## 2 Proposed Static Sound Event Localization and Detection Using Bipartite Matching Loss  
The proposed neural network architecture in this paper for static sound event localization and detection is illustrated in Figure 1. Specifically, it utilizes a stereo audio signal with a sampling rate of 48 kHz, which is converted into a mel-spectrogram by performing a 2048-point short-time Fourier transform (STFT), with the hop length and mel-filterbanks meticulously set to 512 and 128, respectively. The architecture fundamentally adopts a convolutional neural network (CNN) framework, within which the ResNet-50 model is actively utilized [23]. Audio signals are meticulously segmented into 30 s increments. Within this framework, a single audio signal may contain only one distinct sound event, or it may not contain any at all. Additionally, there may be instances where multiple distinct sound events are present, and while these events may overlap, we have scrupulously constructed the audio samples to ensure that no more than two sound events overlap simultaneously. Furthermore, the audio signals have been designed with the capacity to contain up to a maximum of nine distinct sound events. In object detection tasks commonly employed within computer vision, the number of grids and anchors typically results in an abundance of redundant detections, which are then refined to unique results through techniques such as non-maximum suppression (NMS). However, such techniques are tailored to image information like intersection-over-union (IoU) and may not be as effective for audio information, such as mel-spectrograms. Therefore, to adopt anchor- and NMS-free approaches, the neural network is designed to infer the detection of an exact number of \(N\) audio predictions. This circumvents the dependency on traditional image-based post-processing methods, presenting a tailored solution for the auditory domain. Here, \(N\) is typically set to a number greater than the actual count of audio objects present in the mel-spectrogram. If \(N\) is less than the number of audio objects, it becomes challenging to infer all audio objects within that particular audio signal accurately. By choosing a sufficiently large \(N\), it is implied that when there are fewer audio objects than the number set for \(N\), the remaining predictions can be considered as padding with \(\emptyset\) (no objects). This approach ensures that the network is always primed to detect up to the maximum expected number of audio objects without the risk of missing any due to an underestimation of \(N\). If \(N\) is sufficiently large, \(\hat{y}\) will infer a fixed-size set of \(N\) results, some of which may be padded with \(\emptyset\) (no objects) to fill the quota. To align the order of the inferred \(N\) audio predictions with the actual audio objects, \(y\), the bipartite matching is conducted. This process involves finding a permutation of the \(N\) elements that result in the lowest loss. The equation for this matching process is formalized as follows:  
\[\hat{\sigma}=\arg\min_{v}\sum_{i=1}^{N}\mathcal{L}_{\text{match}}(y_{i},\hat{ y}_{\sigma(i)}), \tag{1}\]  
where \(\mathcal{L}_{\text{match}}(y_{i},\hat{y}_{\sigma(i)})\) is a pair-wise matching cost between the actual audio objects and the predictions with index \(\sigma(i)\). This optimal assignment can be computed efficiently with the Hungarian algorithm [22]. In calculating the matching cost, three components can be considered: the classification prediction of the audio object, the inference of localization, and the inference results for detection (onset and offset). For the classification of audio objects, four classes are considered: male, female, child, and \(\emptyset\) (no object). Localization involves estimating the azimuthal angle on a horizontal plane, discretized into ten classes representing directions at 20-degree intervals from 0 to 180 degrees. For detection, onset and offset information is estimated in a regression manner akin to bounding box estimations in the object detection task.