Pseudo HCB for SF Optimization

1) Motivation for Pseudo HCB:

We first look at the MNMR minimization case. The key problem in splitting (3.4) into (3.8) and (3.6) is to choose the correct (optimal) value of bk,i in (3.8). In (3.8), the widk,i or D(sk,i − s^l,i-1) term is unique for a given state or state transition in the SF trellis. However, the value of bk,i depends not only on sk,i associated with the state in the SF trellis; it also depends on the choice of HCB. In the JTB scheme, SF and HCB are chosen simultaneously. Therefore, for each candidate value of SF, all possible bk,i values, corresponding to 12 candidate HCBs, are evaluated. In other words, the chosen value of bk,i for each state Υk,i in the trellis for JTB optimization scheme is optimal [7][8]. But in our sequential optimization scheme, the value of bk,i for the state Υk,i in (3.8) is estimated based upon the available information. The estimated value of b_k,i may not be the optimal value and this may further induce an incorrect (non-optimal) selection in SF optimization. For example, two candidate paths in the SF trellis, A and B, are shown in (3.13). Path A is better than path B because C_{SF_MNMR}^A <CSF_MNMR^B , where C_{SF_MNMR}^A and C_{SF_MNMR}^B are the C_{SF_MNMR} values of path A and path B, respectively. Note that ˆ^A

b and i b^ˆ_i^B in (3.13) are the estimated values of bi for path A and path B. If the decision on SF is made at this point, path A is chosen. Now, let us go one step further. Based on the selected SF sets of path A and path B, we can find their optimal HCBs, hiA

and hiB

respectively, according to the HCB optimization procedure described in Section 3.1.3. Then, their actual bits information b_i^A and b_i^B, for path A and path B, respectively, is obtained. Finally, the total costs C_MNMR^A and C_MNMR^B for two candidate paths are shown in (3.14). The result in (3.14) indicates that path B is actually better than path A when the bits information is correct. With a wrong estimate on b_i, our CTB algorithm would pick up path A for SFs and thus it fails to find the overall optimal path B.

∑

Clearly, with a more accurate estimate on b_k,i, we can select better SFs. For this aim, the concept of “pseudo HCB” is proposed for the trellis-based optimization on SF. The preceding discussions on choosing HCB can be applied to the ANMR minimization case.

2) Design of Pseudo HCB:

When the trellis-based optimization on SF is performed in the pseudo HCB mode, a pseudo HCB with an index set h_k^v_,i needs to be constructed for the state Υk,i to produce bk,i in (3.5) and (3.8). It can be constructed in several ways. For example, h_k^v_,i may contain only one of the 12 candidate HCBs or several codebooks. In order to improve the accuracy of the estimated values of bk,i and h_k^v_,i, we analyze the data collected from the JTB optimization scheme.

For a given value of λ, using the JTB scheme, we can find a set of optimal parameters,

JTB

sopt , h_opt^JTB and b_opt^JTB that minimizes the cost function, CANMR in (3.3) or CMNMR in (3.4).

For comparison purposes, we also construct a reference set of QSC bits, b_min^JTB, in the following way. For the ith SFB, b_{min ,i}^JTB is the minimum number of bits for encoding q_opt^JTB_,i

and is determined byb_min,^JTB_i =min_m{H_m(q_opt^JTB_,i)}, where q_opt^JTB_,i is the QSCs quantized by using

JTB ,i

sopt . In other words, without considering the bits for coding the HCB indices, b_{min ,i}^JTB is the lowest bits number produced by any of the 12 HCBs applied to the QSCs. Because the coded bits for HCB indices, R(hi-1 , hi), are also included in the overall optimization procedure, when comparing coding bits for QSCs only, b_opt^JTB is higher than or equal to b_min^JTB.

By collecting the statistics from the simulations on ten audio sequences, the histogram of the differences between b_opt^JTB and b_min^JTB, denoted by ∆b, is shown in Fig. 3.2. We observe that over 91% of ∆b is less than 3 for both ANMR and MNMR criterions. In general, we can choose the HCB that produces the minimum QSC bits, b_min^JTB.

Fig. 3.2: Histogram on ∆b.

After examining this characteristics of b_opt^JTB, we derive a rule in determining h_k^v_,i and

bk,i. For the state Υk,i, h_k^v_,i is the index set of HCB that satisfies the proposed rule in (3.15);

namely,

{

}

, = n H ≤ H + n∈ L

h_k^v_{i n} ^q_{k,i m m} ^q_k,i δ ^(3.15)

The minm{Hm(qk,i)} term is the minimum number of bits for coding qk,i without considering the coding bits for HCB indices and δ is an offset parameter. For example, if H1(qk,i) and H3(qk,i) are both smaller than or equal to minm{Hm(qk,i)}+δ^{, then} hk^v_,i equals to {1,3}.

Although (3.5) and (3.8) do not include the bits number for coding HCB indices, it is found from experiments that including this term leads to a better estimate of SF. Therefore, we expand (3.5) to approximate (3.3) and expand (3.8) to approximate (3.4) with additional terms.

Based on the above observation, b_k,i is rewritten as:

( )

elements in h_k^v_,i. The symbol R_v is the run-length coding function performed on the pseudo HCB and is defined below.

⎩⎨

Note that the Rv() function is essentially the R() function in (3.11). However, because h_k^v_,iand

v i

hl_,−1 are index sets of HCB, the intersection is used in (3.17).

After having derived (3.15) and (3.16), we still need to determine the proper values for δ and α. The values of δ ^and α can be determined by examining the difference between the JTB scheme and the CTB scheme at different values of δ ^and α and the results are shown in Fig.

3.3 and Fig. 3.4. Note that ^C_ANMR^JTB and C_ANMR^CTB are the CANMR (in (3.3)) derived using the

JTB scheme and CTB scheme, respectively.C_MNMR^JTB and C_MNMR^CTB are the CMNMR (in (3.4)) derived using the JTB scheme and CTB scheme, respectively. We find that for a wide range of δ values, we can achieve a pretty good performance when R_v(h_l^v_,i−1,h_k^v_,i) is included in bk,i

(α > 0). As Fig. 3.3 and Fig. 3.4 indicate, the case that δ^{=1 and} α=0.5 gives the best results.

Hence, we choose 1 for δ and 0.5 for α in our implementation.

Fig. 3.3: ( CTB MNMR^JTB

MNMR C

C − ) vs. (δ ^, α^).

Fig. 3.4: ( CTB ANMR^JTB ANMR C

C − ) vs. (δ ^, α^).