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4x4 Transform and Quantization in H.264/AVC

Revised April 2009; Copyright (c) Iain Richardson, 2007-2009; All rights reserved. Do not reproduce any material without permission.

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 ).

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, Q_step , then rounding the result) (Figure 4a).

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

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

Combine S_f and the quantization process into M_f (Figure 4d), where:

                                                                                                            Equation 1

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 (C_i) and a scaling matrix (S_i) (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 V_i (Figure 5d), where:

                                                                                                            Equation 2

Figure 5 Development of the rescaling and inverse transform process

5       Developing C_f and S_f (4x4 blocks)

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

Y = A×X×A^T                                              

                                                                                                            Equation 3

Where × indicates matrix multiplication and:

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 C_f :

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

The rows of C_f have different norms. To restore the orthonormal property of the original matrix A, multiply all the values c_ij in row r by  :

A₁ = C_f · R­­_f              where

· denotes element-by-element multiplication (Hadamard-Schur product[2]). Note that the new matrix A₁ is orthonormal.

The two-dimensional transform (Equation 3) becomes:

Y  =   A₁×X×A₁^T      =  [C_f · R­­_f]×X×[C_f^T · R­­_f^T ]

Rearranging:

Y       =  [C_f×X×C_f^T] · [R_f · R_f^T] 

=  [C_f×X×C_f^T] · S_f

Where

6       Developing C_i and S_i (4x4 blocks)

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

Z = A^T×Y×A                                               

                                                                                                            Equation 4

Where

        as before.

Choose a particular approximation by scaling each row of A and rounding to the nearest 0.5, giving C_i :

The rows of C_i are orthogonal but have non-unit norms. To restore orthonormality, multiply all the values c_ij in row r by  :

A₂ = C_i · R­­_i               where

The two-dimensional inverse transform (Equation 4) becomes:

Z  =   A₂^T×Y×A₂      =  [C_i^T · R­­_i^T]×Y×[C_i · R­­_i ]

Rearranging:

Z       =  [C_i^T]×[Y · R_i^T · R_i ]×[ C_i]       

=  [C_i^T]×[Y · S_i ]×[ C_i]

Where

The core inverse transform C_i  and the rescaling matrix V_i  are defined in the H.264 standard. Hence we now develop V_i and will then derive M_f .

7       Developing V_i

From Equation 2, V_i = S_i × Q_step × 2⁶

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

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

Q_step(QP) = Q_step(QP%6) × 2^floor(QP/6)

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

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

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

Table 1 Matrix v defined in H.264 standard

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

Denote this as:

V_i = 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 2^floor(QP/6) . In general:

V_i = v (QP%6,n)× 2^floor(QP/6)

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

Z = round ( [C_i^T]×[Y · v (QP%6,n)× 2^floor(QP/6)]×[ C_i] ×  )

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

8       Deriving M_f

Combining Equation 1 and Equation 2:

S_i  , S_f  are known and V_i is defined as described in the previous section. Define M_f as:

The entries in matrix M_f may be calculated as follows (Table 2):

Table 2 Tables v and m

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

Denote this as:

M_f = 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 2^floor(QP/6) . In general:

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

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:

Y = round ( [C_f]×[Y]×[ C_f^T]   · m (QP%6,n)/ 2^floor(QP/6) ] ×  )

(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.

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/