# AMBIQUAL: Towards a Quality Metric for Headphone Rendered Compressed Ambisonic Spatial Audio
## 4 An Objective Model for Coded Spatial Audio QoE Prediction
### Compensating for Empty or Non-Existing Channels  
The B-format Ambisonic audio can contain empty (i.e., zero-padded channels) especially if scenarios where there is no background ambient sound as the sources were synthetically generated (see Figure 9). There are also scenarios where the test file is a lower order than the reference, e.g., a FOA Ambisonic audio test (containing 4-channels) is compared to a 3 OA reference (containing 16-channels). In this case the algorithm must reconcile the fact that 12 channels are not present in the test. Empty or non-existent channels would impact the overall similarity computation by AMBIQUAL. When the reference and the test channels are both empty, the algorithm sets the similarity NSIM score to 1 as the test channel is identical to the reference. However, when a reference channel contains data and the test signal is empty the algorithm computes an invalid NSIM score (NaN) due to zeros in the denominator. Due to the product computation for localization accuracy in eqn (2), this would result in an invalid overall similarity computation.  
To mitigate the effect of empty or non-existing channels, all invalid (NaN) are substituted with a minimum threshold value before applying it to Equation (2). The optimal value for this threshold, \(t_{m}in\), was experimentally obtained using the experiment 1 data.  
The values of \(\alpha\), \(\beta\), \(\gamma\), \(\delta\), \(e\), \(\zeta\), \(\chi\), \(\psi\) and \(t_{m}in\) were chosen experimentally to maximize correlation (both Spearman and Pearson) between AMBIQUAL results and MUSHRA scores obtained during experiment 1 (i.e., listening tests with single point audio sources). A wide range of values were tested. The values presented provided the best results. The results were robust if the relative relationships were similar. The general trend can be described as follows: the first-order horizontalchannels were weighted more than the vertical channel but only by a small amount (4%). The second and third-order vertical channel weightings are a magnitude less than the first-order channel weightings. The horizontal second and third-order weightings were a further magnitude lower and the mixed-orientation channel weights a further magnitude less.  
Both listening quality and localization accuracy were evaluated with various combinations of \(\alpha\), \(\beta\), \(\gamma\), \(\delta\), \(\epsilon\), \(\zeta\), \(\chi\), \(\psi\), and \(\omega\) and were compared to subjective MUSHRA scores collected from experiment 1. Pearson and Spearman correlation were calculated for all the test results except for comparing the reference to itself. This would result in higher ranking scores due to perfect matches for the objective model.  
Using weighting factors' values shown in Figure 10 and \(t_{min}\) set to 0.1 resulted with a good correlation (Pearson = 0.922, Spearman = 0.919) between AMBIQUAL results and subjective MUSHRA scores.  
Figure 10: Values of weighting factors that maximize correlation between AMBIQUAL results and subjective MUSHRA scores.  
Figure 9: Channel occupancy of B-format Ambisonics audio as a function of azimuth (AZ) and elevation (EL) of sound sources. Empty channels are represented here with the ‘\(\times\)’ sign.