Adaptive Gradient Information and BFGS Based Inter Frame Rate Control for High Efficiency Video Coding

Abstract

In order to meet the emerging demands of high-fidelity video services, a new video coding standard — High Efficiency Video Coding (HEVC) is developed to improve the compression performance of high definition (HD) videos and save half of the bitrate for the same perceptual video quality compared with H.264/Advanced Video Coding (AVC). Rate control still plays a significant role in HD video data transmission via the communication channel. However, R-lambda model based HEVC rate control algorithm does not take the relationship between the encoding complexity and Human Visual System (HVS) into account, what’s more, the convergence speed of Least Mean Square (LMS) algorithm is slow. In this paper, an adaptive gradient information and Broyden Fletcher Goldfarb Shanno (BFGS) based R-lambda model (GBRL) is proposed for the inter frame rate control, where the gradient based on Sobel operator can effectively measure the frame-content complexity and BFGS algorithm converges speedily than LMS algorithm. Experimental results show that the proposed GBRL method can achieve bitrate error reduction and peak signal to noise ratio (PSNR) improvement especially for the sequences with large motion, compared to the state-of-the-art rate control methods. In addition, if the optimal initial quantization parameter (QP) prediction model based on linear regression can be incorporated into the proposed GBRL method, the performance of rate control can be further improved.

Appendices

Appendix 1

1.1 Proof of α and β updating in (20) and (21)

The α _old , β _old , bpp _real and λ _real are known variables. According to (6), we have

$$ {\lambda}_{comp}={\alpha}_{old}\cdot {bpp}_{real}^{\beta_{old}} $$

(28)

After the lnoperation

$$ \ln {\lambda}_{comp}= \ln \alpha +\beta \cdot \ln bpp={\alpha}^{\hbox{'}}+\beta \cdot \ln bpp $$

(29)

The squared error between the real λ and caculated λ is

$$ {e}^2={\left( \ln {\lambda}_{real}- \ln {\lambda}_{comp}\right)}^2 $$

(30)

Take the derivative of the eq. (30) with respect to α ^' and β

$$ \frac{\partial {e}^2}{\partial {\alpha}^{\hbox{'}}}=-2\cdot \left( \ln {\lambda}_{real}- \ln {\lambda}_{comp}\right) $$

(31)

$$ \frac{\partial {e}^2}{\partial \beta }=-2\cdot \left( \ln {\lambda}_{real}- \ln {\lambda}_{comp}\right)\cdot \ln bpp $$

(32)

Update α ^' according to BFGS algorithm

$$ {\alpha}_{new}^{\hbox{'}}={\alpha}_{old}^{\hbox{'}}+{\delta}_{armijo}\cdot {d}_{\alpha } $$

(33)

According to (29), we have

$$ \ln {\alpha}_{new}= \ln {\alpha}_{old}+{\delta}_{armijo}\cdot {d}_{\alpha } $$

(34)

Thus,

$$ {\alpha}_{new}={e}^{\ln {\alpha}_{old}}\cdot {e}^{\delta_{armijo}\cdot {d}_{\alpha }} $$

(35)

After Taylor’s expansion and ignore high-order items

$$ {\alpha}_{new}\approx {\alpha}_{old}\cdot \left(1+{\delta}_{armijo}\cdot {d}_{\alpha}\right)={\alpha}_{old}+{\delta}_{armijo}\cdot {d}_{\alpha}\cdot {\alpha}_{old} $$

(36)

It is similar to the updating of β according to BFGS algorithm

$$ {\beta}_{new}={\beta}_{old}+{\delta}_{armijo}\cdot {d}_{\beta } $$

(37)

(36) and (37) form the model of parameter update together.

Appendix 2

1.1 The specific caculation process of δ _armijo

According to (29) and (30), we have

$$ {e}^2={\left( \ln {\lambda}_{real}-{\alpha}^{\hbox{'}}-\beta \cdot \ln bpp\right)}^2 $$

(38)

The initial search step size δ _{InitialArmijo} is set to 0.55, related parameter σ is set to 0.4, the loop count m is initialized to 0, and m _k is set to 0.

Accordin to (31) and (32), we form the error gradient vector ∇f(x _k )

$$ \nabla f\left({x}_k\right)=\left(\begin{array}{c}\hfill \frac{\partial {e}^2}{\partial {\alpha}^{\hbox{'}}}\hfill \\ {}\hfill \frac{\partial {e}^2}{\partial \beta}\hfill \end{array}\right) $$

(39)

According to the equation $d=-{B}_k^{-1}\cdot \nabla f\left({x}_k\right)$, we obtain the search direction vector d

$$ d=\left(\begin{array}{c}\hfill {d}_{\alpha}\hfill \\ {}\hfill {d}_{\beta}\hfill \end{array}\right) $$

(40)

According to (33) and (37), we have

$$ {\alpha}_{new}^{\hbox{'}}={\alpha}_{old}^{\hbox{'}}+{\delta}_{InitialArmijo}\cdot {d}_{\alpha } $$

(41)

$$ {\beta}_{new}={\beta}_{old}+{\delta}_{InitialArmijo}\cdot {d}_{\beta } $$

(42)

According to (B.1), we get the old and new mean square error formulas

$$ {e}_{old}^2={\left( \ln {\lambda}_{real}-{\alpha}_{old}^{\hbox{'}}-{\beta}_{old}\cdot \ln bpp\right)}^2 $$

(43)

$$ {e}_{new}^2={\left( \ln {\lambda}_{real}-{\alpha}_{new}^{\hbox{'}}-{\beta}_{new}\cdot \ln bpp\right)}^2 $$

(44)

The specific caculation steps of δ _armijo are as follows:

• Step 1 Determine whether m is less than 20, if it is, then go to step 2, otherwise, jump to step 4;

• Step 2 Determine whether ${e}_{new}^2$ is less than ${e}_{old}^2+\sigma \cdot {\delta}_{InitialArmijo}^m\cdot \nabla f{\left({x}_k\right)}^T\cdot d$, if it is, then m _k  = m, jump to step 4, otherwise, make m = m + 1;

• Step 3 Jump to step 1;

• Step 4 ${\delta}_{armijo}={\delta}_{InitialArmijo}^{m_k}$.

δ _armijo in step 4 is the so-called search step size.