# Locating Image Sources from Multiple Spatial Room Impulse Responses
## 2 Methods
### Geometry Calibration
#### 2.4.2 Delay-Compensated Fit  
Delay-compensated fit is a method that transforms receiver data to a common coordinate frame. The method consists of two steps. In the first step, one uses the estimated direct sound DoA data and the manually measured source reference positions to estimate the measurement system delays. With the delays at hand, one then approximates the corrected sound source positions relative to each receiver. The second step then applies rigid-body fit to these positions to approximate the receiver orientation. The orientation in turn affects the DoAs, which enables iteratively improving the result. The first step of the delay-compensated fit is illustrated in Figure 3. As a starting point, one knows the reference sound source positions (in grey). Slight rotation of the receiver during measurement and the measurement system delay causes the estimated positions (red squares) to be mislocated from their actual positions. Figure 2: Source array distortion caused by a constant travel distance bias. (**a**) All loudspeakers (gray) have the same delay. When measured from three different receiver positions (red, blue, yellow circles), the detected position distorts differently. (**b**) When the detected positions are overlapped on one loudspeaker, one can distinguish geometry differences in the measured source positions. There are two steps in approximating the measurement system delay. The first step estimates the receiver position from the direct sound DoAs. This is done by generating lines from source positions and their corresponding DoAs and by resolving the closest point to those lines [15]. The second step then approximates the measurement system delay \(t^{\prime}_{e,i,j}\) based on the found receiver position as  
\[t^{\prime}_{e,i,j}=\frac{||\mathbf{s}_{0,i}-\mathbf{r}_{j}||-||\mathbf{s}_{i}- \mathbf{r}_{j}||}{c}, \tag{6}\]  
where \(\mathbf{s}_{i}\) and \(\mathbf{r}_{j}\) are the approximated positions of sound source \(i\) and receiver \(j\), receiver \(j\) and sound source \(i\), respectively; \(\mathbf{s}_{0,i}\) is the reference position of the sound source \(i\); and \(c\) is the assumed speed of sound. Removing these approximated delays practically sets the distance of the approximated source to correspond to the reference source distance from the approximated receiver position. This in turn nudges the detected source array shape closer to the actual shape, see Figure 3. The corrected shape effectively improves the chances to fit the array correctly with the rigid-body fit. The second step applies the rigid-body fit to the distance-corrected measurement data. With rigid-body fit we refer to partial Procrustes analysis, which determines rotation and translation between two point sets [16]. Reorienting the receiver according to the result leads to rotation of all associated DoA data, which can be used in the next iteration to approximate the measurement system delay again. Undoubtedly, there may also be errors in the measured DoAs that have to be treated accordingly. Here, the treatment is implemented by applying Monte Carlo cross-validation. Instead of fitting the data w.r.t. all reference source positions, only a subset of positions are used at a time. Each receiver is fitted multiple times to three or more randomly selected reference positions. The results form fit candidates that are then evaluated by the error  
Figure 3: Line fit model attempts to determine the receiver location by finding the closest points for Direction-of-Arrival (DoA) lines going through reference source positions. (**a**) Sound source with system delay and rotated mic array. (**b**) Least squares line fit assuming that the DoAs and reference source positions are right. (**c**) The measurement system delay is approximated from the distance between the reference source positions and the approximated receiver position. (**d**) The final fit returns a geometry closer to the reference geometry. measure \(f\).