisfied. The eq.(1.6) is as

    below.

    , , , , , and are integersa b c d e f

    (1.6)

    When the kernel of order-8 DCT T is satisfied eq.(1.4), eq.(1.5), and eq.(1.6), it

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    can convert into the order-8 ICT. For example, ICT(5, 3, 2, 1, 3, 1) represents the

    order-8 ICT with a=5, b=3, c=2, d=1, e=3, andf=1.

    1.3.2 Generation of the order-2n ICTs

    The order-n ,n i jT can be extended to the order-2n orthogonal transform

    2 ,n i jT . The process is as below.

    1. The first n basis vectors of 2 ,n i jT : 2 ,2 ,n ni j i jT T and 2 ,2 1 , , for 0 1.n ni j i j j nT T

    (1.7)

    2. The last n basis vectors of 2 ,n i jT :1 2 ,2 ,n ni n j i jT T and

    2 ,2 1 , , for 0,2,4, , 2 .n ni n j i j j nT T

    (1.8)

    2 2 ,2 ,n ni n j i jT T and

    2 ,2 1 , , for 1,3,5, , 1 .n ni n j i j j nT T

    (1.9)

    Any order-2m

    ICT for m>3 can be generated from the order-8 ICT.

    1.4 Performance of order-n ICTs

    1.4.1 Transform efficiency [4]

    One part of a transform coding is using orthogonal transform converts highly

    correlated image data into weakly correlated coefficients. If it uses the optimal

    Karhunen-Loeve transform (KLT), we can say that its transform efficiency is equal to

    100% for all .

    Consider n-dimensional vector X, which is a sample from one-dimensional,

    zero-mean, unit-variance first-order Markov process with adjacent element correlation

    , and Yis a transformed matrix.

    Y T X

    (1.10)

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    11 1

    1

    =

    =

    tY

    tX

    n

    n nn

    EC Y Y

    T TC

    s s

    s s

    (1.11)

    where XC is covariance matrix of the n-dimensional vector X then the (i, j)th

    element of XC isi j

    and YC is covariance matrix of the transformed

    matrix. The efficiency of the kernel Tis defined on the transformed domain:

    1

    1 1

    Efficiency .

    nii

    in n

    pqp q

    s

    s

    (1.12)

    The DCT has been popularly accepted and has the highest transform efficiency

    within the suboptimal transforms when is close to unity. Table 1.5 shows the

    highest transform efficiencies of the twelve order-8 ICTs when is equal to 0.9 and

    a less than or equal to 255. From Table 1.5, we can realize that the transform

    efficiencies of all twelve order-8 ICTs are higher than the order-8 DCT.

    90.221 ICT(230, 201, 134, 46, 3, 1)

    90.220 ICT(175, 153, 102, 35, 3, 1)

    90.219 ICT(120, 105, 70, 24, 3, 1)

    90.217 ICT(185, 162, 108, 37, 3, 1)

    90.217 ICT(250, 219, 146, 50, 3, 1)

    90.215 ICT(65, 57, 38, 13, 3, 1)

    90.213 ICT(55, 48, 32, 11, 3, 1)

    90.213 ICT(205, 180, 120, 41, 3 , 1)

    90.212 ICT(140, 123, 82, 28, 3, 1)

    90.211 ICT(215, 189, 126, 43, 3, 1)

    90.210 ICT(75, 66, 4 4, 15, 3, 1)

    90.208 ICT(235, 207, 138, 47, 3 , 1)

    89.836 DCT

    86.785 CMT

    85.842 slant transform

    84.097 HCT

    77.140 Walshtransform

    Transform

    efficiencyTransform

    Table 1.5 The highest transform efficiency of the twelve order-8 ICTs when is

    equal to 0.9 and a less than or equal to 255.

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    Consider the order-16 ICT of the case, we can find that ICT(246, 222, 147, 50, 3

    , 1) has the highest transform efficiency when a less than or equal to 255 and ICT(

    10, 9, 6, 2, 3, 1) has the highest transform efficiency when a less than or equal to

    128. The transform efficiencies of all two order-16 ICTs are lower than the order-16

    DCT and the slant transform as shown in Table 1.6. Because the order-16 DCT and

    the slant transform have 16-level, and the order-16 ICT only has 8-level then the

    result is happened as the above said.

    82.3 DCT

    74.1 slant transform

    73.9 ICT(246, 222, 147, 50, 3, 1)

    73.8 ICT(10, 9, 6, 2, 3, 1)

    73.7 CMT

    68.4 HCT

    60.9 Walsh transform

    Transform

    efficiency Transform

    Table 1.6 The highest transform efficiencies of the two order-16 ICTs when a less

    than or equal to 255.

    1.4.2 Basis restriction mean square errorTo consider a two-dimensional zero-mean unit-variance nonseparable isotropic

    Markov process with covariance function

    2 2

    , ,, ; ,

    =

    x i j p q

    i p j q

    i j p q E C x x

    (1.13)

    where is the adjacent element correlation in the vertical and horizontal directions.

    Let the nnmatrix [X] be a sample of the Markov process. If [X]is transformed into

    [C]by transform [T], we can get

    .t

    C T X T

    (1.14)

    where the elements of [X] and [C] are,i jx and ,u vc then covariance function of [C]

    is

    , ,, ; ,

    = , ; , , , , , .

    c u v r s

    xi j p q

    u v r s E C c c

    i j p q T u i T u j T r p T s qC

    (1.15)

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    The variance of ,u vc is equal to

    2 , = , ; ,cu v u v u vC

    (1.16)

    Suppose be the set containingMindex pairs (u, v)corresponding to the largestM

    ,c u v then the basis restriction mean square error is defined as:

    2

    ,

    2

    ,

    ( ) 1,

    cu v

    cu v

    u v

    e Mu v

    (1.17)

    where ,c

    u v is variance of matrix

    tC T X T and means the set

    which contains M index pairs ,u v then corresponds to the largest M ,c u v .

    We can use the basis restriction mean-square-error to measure the data compression

    ability of a transform. Comparing the basis restriction mean-square-errors of the ICTs

    with various transforms when 0.95 is shown in Table 1.7. From Table 1.7, we

    can know that the relationship of the basis restriction mean square error:

    Walsh>CMT>ICT>DCT>KLT.

    (1.18)

    1.5 Implementation

    Let T is an ICT and look the eq.(1.3) that we know

    1

    andt t

    T K J T T J K

    (1.19)

    where K is a diagonal matrix and the values of the elements of J are integers

    because T is an ICT. The adaptive 1-D transform coding system uses an order-8

    ICT as shown Fig. 1.1. In upper part of Fig. 1.1 is its transmitter and in lower part of

    Fig. 1.1 is its receiver.

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    2 0.1372 0.1381 0.1381 0.1381 0.1382 0.1387 0.1468

    6 0.0567 0.0572 0.0573 0.0573 0.0573 0.0587 0.0785

    10 0.0406 0.0409 0.0410 0.0410 0.0410 0.0431 0.0541

    14 0.0320 0.0322 0.0323 0.0323 0.0323 0.0348 0.0441

    18 0.0263 0.0264 0.0266 0.0266 0.0266 0.0287 0.0361

    22 0.0221 0.0222 0.0223 0.0223 0.0224 0.0238 0.0300

    26 0.0189 0.0189 0.0190 0.0190 0.0190 0.0198 0.0251

    30 0.0160 0.0160 0.0162 0.0162 0.0162 0.0165 0.0205

    34 0.0136 0.0136 0.0137 0.0137 0.0137 0.0140 0.0170

    no. of

    coefficients

    retained KLT DCT

    ICT(230,

    201, 134,

    46, 3, 1)

    ICT(55,

    48, 32,

    11, 3, 1)

    ICT(10,

    9, 6, 2, 3,

    1) CMT Walsh

    Table 1.7 comparing the basis restriction mean-square-errors of the ICTs with various

    transforms when 0.95 .

    0k

    1k

    7k

    quantizer #0

    quantizer #1

    quantizer #7

    JX

    adaptive bit

    allocation

    0c

    1c

    7c

    0k

    1k

    7k

    quantizer #0

    quantizer #1

    quantizer #7

    t

    J

    'X

    adaptive

    scheme

    '0c'

    1c

    '7c

    overhead

    channel

    overhead

    '0c

    '1c

    '7c

    Fig. 1.1 The Adaptive 1-D transform coding system using an order-8 ICT.

    The fast algorithm of the ICT is like the DCT and only requires 4 iterations as

    shown Fig. 1.2.

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    0

    1

    2

    3

    4

    5

    6

    7

    -

    -

    -

    -

    -

    -

    -

    -

    -

    -q-q

    p

    p

    e

    e

    d

    d

    b

    b

    a

    -a

    c-c

    0c

    4c

    6c

    2c

    1c

    5c

    3c

    7c

    Fig. 1.2 The fast algorithm of the order-8 ICT where ( ) / 2p b c a and

    ( ) / 2q a d c .Equation Chapter (Next) Section 1

    2.Implementation of order-16 integer cosine transform bydyadic symmetry

    2.1 IntroductionIn chapter 1, we introduce a new transform that is simple to implement using

    simple integer arithmetic [8]. On the basis of theory of dyadic symmetry, transformsthe order-8 and cosine transform into a family of integer cosine transforms (ICTs).

    Any order-2m

    ICT for m>3 can be generated from the order-8 ICT. But its

    performance is poorer than the order-16 DCT and the slant transform. Therefore, we

    will introduce the method how directly use the order-16 ICT (denotes as ITs here) to

    achieve the order-16 DCT by dyadic symmetry [9] then its performance is better than

    the order-16 ICT [8] and the slant transform.

    2.2 Definition of dyadic symmetryAs we have been seen in section 1.2, we can know the definition of dyadic

    symmetry. By the definition, the order-16 even dyadic symmetry can be shown as

    Table 2.1.

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    0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15a a a a a a a a a a a a a a a a

    1

    2

    3

    4

    5

    6

    7

    1 2 3 4 5 6 7 8 5 6 7 8 1 2 3 4

    1 2 3 4 5 6 7 8 6 5 8 7 2 1 4 3

    1 2 3 4 5 6 7 8 7 8 5 6 3 4 1 2

    1 2 3 4 5 6 7 8 8 7 6 5 4 3 2 1

    h h h h h h h h h h h h h h h h

    h h h h h h h h h h h h h h h h

    h h h h h h h h h h h h h h h h

    h h h h h h h h h h h h h h h h

    1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8

    1 2 1 2 3 4 3 4 5 6 5 6 7 8 7 8

    1 2 2 1 3 4 4 3 5 6 6 5 7 8 8 7

    1 2 3 4 1 2 3 4 5 6 7 8 5 6 7 8

    1 2 3 4 2 1 4 3 5 6 7 8 6 5 8 7

    1 2 3 4 3 4 1 2 5 6 7 8 7 8 5 6

    1 2 3 4

    h h h h h h h h h h h h h h h h

    h h h h h h h h h h h h h h h h

    h h h h h h h h h h h h h h h h

    h h h h h h h h h h h h h h h h

    h h h h h h h h h h h h h h h h

    h h h h h h h h h h h h h h h h

    h h h h 4 3 2 1 5 6 7 8 8 7 6 5h h h h h h h h h h h h

    1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8

    1 2 3 4 5 6 7 8 2 1 4 3 6 5 8 7

    1 2 3 4 5 6 7 8 3 4 1 2 7 8 5 6

    1 2 3 4 5 6 7 8 4 3 2 1 8 7 6 5

    h h h h h h h h h h h h h h h h

    h h h h h h h h h h h h h h h hh h h h h h h h h h h h h h h h

    h h h h h h h h h h h h h h h h

    8

    910

    11

    12

    13

    14

    15

    VectorsHS

    Table2.1 The fifteen vectors sH

    having Sth even dyadic symmetry.

    If want to get the order-16 dyadic symmetry matrix, only needs to change the signs

    of sH according to the odd dyadic symmetry in section 1.2. It is said that two

    vectors Uand Vare orthogonal ifUand Vhave the same type of dyadic symmetry

    and one is even and another is odd.

    2.3 Generation of the order-16 ICTs (ITs)Let T be the kernel of order-16 DCT and 16 ,i jT is thejth component of the

    ith DCT basis vector as below.

    16

    1, for 0

    16, , 0 15

    0.52cos , for 1 15

    16 16

    i

    i j jTi j

    i

    (2.1)

    We describe the steps that transforms the order-16 DCT kernels into ICT kernels in

    the following.

    Step 1: Express the order-8 DCT kernel T in form of a matrix of variable.

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    Consider the order-16 DCT, we can get its kernel from eq.(2.2):

    0 0 1 1 2 2 3 3 4 4 5 6 6 7 75[ , , , , , , ,,T kk J k J k J k J k J J k J k J

    8 8 9 9 10 10 11 11 12 12 13 14 14 15 1513, , , , , , ],

    t

    kk J k J k J k J k J J k J k J

    (2.2)

    where iJ means ith basis vector and ik denotes a scaling constant then makes

    1iik J . Let , J i j be the jth element of iJ . As shown Table 2.2, all sixteen

    vectors can be expressed as variables a, b, c, , m and n in the same manner.

    l l l l l l l l l l l l l l l l

    a b c d e f g h h g f e d c b a

    i j k l l k j i i j k l l k j i

    b e h f c a d g g d a c f h e b

    m n n m m n n m m n n m m n n m

    c h d b g e a f f a e g b d h c

    j l i k k i l j j l i k k i l j

    d f b h a g c e e c g a h b f d

    l l l l l l l l l l

    l l l l l l

    e c g a h b f d d f b h a g c e

    k i l j j l i k k i l j j l i k

    f a e g b d h c c h d b g e a f

    n m m n n m m n n m m n n m m n

    g d a c f h e b b e h f c a d g

    l k j i i j k l l k j i i j k l

    h g f e d c b a a b c d e f g h

    iJ

    0

    1

    2

    3

    4

    5

    6

    7

    8

    9

    10

    11

    12

    13

    14

    15

    Table 2.2 The 16 scaled basis vectors in J .

    Step 2: Find the conditions which iJ and jJ are orthogonal.

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    ,

    ,

    E E E E E E E E E E E E E E E

    O

    O O E O

    O

    O O E E O O E

    O

    O O E

    O

    O O E E O O E E O O E E O O E

    O

    O O E O

    O

    O O E E O O E

    O

    O O E

    O

    1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

    0

    1

    2

    3

    4

    5

    6

    7

    8

    9

    10

    11

    12

    13

    14

    15

    i

    Dyadic Symmetry S in iJ

    Table 2.3 Sth dyadic symmetry type in each basis vector iJ .

    We find the condition which the ith basis vector iJ and the jth basis vector jJ

    are orthogonal for all i,j. Seeing Table 2.2 and Table 2.3, we can know that 0J has

    even 15th dyadic symmetry and 1J has odd 15th dyadic symmetry. As we

    mentioned the orthogonal property in section 2.2, so 0J and 1J are always

    orthogonal. Because 0J has even 7th dyadic symmetry (or even 8th dyadic

    symmetry, even 15th dyadic symmetry) and 2J has odd 7rd dyadic symmetry (or

    odd 8th dyadic symmetry, odd 15th dyadic symmetry) then 0J and 2J are always

    orthogonal. To make the ith basis vector iJ and the jth basis vector jJ are

    orthogonal for all i,j that must be satisfied eq.(2.3)-eq.(2.6).

    ab be ch df ce dg gh af

    (2.3)

    ac bh ef ag fh cd bd eg

    (2.4)

    ad dh ae fg bf bc cg eh

    (2.5)

    ij jl kl ik

    (2.6)

    Step 3: Set up boundary conditions and generate the new orthogonal transforms.

    From eq.(2.1) and Table 2.2 , the relationship of the magnitude of the variables a, b,

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    c, d, e, andfis for the DCT:

    a i b m c j d e k f n g l h

    (2.7)

    To relax the condition in eq.(2.7), we can get

    ,a b c d e f g h

    (2.8)

    ,i j h k l

    (2.9)

    .m n

    (2.10)

    For eliminating truncation error, the eq.(2.8) also must be satisfied. The eq.(2.8) is as

    below.

    , , , , , , and are integersa b c d e m n

    (2.11)

    When the kernel of order-16 DCT T is satisfied eq.(2.3)-eq.(2.6) and

    eq.(2.8)-eq.(2.11), it can convert into the order-16 ICT. For example, when i=55,j=48,

    k=32, l=11, m=3 and n=1, the computer search can find 119 compositions of variablesa-h (the magnitudes of variables a-h are restricted in0~255).

    2.4 Performance of order-16 ICTs (ITs)In section 1.4, two objective measures are defined to evaluate the performance of

    the new order-16 integer transforms, Transform efficiency and Basis restriction mean

    square error.

    2.4.1 Transform efficiency

    Table 2.4 shows the transform efficiencies of the IT and ICT [8] and otherwell-known transforms for 0.9 . As we have seen, the transform efficiency of ITs

    is higher than the order-16 ICT [8] and other well-known transforms.

    Transform

    efficiencyTransform

    82.3 DCT

    81.9 IT#1

    74.1 slant transform

    73.9 ICT(246, 222, 147, 50, 3, 1)

    73.8 ICT(10, 9, 6, 2, 3, 1)

    73.7 CMT

    68.4 HCT

    60.9 Walsh transform

    IT#1: a=16, b=160, c=148, d=134, e=106,f=80,

    g=46, h=26, i=55,j=48, k=32, l=11, m=3, n=1

    Table 2.4 The transform efficiencies of the IT and ICT [8] and other well-known

    transforms for 0.9 .

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    2.4.2 Basis restriction mean square error

    We can use the basis restriction mean-square-error to measure the data compression

    ability of a transform.Comparing the basis restriction mean-square-errors of the ITs

    with ICT [8] when 0.9 is shown in Table 2.5. From Table 2.5, we can know that

    the relationship of the basis restriction mean square error:

    ICT>IT>DCT.

    (2.12)

    0.4422 0.4421 0.4424

    0.2321 0.2321 0.2330

    0.1748 0.1748 0.1756

    0.1438 0.1436 0.1443

    0.1241 0.1238 0.1245

    0.1103 0.1100 0.1107

    0.1003 0.0999 0.1005

    0.0919 0.0914 0.0920

    0.0853 0.0849 0.0854

    No. of coeff.

    retained

    2

    6

    10

    14

    18

    22

    26

    30

    34

    IT DCT ICT

    Table 2.5 comparing the basis restriction mean-square-errors of the ITs with ICT [8]

    when 0.9 .

    Equation Chapter (Next) Section 1

    3.The other integer sinusoidal transforms3.1 Introduction

    Each member of the sinusoidal family is the optimal Karhunen-Loeve transform

    (KLT) of a particular Markov process. The cosine and sine transforms are widely used

    for image coding. In the paper [10], the authors derive order-8 sinusoidal transforms

    that can be implemented by using the integer arithmetic.

    3.2 Integer sinusoidal transformsEven sine-1 transform

    Let T be the kernel of order-8 even sine-1 transform and ,n i jT is the jth

    component of the ith even sine-1 transform basis vector as below.

    1 12

    , sin , for 0 , 11 2

    n

    i ji j i j nT

    n n

    (3.1)

    Consider the order-8 even sine-1 transform, we can get its kernel from eq.(3.2):

    0 0 1 1 2 2 3 3 4 4 5 6 6 7 75, , , , , ,,t

    T kk J k J k J k J k J J k J k J

    (3.2)

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    where iJ means ith basis vector and ik denotes a scaling constant then makes

    1iik J . As shown eq.(3.3), all eight vectors can be expressed as variables a, b,

    c, and d by using eq.(3.1).

    0

    1

    2

    3

    4

    5

    6

    7

    ( )

    ( )

    ( 0 0 )

    ( )

    ( )

    ( 0 0 )

    ( )

    ( )

    a b c d d c b ak

    b d c a a c d bk

    c c c c c ck

    d a c b b c a d kT

    d a c b b c a d k

    c c c c c ck

    b d c a a c d bk

    a b c d d c b ak

    (3.3)

    As well as derived in section 1.2, the variable a, b, c, and d must be satisfied the

    condition to make the kernel T be orthogonal:

    0c a b d

    (3.4)

    2 0a d b bd c

    (3.5)

    To simplify the eq.(3.4) and eq.(3.5), we can get

    0d a b c

    (3.6)

    2 2c aba b

    (3.7)

    From eq.(3.1) and eq.(3.3), the relationship of the magnitude of the variables a, b, c,

    and dis for the order-8 even sine-1 transform:

    >0.d c b a

    (3.8)

    The other integer sinusoidal transforms can be derived as well as section 1.2 then

    summarizes in Table 3.1.

    The

    integer

    sinusoidal

    transform

    The basis vector of the transform ,n i jT

    kernel and the transform kernel T

    The orthogonal

    condition and

    relationship of the

    magnitude

    Even sine-1

    transform

    ,

    1 12sin , for 0 , 1

    1 2

    n i jT

    i ji j n

    n n

    0d a b c

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    0

    1

    2

    3

    4

    5

    6

    7

    ( )

    ( )

    ( 0 0 )( )

    ( )

    ( 0 0 )

    ( )

    ( )

    T

    a b c d d c b ak

    b d c a a c d bk

    c c c c c ckd a c b b c a d k

    d a c b b c a d k

    c c c c c ck

    b d c a a c d bk

    a b c d d c b ak

    2 2c aba b

    >0.d c b a

    Odd sine-2

    transform

    ,

    2 1 12 sin , for 0 , 1

    2 12 1

    n i jT

    i ji j n

    nn

    0

    1

    2

    3

    4

    5

    6

    7

    T

    a b c d e f g hk

    b d f h g e c ak

    c f h e b a d gk

    d h e a c g f bk

    e g b c h d a f k

    f e a g d b h ck

    g c d f a h b ek

    h a g b f c e d k

    0

    ab bd cf dh

    eg fe gc ha

    0

    ac bf ch de

    eb fa gd hg

    0

    ae bg cb dc

    eh fd ga hf

    0

    d e c f

    b g a h

    Odd sine-1

    transform

    ,

    2 1 12sin , for 0 , 1

    2 11

    n i jT

    i ji j n

    nn

    0

    1

    2

    3

    4

    5

    6

    7

    T

    a b c d e f g hk

    c f h e b a d gk

    e g b c h d a f k

    g c d f a h b ek

    h a g b f c e d k

    f e a g d b h ck

    d h e a c g f bk

    b d f h g e c ak

    0

    a c f b e f

    d g e h g c

    0

    a e g b c g

    d c f h f e

    0

    a h b c f g

    d b h e f g

    h g f e

    d c b a

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    Odd sine-3

    transform

    ,

    1 2 12sin , for 0 , 1

    2 12 1

    n i jT

    i ji j n

    nn

    0

    1

    2

    3

    4

    5

    67

    T

    a b c d e f g hk

    h f d b a c e gk

    b e h g d a c f k

    g c b f h d a ek

    c h e a f g b d k

    f a g e b h d ck

    d g a h c e f bk

    e d f c g b h ak

    0

    ah bf cd db

    ea fc ge hg

    0

    ab be ch dg

    ed fa gc hf

    0

    ac bh ce da

    ef fg gb hd

    0

    e d f c

    g b h a

    Even sine-2

    transform

    ,

    1 2 12sin ,

    2

    for 0 , 1and 1

    1,

    for 0 1and 1

    n

    j

    i jT

    i j

    n n

    i j n i n

    n j n i n

    0

    1

    2

    3

    4

    5

    6

    7

    T

    a b c d d c b ak

    e f f e e f f ek

    b d a c c a d bk

    g g g g g g g gk

    c a d b b d a ck

    f e e f f e e f k

    d c b a a b c d k

    g g g g g g g gk

    0ab bd ac cd

    0d c b a

    f e

    Even sine-3

    transform

    ,

    2 1 2 12sin , for 0 , 1

    4

    n i jT

    i ji j n

    n n

    0

    ab be ch df

    ec fa gd hg

    0

    ac bh cd db

    eg fe ga hf

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    0

    1

    2

    3

    4

    5

    6

    7

    T

    a b c d e f g hk

    b e h f c a d gk

    c h d b g e a f kd f b h a g c ek

    e c g a h b f d k

    f a e g b d h ck

    g d a c f h e bk

    h g f e d c b ak

    0

    ad bf cb dh

    ea fg gc he

    0

    h g f e

    d c b a

    Odd

    cosine-1

    transform

    ,

    2 1 2 12 cos ,

    4 24 1

    for 0 , 1

    n i jT

    i j

    nn

    i j n

    0

    1

    2

    3

    4

    5

    6

    7

    T

    a b c d e f g hk

    b e h g d a c f k

    c h e a f g b d k

    d g a h c e f bk

    e d f c g b h ak

    f a g e b h d ck

    g c b f h d a ek

    h f d b a c e gk

    0

    ab be ch dg

    ed fa gc hf

    0

    ac bh cf ad

    ef fg bg dh

    0

    ae bd cf cd

    eg bf gh ha

    0

    a b c d

    e f g h

    Even

    cosine-2

    transform

    ,

    2 1 2 12cos , for 0 , 1

    4

    n i jT

    i ji j n

    n n

    0

    ab be ch df

    ec fa gd hg

    0

    ac bh cd db

    eg fe ga hf

    0

    ad bf cb dh

    ea fg gc he

    0

    a b c d

    e f g h

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    0

    1

    2

    3

    4

    5

    6

    7

    T

    a b c d e f g hk

    b e h f c a d gk

    c h d b g e a f kd f b h a g c ek

    e c g a h b f d k

    f a e g b d h ck

    g d a c f h e bk

    h g f e d c b ak

    Even

    cosine-1

    transform

    (introduced

    in section

    1.3)

    1,

    for 0, 0 1,

    0.52cos ,

    for 0 , 1 and 0

    n

    n

    i j ni jT

    i j

    n n

    i j n i

    0

    1

    2

    3

    4

    5

    6

    7

    1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1

    T

    k

    a b c d d c b ak

    e f f e e f f ek

    b d a c c a d bk

    k

    c a d b b d a ck

    f e e f f e e f k

    d c b a a b c d k

    ab ac bd dc

    a b c d

    e f

    Table 3.1 The integer sinusoidal transforms.

    4.ConclusionsIn the chapter 1, we introduce a new transform that is simple to implement using

    simple integer arithmetic. On the basis of theory of dyadic symmetry, transforms the

    order-8 cosine transform into a family of integer cosine transforms (ICTs). When

    using longer bit lengths to present the kernel components, we can get better

    performance for the ICT. Furthermore, the fast algorithm of the order-8 ICT is like the

    order-8 DCT that only requires integer multiplication and addition operations. Any

    order-2m ICT for m>3 can be generated from the order-8 ICT but the performance is

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    poorer than the order-16 DCT and the slant transform. Therefore, we have introduced

    the method how directly use the order-16 ICT to achieve the order-16 DCT by dyadic

    symmetry in the chapter 2. The ITs have higher transform efficiency and lower basis

    restriction MSE performance than the ICTs. Furthermore, the performance of the ITs

    is very close to the DCT.

    In the chapter 3, we show the other integer sinusoidal transforms. The order-8

    sinusoidal transforms can be implemented by using the integer arithmetic. Each

    member of the sinusoidal family is the optimal Karhunen-Loeve transform (KLT) of a

    particular Markov process. The cosine and sine transforms are widely used for image

    coding.Finally, an engineer can freely choose to tradeoff the performance and speed

    for the ICTs in designing the transform coding.

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    5.References[1] B.J. FINO, and V.R. ALGAZI, Unified matrix treatment of the fast

    Walsh-Hadamard transform,IEEE Trans., 1976, C-25, (1 l), pp.1142-1146.

    [2] W.K.CHAM and R.J.CLARKE, Dyadic symmetry and Walsh matrices, IEE

    Proc. Commun.,Radar & Signal Process., 1987, 134, (2), pp. 141-144.

    [3] W.K PRATT, W.CHEN, and WELCH, L.R., Slant transform image coding,

    IEEE Trans., 1974, COM-22, (8), pp. 1075-1093.

    [4] W.K.CHAM and R.J. CLARKE, Application of the principle ofdyadic symmetry

    to the generation of orthogonal transforms, IEE Proc. F, Commun., Radar &

    Signal Process., 1986, 133, (3), pp.264-270.

    [5] J H.W.ONES, D.N.HEIN, and K S.C.NAUER, The Karhunen-Loeve discrete

    cosine and related transforms obtained via the Hadamard transform, Proc. Intl.

    Telemetering Conference, November 1978,14, pp. 87-98.

    [6] R.SRINIVASAN and RAO, K.R., An approximation to the discrete cosine

    transform for N = 16,Signal Process., 1983, 5, pp.81-85.

    [7] H.S.KWAK, R. SRINIVASAN, and K.R. RAO, C-matrix transform,IEEE

    Trans., 1983, ASP-31, (5), pp. 1304-1307.

    [8] W.K. CHAM, Development of integer cosine transforms by the principle of

    dyadic symmetry,IEE PROCEEDINGS, Vol. 136, Pt. I, No. 4, AUGUST 1989.

    [9]S. N. KOH, S. J. HUANG, and H. K. TANG , Development of order-16 integertransforms, Signal Processing, Volume 24, Issue 3, September 1991, pp. 283-

    289.

    [10] W.K. CHAM and P.P.C. YIPS, Integer sinusoidal transforms for image

    processing, INT. J. ELECTRONICS, VOL. 70, NO. 6, 1991, pp.1015-1030.


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