Inter-Quantizer Prioritization by Bit Allocation Analysis

Once the attention-based quantizer formation is completed, a prioritization protocol should be used to choose the allocation of bits among competing quantizers up to availability limit $R$, at any given time of the transmission, where: (i) Intra-quantizer prioritization simply follows embedded zerotree coding [6]; and (ii) bit allocation among competing quantizers $Q_1, \cdots, Q_m$ follows bit allocation analysis by finding the efficient combination of allocations $\hat{y}$ up to the bit consumption limitation.

Intra-quantizer prioritization by zerotree coding provides substantial coding gains over the first-order entropy for significance maps. Zerotree coding predicts insignificance across scales and spatio-temporal orientations using a model that is easy for most sequences to satisfy. The zerotree approach can isolate interesting non-zero details by immediately eliminating large insignificant 3D regions from consideration. At very low bit rates, where the probability of an insignificant coefficient must be high and thus, the significant threshold must also be large, expecting the occurrence of zerotrees and encoding significance maps using zerotree coding is reasonable for every competing quantizer $Q_j$: If it is observed that a parent is insignificant with respect to the threshold, a zerotree is expected regardless of the correlation between squares of parents (coefficients) and squares of children. Hence we have that differences in quantization costs $Cost_{Q_j}$ for competing quantizers $Q_j$ are not relevant, at extremely low bit rates, in determining the technological possibilities of the transmission problem. The basic allocation processes

$$a^j = ( 0, \cdots, 0, \overbrace{1}^{j} , 0, \cdots, 0; \overbrace{-1/Cost_{Q_j}}^{m+1} )^T \ , \ \ j=1, 2, \cdots, m$$

may then be simplified to

$$a^j = ( 0, \cdots, 0, \overbrace{1}^{j} , 0, \cdots, 0; \overbrace{-1}^{m+1} )^T, \ \ j = 1,2, \cdots, m$$

where the bit consumption is $-1$ when using $a^j$ at its basic activity level.

In the maximization of $p \cdot y$ with respect to $y$ over the set of allocation processes $Y$ at any given time $t$, we have that

$$p = ( p_1, \cdots, p_m, p_{m+1})^T$$

is a profit vector and $p \cdot y$ represents the profit from $y$ at time $t$; where $p_j$ denotes the gross profit per bit for quantizer $Q_j$ at $t$; and $p_{m+1}$ denotes the price per bit for available consumption at $t$. Following Theorem 1, we make use of the imputed profit vector $p$ at any time $t$ in order to make a choice from the set of efficient points at this particular truncation time of the progressive transmission.

From Chapter 1 in Reference [10], it follows that, within a rational approach for progressive transmission, the gross profit per bit $p_j$ for any quantizer $Q_j$ at any time $t$ can have the form of its expected increase in utility per coding bit $\frac{ I[S(Q_j,t) / {\cal Q} ]}{\char93 S(Q_j,t)}$ as given by Proposition 3 in [11], with $S(Q_j,t)$ denoting the maximum availability of bit resources for quantizer $Q_j$ at time $t$ and $\char93 S(Q_j,t)$ being the number of bits in $S(Q_j,t)$. For example, $S(Q_j,t)$ may be a fixed number of sorting and refinement bit streams from the $Q_j$ intra-quantizer prioritization which follows the embedded zerotree coding scheme. Reference  [11] demonstrates that this definition avoids certain forms of behavioral inconsistency within a rational approach which first states some general principles that the solution of this problem must obey and then derives the solution that satisfies exactly the principles.

In any case, the theory of bit allocation analysis allows to achieve efficient allocations which may be descriptive of reality at any given time of the progressive transmission, since the profit vector can change its value over time to give a good predictor of target saliency for humans performing visual search and detection tasks at any given bit rate. An example is presented in the following section of experimental results.

One possible use of our results would be for a video codec to possess several alternative definitions of a profit vector $p$ in such a way that a user of the system may choose the appropriate strategy for computing vector $p$ at different bit rates to the application of interest.

For example, at extremely low bit rates (e.g., up to 10 kbps) a profit vector

$$p = ( p_1, \cdots, p_m, p_{m+1})^T$$

may be defined such that the profit per bit for quantizer $Q_j$ is based on the expected increase in utility per coding bit: $p_j= \frac{ I[S(Q_j,t) / {\cal Q} ]}{\char93 S(Q_j,t)}$, with $j=1, \cdots, m$. The price per bit for consumption is $p_{m+1} = 0$. At higher bit rates the profit vector may then change its definition in such a way that the profit per bit for quantizer $Q_j$, with $j=1, \cdots, m$, is now based on the mean square error between the original and the reconstruction for quantizer $Q_j$.

Thus, in bit allocation analysis, the system makes a choice from the set of efficient allocations at any given time by using the appropriate strategy for computing the profit vector. It may allow to attend to different parameters of interest at different bit rates within the same spatial locations.