# A Psychoacoustic-Based Multiple Audio Object Coding Approach via Intra-Object Sparsity
## 2 Proposed Compression Framework
### NPTF Allocation Strategy  
Allocating the _NPTF_ for each active object signal can be actualized with various manners according to realistic application scenarios. As a most common used means called simplified average distribution method, all active objects share the same _NPTF_ has been employed in [19; 21]. This allocation method balances a tradeoff between computational complexity and perceptual quality. Therefore, it is a simple and efficient way. Nonetheless, this allocation strategy cannot guarantee all decoded objects with similar perceptual quality. Especially, the uneven quality can be emerged if there exists big difference of intra-object sparseness amongst objects. To conquer the above-mentioned issue, an Analysis-by-Synthesis (ABS) framework was proposed to balance the perceptual quality for all objects through solving a minimax problem via the iterative processing [20]. The test results show that this technique yields the approximate evenly distributed Frame Energy Preservation Ratio (_FER_, defined in Appendix A) for all objects. Despite the harmonious perceptual quality can be maintained, the attendant problem which is the sharp increase in computational complexity cannot be neglected. Accordingly, relied on the TF sorting result obtained in Section 2.2, an _NPTF_ allocation strategy for obtaining a balanced perceptual quality of all inputs is proposed in this work. In the \(n^{\text{th}}\) frame, we assume that the \(q^{\text{th}}\) object will be distributed \(k_{q}\)_NPTF_, i.e., \(k_{q}\) TF instants will be extracted for coding. An Individual Object Energy Retention ratio (_IOER_) function for the \(q^{\text{th}}\) object is defined by:  
\[f_{IOER}(k,q)=\frac{\sum\limits_{i=1}^{k}S_{q}\Big{(}n,l_{i}^{q}\Big{)}}{\sum \limits_{l=1}^{k}S_{q}(n,l)} \tag{7}\]  
where \(l_{i}^{q}\) is the reorder frequency index obtained in the previous section. _IOER_ function represents the energy of the \(k\) perceptual significant elements against the original signal \(S_{q}(n,l)\). Thus, \(k_{q}\) will be allocated for each object with approximate _IOER_. Under the criterion of minimum mean-square error, for all \(q\in\ \{1,2,\ldots,Q\}\) the \(k_{q}\) can be attained via a constrained optimization equation as follow:  
\[ \begin{split}\min\limits_{k_{1},k_{2},\ldots,k_{Qq}=1}\left\|f_{ IOER}(k_{q},q)-\overline{f}\right\|^{2}\\ \text{s.t.}\ \sum\limits_{q=1}^{Q}k_{q}=L\end{split}  \tag{8}\]  
where \(\overline{f}=\frac{1}{Q}\sum\limits_{q=1}^{Q}f_{IOER}(k,q)\) represents the average energy of all objects. The optimal solution \(k_{1}\), \(k_{2}\),..., \(k_{Q}\) for each object are the desired _NPTF\({}_{1}\)_, _NPTF\({}_{2}\)_,..., _NPTF\({}_{Q}\)_, which can be searched by our proposed method elaborated in Algorithm 1. The proposed _NPTF_ allocation strategy allows different reserved TF instants (i.e., MDCT coefficients) for each object among a certain group of multi-track audio objects without iterative processing, therefore, the computational complexity decrease rapidly through the dynamic TF instants distribution algorithm. In addition, a sub-equal perception quality for each object can be maintained via our proposed _NPTF_ allocation strategy rather than pursuit the quality of a particular object. Thereafter, vector \(\mathbf{P}_{q}^{\prime}\) needs to be extract the _NPTF\({}_{q}(k_{q})\)_ elements to forming a new vector \(\mathbf{\widetilde{p}}_{q}\equiv\big{[}P_{q}(n,l_{1}^{q}),\ldots,P_{q}(n,l_{NPTF _{q}}^{q})\big{]}\).. It should be note that \(l_{1}^{q},l_{2}^{q},\ldots,l_{NPTF_{q}}^{q}\) indicate the origin of \(S_{q}\Big{(}n,l_{1}^{q}\Big{)},S_{q}\Big{(}n,l_{2}^{q}\big{)},\ldots,S_{q}\Big{(} n,l_{NPTF_{q}}^{q}\Big{)}\), respectively.