The H.264 Integer Transform

Video compression design, analysis, consulting and research

White Paper:
4x4 Transform and Quantization in H.264/AVC

© Iain Richardson / VCodex Limited

Version 1.1a Revised April 2009

Vcodex White Paper: 4x4 Transform and Quantization in H.264/AVC
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H.264 Transform and Quantization

1 Overview

In an H.264/AVC codec, macroblock data are transformed and quantized prior to
coding and rescaled and inverse transformed prior to reconstruction and display
(Figure 1). Several transforms are specified in the H.264 standard: a 4x4 “core”
transform, 4x4 and 2x2 Hadamard transforms and an 8x8 transform (High profiles
only).

Figure 1 Transform and quantization in an H.264 codec

This paper describes a derivation of the forward and inverse transform and
quantization processes applied to 4x4 blocks of luma and chroma samples in an H.264
codec. The transform is a scaled approximation to a 4x4 Discrete Cosine Transform
that can be computed using simple integer arithmetic. A normalisation step is
incorporated into forward and inverse quantization operations.

2 The H.264 transform and quantization process

The inverse transform and re-scaling processes, shown in Figure 2, are defined in the
H.264/AVC standard. Input data (quantized transform coefficients) are re-scaled (a
combination of inverse quantization and normalisation, see later). The re-scaled values
are transformed using a “core” inverse transform. In certain cases, an inverse
transform is applied to the DC coefficients prior to re-scaling. These processes (or their
equivalents) must be implemented in every H.264-compliant decoder. The
corresponding forward transform and quantization processes are not standardized but
suitable processes can be derived from the inverse transform / rescaling processes
(Figure 3).

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Figure 2 Re-scaling and inverse transform

Figure 3 Forward transform and quantization

3 Developing the forward transform and quantization process

The basic 4x4 transform used in H.264 is a scaled approximate Discrete Cosine
Transform (DCT). The transform and quantization processes are structured such that
computational complexity is minimized. This is achieved by reorganising the processes
into a core part and a scaling part.

Consider a block of pixel data that is processed by a two-dimensional Discrete Cosine
Transform (DCT) followed by quantization (dividing by a quantization step size, Qstep ,
then rounding the result) (Figure 4a).

Rearrange the DCT process into a core transform (Cf) and a scaling matrix (Sf) (Figure
4b).

Scale the quantization process by a constant (215) and compensate by dividing and
rounding the final result (Figure 4c).

Combine Sf and the quantization process into Mf (Figure 4d), where:

Sf # 215
Mf "
Qstep
Equation 1

! (The reason for the use of ≈ will be explained later).


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Figure 4 Development of the forward transform and quantization process

4 Developing the rescaling and inverse transform process

Consider a re-scaling (or “inverse quantization”) operation followed by a two-
dimensional inverse DCT (IDCT) (Figure 5a).

Rearrange the IDCT process into a core transform (Ci) and a scaling matrix (Si) (Figure
5b).

Scale the re-scaling process by a constant (26) and compensate by dividing and
rounding the final result (Figure 5c)1.

Combine the re-scaling process and S into Vi (Figure 5d), where:

Vi = S i " 2 6 " Qstep
Equation 2

!

1
This rounding operation need not be to the nearest integer.


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Figure 5 Development of the rescaling and inverse transform process

5 Developing Cf and Sf (4x4 blocks)

Consider a 4x4 two-dimensional DCT of a block X:

Y = A⋅X⋅AT
Equation 3

Where ⋅ indicates matrix multiplication and:

#a a a a & a = 12
% (
b c "c "b(
A=% , b = 1 2 cos ) 8 = 0.6532....
%a "a "a a (
% (
$c "b b "c ' c = 1 2 cos 3) 8 = 0.2706....

The rows of A are orthogonal and have unit norms (i.e. the rows are orthonormal).
Calculation of Equation 3 on a practical processor requires approximation of the
! irrational numbers b and c. A fixed-point approximation is equivalent to scaling each
row of A and rounding to the nearest integer. Choosing a particular approximation
(multiply by 2.5 and round) gives Cf :


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#1 1 1 1 &
% (
Cf = %2 1 "1 "2(
%1 "1 "1 1 (
% (
$1 "2 2 "1'

This approximation is chosen to minimise the complexity of implementing the
transform (multiplication by Cf requires only additions and binary shifts) whilst
! maintaining good compression performance.

The rows of Cf have different norms. To restore the orthonormal property of the
1
original matrix A, multiply all the values cij in row r by :
2
" c rj
j

" 1 2 12 12 %
12
$ '
$1 ! 10 1 10 1 10 '
10 1
A1 = Cf • Rf where R f = $ '
$ 1 2 12 12 12 '
$ '
$1
# 10 1 10 1 10 1 10 '
&

• denotes element-by-element multiplication (Hadamard-Schur product2). Note that
!
the new matrix A1 is orthonormal.

The two-dimensional transform (Equation 3) becomes:

Y = A1⋅X⋅A1T = [Cf • Rf]⋅X⋅[CfT • RfT ]

Rearranging:

Y = [Cf⋅X⋅CfT] • [Rf • RfT]
= [Cf⋅X⋅CfT] • Sf

Where

" %
$ 1 4 1 2 10 1 4
1 2 10 '
$ '
T 1 2 10 1 10 1 2 10 1 10 '
Sf = R f • R f =$
$ 1 4 1 2 10 1 4 1 2 10 '
$ '
$1 2 10 1 10 1 2 10 1 10 '
# &

2
P = Q•R means that each element pij = qij⋅rij
!


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6 Developing Ci and Si (4x4 blocks)

Consider a 4x4 two-dimensional IDCT of a block Y:

Z = AT⋅Y⋅A
Equation 4

Where

#a a a a & a = 12
% (
b c "c "b(
A=% , b = 1 2 cos ) 8 = 0.6532.... as before.
%a "a "a a (
% (
$c "b b "c ' c = 1 2 cos 3) 8 = 0.2706....

Choose a particular approximation by scaling each row of A and rounding to the
nearest 0.5, giving Ci :
!
#1 1 1 1 &
% (
% 1 1 2 "1 2 "1 (
Ci = % (
%1 "1 "1 1 (
% (
%1 2 "1
$ 1 "1 2(
'

The rows of Ci are orthogonal but have non-unit norms. To restore orthonormality,
1
! multiply all the values cij in row r by :
2
"c rj
j

"1 2 12 12 12%
$ '
! $ 2 5 2 5 2 5 2 5'
A2 = Ci • Ri where R i = $ '
$1 2 12 12 12'
$ '
$ 2 5
# 2 5 2 5 2 5'
&

The two-dimensional inverse transform (Equation 4) becomes:
!
Z = A2T⋅Y⋅A2 = [CiT • RiT]⋅Y⋅[Ci • Ri ]

Rearranging:

Z = [CiT]⋅[Y • RiT • Ri ]⋅[ Ci]
= [CiT]⋅[Y • Si ]⋅[ Ci]


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Where

" %
$ 1 4 1 10 1 4
10 '1
$ '
T 1 10 2 5 1 10 2 5 '
Si = R i •Ri = $
$ 1 4 1 10 1 4 1 10 '
$ '
$1 10 2 5 1 10 2 5 '
# &

The core inverse transform Ci and the rescaling matrix Vi are defined in the H.264
standard. Hence we now develop Vi and will then derive Mf .
!

7 Developing Vi

From Equation 2, Vi = Si ⋅ Qstep ⋅ 26

H.264 supports a range of quantization step sizes Qstep . The precise step sizes are not
defined in the standard, rather the scaling matrix Vi is specified. Qstep values
corresponding to the entries in Vi are shown in the following Table.

QP Qstep
0 0.625
1 0.702..
2 0.787..
3 0.884..
4 0.992..
5 1.114..
6 1.250
… …
12 2.5
… …
18 5.0
… …
48 160
… …
51 224

6
The ratio between successive Qstep values is chosen to be 2 = 1.2246... so that Qstep
doubles in size when QP increases by 6. Any value of Qstep can be derived from the first
6 values in the table (QP0 – QP5) as follows:

Qstep(QP) = Qstep(QP%6) ⋅ 2floor(QP/6) !

The values in the matrix Vi depend on Qstep (hence QP) and on the scaling factor
matrix Si . These are shown for QP 0 to 5 in the following Table.


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QP Qstep ⋅ 26 Vi = round ( Si ⋅ Qstep ⋅ 26 )
0 40 "10 13 10 13%
$ '
$13 16 13 16'
$ '
$10 13 10 13'
$ '
$13
# 16 13 16'
&

1 44.898 "11 14 11 14%
$ '
! $14 18 14 18'
$ '
$11 14 11 14'
$ '
#14 18 14 18&

2 50.397 "13 16 13 16%
$ '
! $16 20 16 20'
$ '
$13 16 13 16'
$ '
#16 20 16 20&

3 56.569 "14 18 14 18%
$ '
! $18 23 18 23'
$ '
$14 18 14 18'
$ '
#18 23 18 23&

4 63.496 "16 20 16 20%
$ '
! $20 25 20 25'
$ '
$16 20 16 20'
$ '
#20 25 20 25&

5 71.272 "18 23 18 23%
$ '
! $23 29 23 29'
$ '
$18 23 18 23'
$ '
#23 29 23 29&

For higher values of QP, the corresponding values in Vi are doubled (i.e. Vi (QP=6) =
!
2Vi(QP=0) , etc).

Note that there are only three unique values in each matrix Vi . These three values are
defined as a table of values v in the H.264 standard, for QP=0 to QP=5 :


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Table 1 Matrix v defined in H.264 standard
QP v (r, 0): v (r, 1): v (r, 2):
Vi positions (0,0), Vi positions (1,1), Remaining
(0,2), (2,0), (2,2) (1,3), (3,1), (3,3) Vi positions
0 10 16 13
1 11 18 14
2 13 20 16
3 14 23 18
4 16 25 20
5 18 29 23

Hence for QP values from 0 to 5, Vi is obtained as:

"v (QP , 0) v (QP , 2) v (QP , 0) v (QP , 2)%
$ '
$v (QP , 2) v (QP ,1) v (QP , 2) v (QP ,1)'
Vi = $ '
$v (QP , 0) v (QP , 2) v (QP , 0) v (QP , 2)'
$ '
$v (QP , 2)
# v (QP ,1) v (QP , 2) v (QP ,1)' &

Denote this as:
!
Vi = v(QP, n)
Where v (r,n) is row r, column n of v.

For larger values of QP (QP>5), index the row of array v by QP%6 and then multiply
by 2floor(QP/6) . In general:

Vi = v (QP%6,n)⋅ 2floor(QP/6)

The complete inverse transform and scaling process (for 4x4 blocks in macroblocks
excluding 16x16-Intra mode) becomes:

1
Z = round ( [CiT]⋅[Y • v (QP%6,n)⋅ 2floor(QP/6)]⋅[ Ci] ⋅ )
26

(Note: rounded division by 26 can be carried out by adding an offset and right-shifting
by 6 bit positions).
!

8 Deriving Mf

Combining Equation 1 and Equation 2:


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S i # Sf # 221
Mf "
Vi

Si , Sf are known and Vi is defined as described in the previous section. Define Mf as:

!
# S " S " 221 &
Mf = round% i f
% (
(
$ Vi '

# 131072 104857.6 131072 104857.6&
! % (
%104857.6 83886.1 104857.6 83886.1 (
S i " Sf " 221 =% (
% 131072 104857.6 131072 104857.6(
% (
$104857.6 83886.1 104857.6 83886.1 '

The entries in matrix Mf may be calculated as follows (Table 2):
!
Table 2 Tables v and m
QP v (r, 0): v (r, 1): v (r, 2): m (r, 0): m (r, 1): m (r,2):
Vi positions Vi positions Remaining Mf positions Mf positions Remaining
(0,0), (0,2), (1,1), (1,3), Vi positions (0,0), (0,2), (1,1), (1,3), Mf positions
(2,0), (2,2) (3,1), (3,3) (2,0), (2,2) (3,1), (3,3)
0 10 16 13 13107 5243 8066
1 11 18 14 11916 4660 7490
2 13 20 16 10082 4194 6554
3 14 23 18 9362 3647 5825
4 16 25 20 8192 3355 5243
5 18 29 23 7282 2893 4559

Hence for QP values from 0 to 5, Mf can be obtained from m , the last three columns
of Table 2:

"m(QP , 0) m(QP , 2) m (QP , 0) m (QP , 2)%
$ '
$m(QP , 2) m(QP ,1) m (QP , 2) m (QP ,1)'
Mf = $ '
$m(QP , 0) m(QP , 2) m (QP , 0) m (QP , 2)'
$ '
$m(QP , 2)
# m(QP ,1) m (QP , 2) m (QP ,1)' &

Denote this as:
!
Mf = m(QP, n)
Where m (r,n) is row r, column n of m.

For larger values of QP (QP>5), index the row of array m by QP%6 and then divide by
2floor(QP/6) . In general:

Mf = m (QP%6,n)/ 2floor(QP/6)


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Where m (r,n) is row r, column n of m.

The complete forward transform, scaling and quantization process (for 4x4 blocks and
for modes excluding 16x16-Intra) becomes:

1
Y = round ( [Cf]⋅[Y]⋅[ CfT] • m (QP%6,n)/ 2floor(QP/6) ] ⋅ )
215

(Note: rounded division by 215 may be carried out by adding an offset and right-
shifting by 15 bit positions).
!

9 Further reading

ITU-T Recommendation H.264, Advanced Video Coding for Generic Audio-Visual
Services, November 2007.

H. Malvar, A. Hallapuro, M. Karczewicz, and L. Kerofsky, Low-complexity transform and
quantization in H.264/AVC, IEEE Transactions on Circuits and Systems for Video
Technology, vol. 13, pp. 598–603, July 2003.

T. Wiegand, G.J. Sullivan, G. Bjontegaard, A. Luthra, Overview of the H.264/AVC video
coding standard, IEEE Transactions on Circuits and Systems for Video Technology, vol.
13, No. 7. (2003), pp. 560-576.

I. Richardson, The H.264 Advanced Video Compression Standard, to be published in
late 2009.

See http://www.vcodex.com/links.html for links to further resources on H.264 and
video compression.

Acknowledgement

I would like to thank Gary Sullivan for suggesting a treatment of the H.264 transform
and quantization processes along these lines and for his helpful comments on earlier
drafts of this document.

About the author

As a researcher, consultant and author working in the field of video compression
(video coding), my books on video codec design and the MPEG-4 and H.264 standards
are widely read by engineers, academics and managers. I advise companies on video
coding standards, design and intellectual property and lead the Centre for Video
Communications Research at The Robert Gordon University in Aberdeen, UK and the
Fully Configurable Video Coding research initiative.


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Using the material in this document

This document is copyright – you may not reproduce the material without permission.
Please contact me to ask for permission. Please cite the document as follows:

Iain Richardson, 4x4 Transform and Quantization in H.264/AVC, VCodex Ltd White
Paper, April 2009, http://www.vcodex.com/


The H.264 Integer Transform

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