# Interaural Level Difference Optimization of Binaural Ambisonic Rendering
## 3 Objective Evaluation
### Change in ILD  
To assess the effect of AIO on binaural Ambisonic ILD reproduction, the ILD of all datasets: \(\mathbf{\hat{H}}^{std}\) and \(\mathbf{\hat{H}}^{AIO}\) was estimated for all measurement locations \(\{e_{1},e_{2},...,e_{16,020}\}\) and orders of Ambisonics \(\{M=1,M=2,...,M=5\}\) according to (8). The absolute difference between the estimated ILD values for each measurement location was then calculated as  
\[\Delta\mathrm{ILD}_{\mathrm{e}}=|\mathrm{ILD}(H_{\mathrm{e}})-\mathrm{ILD}( \hat{H}_{\mathrm{e}})| \tag{15}\]  
Figure 5 shows \(\Delta\mathrm{ILD}\) for all measurement locations over the left hemisphere, both without (top) and with (bottom) AIO, for orders of Ambisonics \(\{M=1,M=2,...,M=5\}\). Smaller values of \(\Delta\mathrm{ILD}\) indicate ILD rendering closer to the HRIR. It is clear that ILD is improved for most locations on the sphere, for all tested orders of Ambisonics, though the effect is most pronounced at first and second order.  
Figure 4: Voronoi sphere plot of an \(8^{\circ}\) Gauss-Legendre grid.  
Figure 5: \(\Delta\mathrm{ILD}\) between HRIRs and binaural Ambisonic rendering (Ambisonic orders \(\{M=1,M=2,...,M=5\}\)) for all directions over the left hemisphere, without and with AIO.  
A single value of \(\Delta\)ILD for all locations on the sphere, \(\overline{\Delta\Pi\text{LD}}\), was calculated from the solid-angle weighted sum of all \(\Delta\Pi\text{LD}\) values as  
\[\overline{\Delta\Pi\text{LD}}=\sum_{\varrho=1}^{16020}\Omega_{\varrho}\Delta \Pi\text{LD}_{\varrho} \tag{16}\]  
Values of \(\overline{\Delta\Pi\text{LD}}\) for orders of Ambisonics \(\{M=1,M=2,...,M=5\}\) are presented in Figure 6. This shows that with AIO, ILD is reproduced with greater accuracy for all tested orders of Ambisonics; indeed, a greater accuracy than \(M+1\) without AIO for all tested orders apart from the \(M=4\) instance. The improvement is greatest at orders \(M=1\) and \(M=2\) though, where Ambisonic ILD reproduction is inherently the least accurate.  
To look closer at \(\Delta\Pi\text{LD}\) between binaural Ambisonic rendering and HRIRs, a second ILD calculation was made to observe how \(\Delta\Pi\text{LD}\) changes with frequency. Instead of producing one value of ILD for all frequencies using 30 ERB bands as in (8), ILD was calculated separately for 5 frequency bands with center frequencies of 1 kHz, 2 kHz, 4 kHz, 8 kHz and 16 kHz. Figure 7 illustrates the median value of \(\Delta\Pi\text{LD}\) over all measurement locations \(\{\varrho_{1},\varrho_{2},...,\varrho_{16,020}\}\) between \(\mathbf{H}\) and \(\mathbf{\hat{H}}\) for Ambisonic orders \(\{M=1,M=2,...,M=5\}\), both without (top) and with (bottom) AIO, across 5 frequency bands. 25% and 75% percentile bars are included to demonstrate the divergence from the median. Observations of the graph show that in general, \(\Delta\Pi\text{LD}\) between binaural Ambisonic rendering and HRIRs increases with frequency, and AIO improves ILD reproduction over all frequency bands fairly evenly.  
Figure 6: Solid-angle weighted value of \(\Delta\Pi\text{LD}\) for Ambisonic orders \(\{M=1,M=2,...,M=5\}\), without and with AIO.