# A Psychoacoustic-Based Multiple Audio Object Coding Approach via Intra-Object Sparsity
## 2 Proposed Compression Framework
### Psychaocustic-Based TF Instants Sorting  
In Appendix A, it is proved that the majority of the frame energy concentrates in finite \(k\) time-frequency instants for each audio object. For this reason, we can extract these \(k\) dominant TF instants for compression. In our previous work [19; 20; 21], TF instants are sorted and extracted by natural ordering via the magnitude of the normalized energy. However, this approach does not take into account HAS. It is well-known that HAS is not equally sensitive to all frequencies within the audible band since it has a non-flat frequency response. This simply means that we can hear some tones better than others. Thus, tones played at the same volume (intensity) at different frequencies are perceived as if they are being played at different volumes. For the purpose of enhance perceptual quality, we design a novel method through absolute auditory masking threshold to extract the dominant TF instants. The absolute threshold of hearing characterizes the amount of energy needed in a pure tone such that it can be detected by a listener in a noiseless environment and it is expressed in terms of dB Sound Pressure Level (SPL) [24]. The quiet threshold is well approximated by the continuous nonlinear function, which is based on a number of listeners that were generated in a National Institutes of Health (NIH) study of typical American hearing acuity [25]:  
\[T(f)=3.64\times(f/1000)^{-0.8}-6.5\times e^{-0.6(f/1000-33)^{2}}+10^{-3}\times( f/1000)^{4} \tag{3}\]  
where \(T(f)\) reflects the auditory properties for human ear in the STFT domain. Hence, the \(T(f)\) should be discretized and converted into the MDCT domain. The whole processing procedure includes two steps: inverse time-frequency transform and MDCT [26]. After these operations, absolute auditory masking threshold in the MDCT domain is denoted as \(T_{mdct}\) (\(l\)) (dB expression), where \(l\) = 1, 2,..., \(L\). Then, an \(L\)-dimensional Absolute Auditory Masking Threshold (AAMT) vector \(\mathbf{T}\equiv[T_{mdct}(1),T_{mdct}(2),\)..., \(T_{mdct}(L)]\) is generated for subsequent computing. From psychoacoustic theory, it is clear that if there exists a TF bin (\(n_{0}\), \(l_{0}\)) that the difference between \(S_{dB}(n_{0}\), \(l_{0})\) (dB expression of \(S(n_{0}\), \(l_{0})\)) and \(T_{mdct}(l_{0})\) is larger than other TF bins, which means that \(S(n_{0}\), \(l_{0})\) can be perceived more easily than other TF components, but not vice versa. Specifically, any signals below this threshold curve (i.e., \(S_{dB}(n_{0}\), \(l_{0})\) -- \(T_{mdct}(l_{0})\) < 0) is imperceptible (because \(T_{mdct}\) (\(l\)) is the lowest limit of HAS). Rely on this phenomenon, the AAMT vector \(\mathbf{T}\) is used for extracting the perceptual dominant TF instants efficiently. For \(q^{\text{th}}\) (\(1\leq q\leq Q\)) audio object \(S_{q}(n\), \(l)\), whose dB expression is written as \(S_{q\_dB}(n\), \(l)\). An aggregated vector can be attained by converging each \(S_{q\_dB}(n\), \(l)\) denoted as \(S_{q\_dB}\equiv[S_{q\_dB}(n\), \(1)\), \(S_{q\_dB}(n,2)\),..., \(S_{q\_dB}(n,L)]\). Subsequently, a perceptual detection vector is designed as:  
\[\mathbf{P}_{q}=\mathbf{S}_{q\_dB}\mathbf{-}\mathbf{T}\equiv\ \left[P_{q}(n,1),P_{q}(n,2), \cdots,P_{q}(n,L)\right] \tag{4}\]  
where \(P_{q}(n\),\(l)=S_{q\_dB}(n\),\(l)-T_{mdct}(l)\). To sort each element in \(\mathbf{P}_{q}\) according to the magnitude in descending order, mathematically, a new vector can be attained as:  
\[\mathbf{P}^{\prime}_{q}\equiv\left[P_{q}(n,l_{1}^{q}),\cdots,\ P_{q}(n,l_{L}^{q})\right] \tag{5}\]  
the elements in \(\mathbf{P}^{\prime}_{q}\) satisfy:  
\[P_{q}(n,l_{i}^{q})\geq P_{q}(n,l_{j}^{q}),\ \forall i<j,\ i,j\in\{\,1,2, \cdots,L\} \tag{6}\]  
where \(l_{1}^{q}\), \(\cdots\), \(l_{L}^{q}\) is the reorder frequency index which represent the perceptual significantly TF instants in order of importance for HAS. In other words, \(S_{q}(n,l_{1}^{q})\) is the most considerable component with respect to HAS.