# Locating Image Sources from Multiple Spatial Room Impulse Responses
## 2 Methods
### Estimate Directions  
The directional estimation stage aims at determining the sample-wise Direction-of-Arrival (DoA) for each source-receiver measurement. To do so, we employ the directional analysis algorithm found in the SDM --a parametric SRIR processing method that uses a broadband directional estimate for every sample of the response. It has first been introduced by [PERSON] et al. [1] and is distributed as a MATLAB toolbox [10]. The algorithm can use any microphone array with at least four 4 microphones that are not in the same plane. In addition to the DoA estimates, also an omnidirectional response \(h\) is needed for further processing. In case the used microphone array is comprised of omnidirectional receivers, simply one of the microphones is used. It is important to note that the particular estimation method operates immediately on the responses measured by a microphone array (A-Format). Nevertheless, also spherical harmonics domain (B-Format) responses could be processed by using a different directional estimator.  
As an input, the directional analysis takes a set of \(M\) impulse responses \(\mathbf{h}_{i;j}[n]\) measured by each of the \(R\) receiver microphone arrays in response to each of the \(S\) sources. The method then segments the responses into blocks of \(N\) samples with a hop size of one sample, and computes the cross-correlation between all microphone pairs. For instance, 15 correlation functions are obtained for the six-capsule open array used in this study. After applying Gaussian interpolation to each cross-correlation function, the location of its highest peak reveals the TDoA between the corresponding microphone pair. The DoA of the strongest sound event in the analysis block is computed from all TDoAs and the known microphone positions within the array. This way, each source-receiver measurement is associated with the DoA estimate \(\mathbf{u}_{i;j}[n]\) for each sample \(n\).  
The minimum possible length of the analysis block is determined by the spatio-temporal limit \(\Delta T_{min}\), that is, the time difference at which two reflections can still be separated. This limit is proportional to the largest distance within the receiver array \(l_{max}\)[1]:  
\[\Delta T_{min}=\frac{2\,l_{max}}{c_{0}}, \tag{1}\]  
where \(c_{0}\) is the speed of sound inside the measured room. For instance, the measurement array in Section 3.1 has \(l_{max}=5\) cm, which corresponds to a spatio-temporal limit of \(\Delta T_{min}\approx 0.29\) ms. In practice, window sizes are selected to be even slightly larger \(N=\Delta T_{min}f_{s}+C\) (with some constant \(C\)) in order to stabilize the DoA estimates. Of course, the increased stability comes at the cost of a lower time resolution. This means that the selected block length limits the separability of reflections. Should two or more reflections arrive at the array within the same analysis block, they cannot be separated reliably.  
Clearly, apart from the room geometry, the Time-of-Arrival (ToA) of reflections depends on the source and receiver positions. In rooms with simple geometries, one can try to set up sources and receivers so that at least the first-order reflections are separated well in time, and not more than one reflection falls within one analysis block. For example, if either source or receiver is placed at the center of the room, the symmetry axes of the room should be avoided for the counterpart. For rooms with more complex geometries, attempting such informed placement represents a larger effort and makes the measurement-based detection of reflections obsolete. Hence, it must be assumed that some individual reflections fall below the spatio-temporal limit and cannot be detected reliably, especially in small rooms. In these cases, one greatly benefits from the proposed measurement combination algorithm. By using multiple receiver positions, it is less probable that the same reflections overlap in all positions. This in turn increases the probability of detecting the true direction of each individual sound event.