# Deep Learning-Based Low-Frequency Passive Acoustic Source Localization
## 4 Results
### Case 1  
For case 1, the model was trained to detect six monopole sources that were spread across a 12 \(\times\) 12 scanning grid plane by randomly assigning coordinates. The results are \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Cases** & **Conv. Layers, Dimension** & **Max-Pool Layers, Dimension** & **Dense Layers (Nodes), Output Dimension** \\ \hline
1 & 3, (3 \(\times\) 3), Filters: 32, 64, 64 & 2, (2 \(\times\) 2) & 4 (128), Output: 144 \\ \hline
2 (i): S1 & 2, (2 \(\times\) 2), Filters: 128, 64 & None & 1 (128), Output: \(N\) \\ \hline
2 (ii): S2 & 3, (2 \(\times\) 2), Filters: 128, 64, 64 & 2, (2 \(\times\) 2) & 1 (128), Output: \(N\) \\ \hline
3 (i): S1 & 2, (2 \(\times\) 2), Filters: 128, 64 & None & (\(\theta\)): 1 (128), \(N_{1}\) (a): 1 (128), \(N_{2}\) \\ \hline
3 (ii): S2 & 2, (2 \(\times\) 2), Filters: 128, 64, 64 & 2, (2 \(\times\) 2) & (\(\theta\)): 1 (128), \(N_{1}\) (a): 1 (128), \(N_{2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of the convolutional neural network architecture used in all the cases. Figure 11: A multi-task CNN used for three-dimensional ASL. The main branch performed the convolution and pooling operations on the input feature. The two output branches peeled off the main branch using the flattened feature vector as the input to give separate predictions for \(\theta\) and \(\alpha\). presented and compared for source frequencies of 100 Hz and 300 Hz. [PERSON] et al. [18], who also presented results for six monopole source localization using DenseNet, generated a batch of 512 training samples and 1000 validation samples for each epoch and ran the model for almost 3000 epochs. This means the training was carried out with more than 4.5 million samples. In contrast, our shallow CNN model trained for 600 epochs on 50,000 training samples. The training data are 0.00045% of the total number of possible samples \(\binom{14}{6}\), which is in the order of \(\approx\)10\({}^{10}\), thus exemplifying the limited nature of the data. The CNN also had a significantly smaller number of learnable parameters (\(\approx\)1 million) compared with DenseNet (\(\approx\)20 million). The details of the CNN model are given in Table 1. The learning rate for the optimizer was fine-tuned to 0.0001. This case took approximately 100 min to train. The obtained results were post-processed to highlight the strongest sources (predicted source strengths of 0.3 and higher) and allow us to derive qualitative and quantitative trends. Figures 12 and 13 show two source configurations as predicted by the model for monopole sources at 300 Hz. Figures 14 and 15 show similar results for monopole sources at 100 Hz. The red dots represent the ground truth source configuration. These results are followed by Table 2, which gives the source detection statistics at the two source frequencies for 1000 test samples. Figure 14: Sparse sources (100 Hz). The model struggles at lower frequencies when the concentration of sources is less. \begin{table}
\begin{tabular}{c c c} \hline
**Sources Detected** & **300 Hz** & **100 Hz** \\ \hline
6 & 9 & 1 \\ \hline
5 & 70 & 10 \\ \hline
4 & 272 & 39 \\ \hline
3 & 397 & 142 \\ \hline
2 & 210 & 344 \\ \hline
1 & 42 & 348 \\ \hline
0 & 0 & 116 \\ \hline Total & 1000 & 1000 \\ \hline \end{tabular}
\end{table} 
Table 2: Source detection statistics for 1000 test samples. The frequency-dependence of model performance is evident—the model was able to detect more sources when they were oscillating at a higher frequency (300 Hz). Figure 15: Clustered sources (100 Hz). The model performs relatively better when low-frequency sources are highly concentrated. Discussion  
As mentioned, [PERSON] et al. (2018) used a batch of 512 training and 1000 validation samples each epoch while running the model for around 3000 epochs, the total number of samples thus amounting to \(\approx\)4.5 million. We limited our training data and epochs to 50,000 samples (\(\approx\)1% of [PERSON] et al.