# Locating Image Sources from Multiple Spatial Room Impulse Responses
## 2 Methods
### Geometry Calibration
#### 2.4.1 Time-of-Arrival Bias  
Knowing the ToA enables estimating the distance of a sound event. In this way, the ToA of the direct sound helps to determine the array geometry. Additionally, the ToAs associated with the ER objects are ultimately used to estimate the image source positions in space. However, determining distances based on the ToA data is easily biased by errors in the assumed speed of sound \(c_{0}\) and any uncompensated measurement system delay \(t_{e}\). Depending on the actual distance of the sound event \(d_{i}\) and the speed of sound \(c_{0}\) during the measurements, the ToA of the sound event \(t_{i}\) will be registered as  
\[t_{i}=\frac{d_{i}}{c_{0}}+t_{e}, \tag{4}\]  
where \(t_{e}\) is the measurement system delay biasing the ToA. When translating this measured ToA back into its associated travel distance, using the assumed speed of sound \(c\) will in turn bias the travel distance estimate \(d_{i}^{\prime}\):  
\[d_{i}^{\prime}=\frac{c}{c_{0}}d_{i}+ct_{e}. \tag{5}\]  
Clearly, the errors have different effects on the reconstructed travel distance \(d_{i}^{\prime}\). While the measurement system delay adds a constant distance bias, a ratio of assumed vs. real speed of sound scales the distance.  
The scale of the two biases is very different. Usually, the speed of sound deviation does not affect the result much. By default, the SDM toolbox [10] fixes \(c\) to 345 m/s, while \(c_{0}\) varies between 343.8-347.2 m/s in regular room conditions (20-25.5 \({}^{\circ}\)C, 1013.25 hPa, 50 % Rh) [14]. This means that the deviation between them varies only \(\pm\)2 m/s, which is small w.r.t. the speed of sound itself. This error is negligible for small rooms, but it starts to take effect with longer distances. For example, the farthest distance seen in the concert hall data presented in Section 3 is roughly 35 m, which would result in a difference of about \(\pm\)20 cm. In contrast to that, the measurement system delay heavily impacts the estimation of small distances. For instance, a 2 ms delay on the signal onset corresponds to roughly a 69 cm offset on the measured sound event distance, independent of \(d_{i}\).  
The biased travel distance causes position-dependent warping of the detected measurement layout. This phenomenon has been illustrated in Figure 2. There, three receivers (red, blue and yellow circles) have been set up to measure three sound sources (grey rectangles). Each receiver has managed to capture the DoAs correctly, but the ToA has been biased by constant delay of the sound sources. Therefore, each receiver has located the sound source further away from the measurement point than what it actually is (colored rectangles). This affects the detected shape of the sound source array. For this very reason, one cannot apply a simple rigid-body fit to find the receiver array orientations, which can only identify a common rotation, translation and scaling. When the rigid-body fit finds the optimal rotation and translation based on the warped array, the position and orientation of the receiver are not accurate. Accordingly, it is necessary to compensate for the delays first.