The Vault

An Improved Parametric Bit Rate Model for Frame-level Rate Control in Video Coding
Research Paper / Jan 2011

1
An Improved Parametric Bit Rate Model for
Frame-level Rate Control in Video Coding

Zhifeng Chen∗, Serhad Doken∗, and Dapeng Wu†
∗InterDigital, Inc., 781 Third Avenue, King of Prussia, Pennsylvania, 19406 USA

†University of Florida, Gainesville, Florida 32611 USA

Abstract

In a wireless video communication system, the video encoder needs to dynamically control the
coding parameters based on the instantaneous video statistics and channel condition to achieve the best
video quality. An accurate bit rate model can help the encoder to achieve accurate bit rate control and
good rate-distortion (R-D) performance. In this paper, we improve the bit rate model by modeling the
component of run-level mapping plus entropy coding as the process of choosing different codebooks for
different quantized transform coefficients. We also compensate the mismatch between the true histogram
and the assumed Laplacian distribution in a parametric model by utilizing the estimation deviation of
previous frames. The experimental results show that our method achieves more accurate estimation of bit
rate than existing models. We then apply our bit rate model to frame-level rate control in the H.264/AVC
JM reference software. The experimental results show that our rate control algorithm achieves better
R-D performance than the existing rate control algorithm in JM.

I. INTRODUCTION

Most practical video compression algorithms reduce spatial and temporal redundancy via
transform coding and motion estimation, respectively. However, the degree of redundancy, and
therefore the resulting rate for a given distortion, can fluctuate widely from scene to scene. For
example, scenes with high motion content will require more bits than more stationary ones [1]. In
real-time video communications, the end-to-end delay for transmitting video data needs to be very
small, particularly in two-way interactive applications such as video calls or videoconferencing.
If the encoded video is transmitted through a fixed-rate channel, the coded bits are placed
into a small buffer and a finite number of bits can be sent from the buffer during each frame
interval [2]. Given the end-to-end delay constraint and the buffer fullness, the video encoder
needs to dynamically control the coding parameters (or an operating point in an R-D sense)
based on the instantaneous video statistics and channel bandwidth. The instantaneous coding
parameters control problem becomes even more important and challenging under a variable
bit-rate channel, e.g. fading channels in 3G/LTE systems.

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In general, the coding parameters involve three aspects, that is, 1) spatial domain parameter,
e.g., spatial sampling rate, 2) temporal domain parameter, e.g., frame rate, and 3) coefficient
domain parameter, e.g., quantization step size. In most typical video encoders, e.g., H.263/264
and MPEG-2/4 encoders, usually only the coefficient domain parameter is adjusted in the encoder
to meet the delay and bit rate constraints.1 Since the video statistics vary between frames and
within each frame, a desirable method is to choose different quantization step sizes for encoding
different frames or different regions within one frame (or basic units), which highlights the
importance of the bit allocation problem. Under certain end-to-end delay constraints, the bit
allocation and quantization step size control problems, which together are called the encoder
rate control problem, can be deployed at different levels, e.g., GOP level, frame level, and basic
unit level. These different-level rate control methods are indeed adopted in the H.264/AVC JM
reference software [3], [4], [5].

For the bit allocation problem, a rate-distortion model is required to minimize the overall dis-
tortion under an overall bit constraint. For the quantization step size control problem, the encoder
needs to do a one-to-one mapping between the quantization step size and allocated bits given
the video statistics. In a practical encoder design, solving both of these two problems demands
a prior knowledge of bit rate as a function of video statistics and coding parameters. Plenty of
bit rate models have been developed in existing literature. Most of the existing works derive bit
rate as a function of video statistics and quantization step size [2], [6], [7], [8], while others
model bit rate as a function of video statistics and other parameters [9]. In general, these models
come from either experimental observation [9], [7], [10], [11] or parametric modeling [12], [8],
[13]. However, both of them have some limitations. The experimental modeling usually induces
some model parameters which can only be estimated from previous frames. Therefore, the model
accuracy depends not only on the statistics and coding parameters but also on the estimation
accuracy of those model parameters. However, in theory, the instantaneous frame bit rate should
be independent of previous frames given instantaneous frame statistics and coding parameters.
In addition, the estimation error of those model parameters may have a significant impact on the
model accuracy, which can be observed in the H.264/AVC JM reference software [3] and will be
explained in detail in the experimental section of this paper. On the other hand, the parametric
modeling has the following two limitations: 1) the assumed residual probability distribution,
e.g., Laplacian distribution, may deviate significantly from the true histogram; 2) the implicit
assumption of all transform coefficients being identically distributed is not valid if run-length
coding is conducted before the entropy coding as in most practical encoders. Since the model-

1In a hybrid video encoder with block-based coding, the reference points and prediction modes can also be adjusted accordingly
for a given quantization step size using rate-distortion optimization.

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selection problem may often be more important than having an optimized algorithm [1], simply
applying these parametric models to a real encoder may result in poor R-D performance. To
compensate the inaccuracy of those parametric models, Refs. [14], [8] introduce some model
parameters, which are determined by heuristic values.

In this paper, we improve the bit rate model by modeling the component of run-level mapping
plus entropy coding as the process of choosing different codebooks for different quantized
transform coefficients. We also compensate the mismatch between the true histogram and the
assumed Laplacian distribution in the parametric model by utilizing the estimation deviation
of previous frames. The experimental results show that our method achieves a more accurate
estimate of bit rate compared to existing models. We then apply our bit rate model to frame-level
rate control in the H.264/AVC JM reference software [3]. The experimental results show that our
rate control algorithm achieves better R-D performance than the existing rate control algorithm
in JM.

The rest of this paper is organized as follows. Section II derives our bit rate model as a function
of video statistics and quantization step size. Section III shows the experimental results, which
demonstrates both the higher accuracy of our model and the better R-D performance achieved
with it over existing models. Section IV concludes the paper.

II. BIT RATE MODELING FOR A HYBRID VIDEO CODER WITH BLOCK-BASED CODING
SCHEME

In this section, we first derive residual bit rate as a function of video statistics and quanti-
zation step size, and then design the estimation algorithm for model parameters with practical
consideration.

A. Derivation of residual rate function
1) The Entropy of Quantized Transform Coefficients for i.i.d. Zero-mean Laplacian Source

under Uniform Quantizer: For transform coefficients with independent and identically distributed
(i.i.d.) zero-mean Laplacian distribution, the probability density function (pdf) is f(x) = λ

2
·eλ·|x|,

where λ =

2

σ
and σ is the standard deviation. For the uniform quantizer with quantization step

size Q and quantization offset θ2, the probability of zero after quantization is

P0 = 2

∫ Q·(1−θ2)
0

p(x)dx = 1− e−θ1·(1−θ2), (1)

and the probability of level n after quantizion is

Pn =

∫ Q·(n+1−θ2)
Q·(n−θ2)

p(x)dx =
1

2
(1− e−θ1) · eθ1·θ2 · e−λ·Q·n, (2)

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where

θ1 =


2 ·Q
σ

. (3)

As a result, the entropy is

H = −P0 · log2 P0 − 2
∞∑

n=1

Pn · log2 Pn

= −P0 · log2 P0 + (1− P0) · (
θ1 · log2 e
1− e−θ1 − log2(1− e

−θ1)− θ1 · θ2 · log2 e + 1).
(4)

2) Improve with run length model: In a video encoder, the quantized transform coefficients are
actually not i.i.d. Although we may assume the DCT transform or integer transform [15] highly
de-correlates the correlation among neighboring pixels, different transform coefficients have very
different variances in statistics. For example, in a 4x4 integer transform, the 16 coefficients show
a decreasing variance in the well-known zigzag scan order as used in H.264. As a result, the
coefficients with higher frequency have higher probability of being zeroes after quantization. On
the other hand, the coefficients with lower frequency show more randomness in different levels
even after quantization. Such characteristics are exploited by the run-level mapping after zigzag
scan to further increase the compressibility for entropy coding. We may regard the component of
run-level mapping plus entropy coding as choosing different codebooks for different quantized
transform coefficients. From information theory, we know the concavity of the entropy as a
function of the distribution (Theorem 2.7.3 in Ref [16]). Therefore, not considering the mixture
of 16 coefficients with different variances will overestimate the entropy of quantized transform
coefficients [12], [8], [13].

To derive the joint entropy for 16 coefficients with different variances, we need to model
the variance relationship among those 16 coefficients. Doing extensive experiments, we find an
interesting phenomenon2, that is, the variance is approximately a function of position in the
two-dimensional transform domain as follows

σ2(x,y) = 2
−(x+y) · σ20, (5)

where x and y is the position in the two-dimensional transform domain, and σ20 is the variance
of the coefficient at position (0, 0).

By using this property, we can derive the variance σ2(x,y) for all positions given the average
variance σ2. For a 4x4 integer transform with average variance σ2, the variance for each transform

2This phenomenon is found from samples in one frame or one GOP for CIF sequences, i.e., the number of sample is larger
than 101376.

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coefficient can be calculate by (5) as

σ2 =
1

16

4∑
x=0

4∑
y=0

σ2(x,y) =
225

1024
· σ20 . (6)

Therefore, we have

σ2(x,y) = 2
−(x+y) · 1024

225
· σ2. (7)

Then, the estimated joint entropy of 16 non-identical transform coefficients by compensating
the run length coding model is

Hrlc =
1

16

3∑
x=0

3∑
y=0

H(x,y), (8)

where H(x,y) is the entropy for coefficient position (x, y), and can be calculated by (7), (1), (3)
and (4) with their own σ2(x,y) and θ1(x,y).

3) Improvement by considering the model mismatch: The assumed residual probability distri-
bution, e.g., Laplacian distribution, may deviate significantly from the true histogram especially
when the number of samples are not sufficient. Therefore, we need to compensate the mismatch
between the true residual histogram and assumed Laplacian distribution to obtain a better
estimate. Denote Hl as the entropy for the case with a Laplacian distribution, Ht as the entropy
for the case with true histogram and ν = Hl

Ht
. In a video sequence, the changes of residual

statistics and quantization step size between adjacent frames have almost the same effect on Hl
and Ht. Therefore, we may use the previous frame statistics to compensate the estimated result
from (8). Assume the ratio between Hkl and Hkt approximate νk−1, we have H

k

l

Hk
t

=
Hk−1

l

Hk−1
t

. As a
result, (8) can be further compensated as

Hˆk =
Hk−1t ·Hk

Hk−1l
. (9)

(8) and (9) significantly improve the estimation accuracy of residual entropy as shown in
Fig. 1.

B. Parameter estimation with practical quantization step size control consideration

1) Encoding bit rate estimation for the H.264 encoder: For a hybrid video coder with block-
based coding scheme, e.g., H.264 encoder, the encoded bit rate R consists of residual bits Rresi,
motion information bits Rmv , prediction mode bits Rmode, and syntax bits Rsyntax. That is,

Rk = Hˆk ·Nresolution ·Nfps +Rkmv +Rkmode +Rksyntax, (10)
where Nresolution is the normalized video resolution considering color components, and Nfps
means frame per second. Compared to Rkresi, Rkmv , Rkmode, and Rksyntax are less affected by Q.
Therefore, Rkmv, Rkmode, Rksyntax can be estimated from the statistics in the previous frames.

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2) Estimation of σ2 in the k-th frame: To control Qk by (3) before the real encoding
processing, the residual variance σ2 in the k-th frame should be estimated first. A simply method
is to estimate it by σ2 in the k − 1-th frame. This method is valid for most P-frames with inter
prediction. However, for some scene change frames, such an estimate may cause a very inaccurate
estimation result. Therefore, we need to treat the scene change frames differently. Note that in
a practical video encoder with rate-distortion optimization (RDO), most macroblocks in the
scene change frames would be encoded with intra prediction mode. That is, there is no motion
information bits from (10). On the other hand, σ2 cannot simply be estimated from the k − 1-
th frame. Note that the residual variance after the intra mode prediction should be less than
the frame difference from the previous frame. We can approximately estimate σ2 for the scene
change frame by

(σ2)k =
(fk − fˆk−1)2

Ck
, (11)

where fk means the original pixel value in the k-th frame, fˆk−1 means the reconstructed pixel
value in the k − 1-th frame, and Ck is a normalizing factor, which is estimated from the
normalizing factor in previous scene change frames.

Another special frame is the P-frame right after the scene change frame. In these P-frames,
apparently σ2 may be much smaller than that of the previous frame. In such a case, we may
estimate σ2 by

(σ2)k = min((σ2)k−1,
(fk − fˆk−1)2

Ck
). (12)

3) Practical consideration of Laplacian assumption: Statistically speaking, (8) is only valid
for sufficiently large samples. When there are not enough samples or the sample variance is very
small, e.g., Q > 3σ, the Laplacian assumption for individual coefficients is not accurate. In such
cases, we may use (4) as the estimate instead of (8). That is,

Hk =




estimated by (8), Q ≤ 3σ
estimated by (4), otherwise.

(13)

4) Practical consideration of encoder setting: In the R-D sense, if there are only a few
coefficients with small values in one macroblock (MB) to be coded, they may be discard as in
the H.264/AVC JM reference software [3]. For example, ‘sum cnt nonz’ is the accumulation
of coeff cost over a whole macro block. If sum cnt nonz ≤ LUMA COEFF COST = 5 for
the whole MB, all nonzero coefficients are discarded for the MB and the reconstructed block
is set equal to the prediction. Therefore, the skip mode may increase and the resulting bit rate
becomes a little less than that calculated from (4) for a Laplacian source. Note that this is a
pure encoder issue and hence does not have any implication on the standard.

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Note that in Refs. [14], [8], authors try to improve their model accuracy by taking the skip
mode into consideration. Authors claim that both P0 and Pn, in their analytical formula, should
be normalized by the probability of skip blocks, which indeed decreases the estimated bit rate
for a given σ2 and Q for a Laplacian source. However, this is not valid in theory since (4)
is the lower bound for the i.i.d. case. In fact, the probability of skip blocks depends on the
preset value of LUMA COEFF COST 3. As a result, we should analyze the bit rate for those
skip blocks and non-skip blocks separately with their probability rather than simply normalizing
P0 and Pn by the probability of non-skip blocks. In addition, PsP0 is a function of Q both in
theory and by simulation. However, Refs. [14], [8] use a constant Ps

P0
. Since the preset values

of LUMA COEFF COST and other similar parameters are a pure encoder issue, we set them
equal to zero in order to compare the bit rate model accuracy between our model and existing
models.

III. EXPERIMENTAL RESULTS

In this section, we first verify the accuracy of our proposed bit rate model. Then, we compare
the R-D performance of the frame-level quantization step size control algorithm with our model
to existing frame-level rate control algorithms 4.

A. Experimental Setup

The JM16.0 encoder and decoder [3] are used in the experiments. All the tested video
sequences are in CIF resolution at 30fps. Each video sequence is encoded for its first 100
frames where the first frame is an I-frame and the following frames are P-frames. The encoder
setting is given as below: the number of reference frames is 3; B slices are not included; only
4x4 transform is used; CABAC is enabled for entropy coding; for all rate control algorithms,
the first frame uses a fixed quantization parameter (QP), i.e., QP=28.

B. Model Accuracy

Fig. 1 shows the true residual bit rate and estimated residual bit rate for ‘foreman’ and ‘mobile’
for the first 20 frames in order to have a distinguishable comparison. In Fig. 1, ‘True bpp’ means
the true bit per pixel (bpp) produced by the JM16.0 encoder; ‘without rlc’ means bpp estimated
by (4); ‘with rlc’ means bpp estimated by (8); ‘without compensation’ means bpp estimated
by (13); ‘with compensation’ means bpp estimated by (13) and (9); ‘Rho-domain’ means bpp
estimated by Refs. [17], [18]; ‘Xiang’s model’ means bpp estimated by Refs. [14], [8].

3There are some other similar parameters setting, e.g., CHROMA COEFF COST , LUMA MB COEFF COST and
LUMA 8x8 COEFF COST in the reference software.

4For single pass rate control algorithms, they are indeed the quantization step size control algorithm.

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8
2 4 6 8 10 12 14 16 18 20
0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Frame Index

bi
ts

p
er

p
ix

el
(b

pp
)

foreman−cif−800000bps





True bpp
Estimated bpp without compensation
Estimated bpp with compensation
Estimated bpp without rlc
Estimated bpp with rlc
Estimated bpp by Rho−domain
Estimated bpp by Xiang’s model

2 4 6 8 10 12 14 16 18 20
0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Frame Index

bi
ts

p
er

p
ix

el
(b

pp
)

mobile−cif−800000bps





True bpp
Estimated bpp without compensation
Estimated bpp with compensation
Estimated bpp without rlc
Estimated bpp with rlc
Estimated bpp by Rho−domain
Estimated bpp by Xiang’s model

(a) (b)
Fig. 1. bpp vs. Frame index: (a) foreman, (b) mobile.

We can see that the estimation accuracy is improved by (8) when true bpp is relatively large.
However, when true bpp is small, ‘without rlc’ gives higher estimation accuracy. By utilizing
the statistics of the previous frame from (9), the estimation accuracy is further improved. We
also find that ‘Rho-domain’ is accurate at low bpp; however, it is not accurate at high bpp. For
‘Xiang’s model’, the estimated bpp is smaller than the true bpp in most cases. Note that we also
want to compare the bit rate model used in JM16.0. However, due to the estimation error of its
model parameters, the first few frames may abnormally underestimate the quantization step size
Q. Therefore, the rate control algorithm in JM16.0 use three parameters, i.e., RCMinQPPSlice,
RCMaxQPPSlice and RCMaxQPChange, to reduce the effect of the estimation error. Their
default values are 8, 42, 4, respectively. However, we believe a good rate control algorithm
should depend mainly on the model accuracy rather than those manually chosen thresholds.
When those parameters are set as 0, 51, 51, the estimated QP could even be 0 in the first few
frames. That is, the first few frames consume most of the allocated bits, and there are only few
bits available for the remaining frames in JM. Therefore, we do not test its model accuracy in
this subsection. Instead, we will plot the R-D performance for it in Section III-C.

C. Performance Comparison
Fig. 2 shows Y-component PSNR vs. bit rate for ‘foreman’ and ‘mobile’5. We test three

different settings of (RCMinQPPSlice, RCMaxQPPSlice, RCMaxQPChange), i.e., (8, 42, 4), (0,
51, 16) and (0, 51, 51) for JM and our proposed rate control algorithms. We see that the R-
D performance of our model is better than, or at least similar to, that of JM for all the cases

5In our rate control algorithm, only the frame-level rate control is used. Therefore, we average the overall bit rate over all
frames except the first frame since it uses a constant QP.

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compared. To be more specific, when RCMinQPPSlice, RCMaxQPPSlice and RCMaxQPChange
are set to the default values, i.e., 8, 42, 4, JM’s rate control algorithm performs almost the same
as our model. However, when those parameters are set to 0, 51, and 51, our model performs
much better than JM’s rate control algorithm. From Fig. 2, we see that the PSNR at 600kbps is
even lower than the PSNR at 400kbps for ‘JM’. This is because the first few frames consume
most of bits at 600kbps due to the estimation error of the model parameters; therefore, the overall
PSNR becomes worse.

3 4 5 6 7 8 9 10
x 105

28

30

32

34

36

38

40

42

Bit rate (bit/sec)

PS
NR

(d
B)

RD−foreman−cif−Y





JM−(8,42,4)
proposed−(8,42,4)
JM−(0,51,16)
proposed−(0,51,16)
JM−(0,51,51)
proposed−(0,51,51)

3 4 5 6 7 8 9 10 11
x 105

25

26

27

28

29

30

31

32

Bit rate (bit/sec)

PS
NR

(d
B)

RD−mobile−cif−Y





JM−(8,42,4)
proposed−(8,42,4)
JM−(0,51,16)
proposed−(0,51,16)
JM−(0,51,51)
proposed−(0,51,51)

(a) (b)
Fig. 2. PSNR vs. Bit rate: (a) foreman, (b) mobile.

From Fig. 2, we also find that the R-D performance of the proposed model without the QP
constraint is very similar to that of the proposed model with the QP constraint. In fact, the bit
rate achieved by the proposed model without the QP constraint is more accurate than the bit
rate achieved by the proposed model with the QP constraint. Actually, there is no reason to
control the QP change to be within a certain limit. Instead, it is much more important to control
the distortion level change to be within a certain limit. In other words, QP can be changed
dramatically to accommodate the residual statistics change from frame to frame; and this can
be achieved with an accurate bit rate model and distortion model.

IV. CONCLUSION

In this paper, we improved the bit rate model by modeling the component of run-level
mapping plus entropy coding as choosing different codebooks for different quantized transform
coefficients. We also compensated the mismatch between the true histogram and the assumed
Laplacian distribution in a parametric model by utilizing the estimation deviation of previous
frames. We considered several practical factors in model parameter estimation for the design of
a quantization step size control algorithm in practical video encoders. The experimental results
showed that 1) our method achieves more accurate estimation of bit rate than existing models;
and 2) the rate control algorithm with our model achieves better R-D performance than the

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existing rate control algorithm in H.264/AVC JM reference software. In our future work, we
will use the same compensation technique for a parametric distortion model and apply both the
bit rate model and distortion model to solving the R-D optimized bit allocation problem.

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