\section{Experimental Results}
\label{sec:experiments}
%Experimental environments were created to evaluate the performance of the proposed inference algorithm with multiple reflectors. Experiments 1--4 consider simulated impulse responses where the amount of prior information available to the algorithm is varied. This provides insight into the performance of the individual processing stages as errors propagate through them. Experiment 5 considers impulse responses obtained from real-world measurements to demonstrate its validity in practical scenarios. In all cases a sampling frequency $f_\mathrm{s}=44.1$~kHz was employed and the microphone array consisted of 4 microphones for the simulated experiments and 5 microphones for the real-world data.
We evaluate the performance of the proposed inference algorithm in the following way:
\begin{itemize}
\item We first consider the SIMO case using simulated AIRs to perform geometric inference of a rectangular room.
\item We then demonstrate the robustness to errors in the TOA information by considering the MISO case using simulated AIRs to perform inference of a single reflector.
\item Finally, we perform geometric inference in a real conference room considering the MIMO case achieved by clustering multiple reflector estimates.
\end{itemize}
\subsection{Evaluation Criteria}
Given reference source location $\rs$ and estimated source location $\hat{\mathbf{r}}_ \textrm{s}$, the source localization error is given by the Euclidian distance $\epsilon_{\textrm{s}}=\|\hat{\mathbf{r}}_ \textrm{s}-\rs\|$. Let $\lb$ and $\hat{\lb}$ be the true and estimated reflector lines, respectively. From these we can evaluate the distance $\rho$ from $\mathbf{r}_{0}$ to a point on each line and the orientation $\alpha$. The distance can be evaluated by projecting $\mathbf{r}_{0}$ onto the line such that
\begin{equation}
\rho=\frac{\left | l_1 x_0 + l_2 y_0 + l_3 \right |}{\sqrt{l_1^{2}+l_2^{2}}},
\end{equation}
and the orientation from
\begin{equation}
\alpha=2\arctan \frac{\sqrt{l_1^{2}+l_2^{2}}-l_1}{l_2}.
\end{equation}
The accuracy of the reflector localization is measured using:
\begin{itemize}
\item distance error $\epsilon_{\textrm{d}} = \left | \rho - \hat{\rho} \right |$;
\item angular error $\epsilon_{\textrm{a}} = \left | \alpha - \hat{\alpha} \right |$;
%\item alignment error $\epsilon_{\textrm{l}}=\frac{\hat{\lb}^T\lb}{\|\hat{\lb}\|\lb\|}$, where values closer to 1 indicate the angle between the lines is small.
\end{itemize}
%Note that in some cases it is not possible to match an estimated reflector to a reference reflector. In the case of such an ambiguity we set $\epsilon_{\textrm{l}}$ to zero.
\subsection{Simulated Experiment}
Simulated AIRs were obtained with the source-image method~\cite{Allen1979, Peterson1986}, taking fractional delays into account, for random source and receiver placement in a rectangular room of random dimensions (X~$\times$~Y)~m, with $\{X,Y\}$ being uniformly distributed in the range 3--5 $m$ and 4--6 $m$ respectively. The floors and ceilings were perfectly absorbing. An example simulation is depicted in Fig.~\ref{fig:sim_ex_result}, showing the source, microphones, ellipses and estimated reflectors.
The performance was assessed by averaging the results of 100 Monte Carlo runs. The mean and variance of $\epsilon_{\textrm{s}}$, $\epsilon_{\textrm{d}}$ and $\epsilon_{\textrm{a}}$ were calculated considering all located reflectors and individual reflectors ranked in order of error. In some cases not all reflectors are identified with the same degree of accuracy; ranking the error in this way provides insight into the distribution of errors as a function of the number of identified reflectors.
The sound source ($\rs$) and the microphones ($\ri$) were uniformly distributed inside the room, constraining the positions to be at a distance of at least 0.5 m from each wall and with each microphone being kept at a minimum distance of 0.5 m from the source. We exclude those cases in which the inference algorithm fails due either to the inseparability of neighbouring peaks in the AIR or if the matrices involved in the source localization, particularly in \eqref{first_estimate}, are rank deficient. In other words, if the microphones are arranged as a linear array it might not be possible to estimate the source location, because of the front-back ambiguity~\cite{Kuttruff2000}. Additionally the simulation was limited to include only first-order reflections.
The sampling frequency is 44.1 kHz using $M=4$ microphones. We consider unsynchronized AIRs, i.e. the direct-path propagation is removed. Source localization is applied as described in Section~\ref{unsupervised_ID} to estimate TOAs from the TDOAs. Inference is performed using the SIMO approach outlined in Section~\ref{sec:mult_ref_extension}.
The results of the source localization are $\mu(\epsilon_{\textrm{s}}) = 0.92$ cm and $\sigma(\epsilon_{\textrm{s}}) = 1.62$ cm. The distance and angular error for the reflector inference are given in Table~\ref{table:dis_ang_results}. Averaged across all walls our approach achieves a $\mu(\epsilon_{\textrm{d}})$ and $\mu(\epsilon_{\textrm{a}})$ of around one cm and less than half degree respectively.
\begin{table}[!t]
\caption{Distance and angular error results for simulated data}
\label{table:dis_ang_results}
\begin{center}
\scriptsize{
\begin{tabular}{l|cccc}
\hline\hline
Walls & $\mu(\epsilon_{\textrm{d}})$ [cm] & $\sigma(\epsilon_{\textrm{d}})$ [cm] &$\mu(\epsilon_{\textrm{a}})$ [$^{\circ}$] & $\sigma(\epsilon_{\textrm{a}})$ [$^{\circ}$] \\ \hline
\bf{All} & \bf{0.926} & \bf{1.169} &\bf{0.215} &\bf{0.426}\\
Best & 0.206 & 0.210 & 0.034 & 0.030\\
2nd best & 0.505 & 0.295 & 0.091 & 0.057\\
2nd worst & 0.884 & 0.421 & 0.179 & 0.138\\
Worst & 2.109 & 1.756 & 0.555 & 0.737\\
\end{tabular}}
\end{center}
\end{table}
%\begin{table}[!t]
% \caption{Source Localization Results with Simulated Data}
% \label{table:sim_src_loc}
% \begin{center}
% \scriptsize{
% \begin{tabular}{cc}
% \hline\hline
%$\mu(\epsilon_{\textrm{s}})$ [cm] & $\sigma(\epsilon_{\textrm{s}})$ [cm] \\ \hline
% 0.92 & 1.62 \\
% \end{tabular}}
% \end{center}
%\end{table}
%The experiments were as follows:
%\begin{enumerate}
% \item \label{en:sim_case1} \emph{Simulated AIRs with known source location.} Peaks in the AIRs correspond to the TOAs of each reflection from source to receiver. Inference is performed on the estimated TOAs.
% \item \label{en:sim_case2} \emph{Simulated AIRs with unknown source location.} Source localization is applied as described in~\ref{unsupervised_ID} in order to parameterize the ellipses and enable inference with the TOAs as in Experiment~\ref{en:sim_case1}.
% \item \label{en:sim_case3} \emph{Simulated AIRs with direct-path propagation removed to render them unsynchronized.} The source location is used to estimate TOAs from the TDOAs. Inference is then applied as in Experiments~\ref{en:sim_case1} and~\ref{en:sim_case2}.
% \item \label{en:sim_case4} \emph{Blindly identified AIRs.} The simulated AIRs are convolved with a WGN signal of duration 5 s and the channels estimated with the RNMCFLMS algorithm~\cite{Haque2008} with parameters $\rho=0.2,~\lambda=0.98$. In order to prevent overmodelling of the AIRs, the effective length of the channels is estimated by $\max(\tau_{i,k})-\min(\tau_{i,k})$, where ground-truth $\tau_{i,k}$ are used; the observed signals $x_i(n)$ are otherwise the only signals available to the BSI algorithm. Uncorrelated sensor noise is added to give SNR $\{-5, -4,\dots,40\}$~dB, providing insight into the behaviour of the inference algorithm with different levels of noise. TDOAs are identified and applied in the same manner as Experiment~\ref{en:sim_case3}.
%\end{enumerate}
%The four experiments are summarized in Table~\ref{table:exp1-4} and were conducted in 200 Monte Carlo runs.
%\begin{table}[!h]
% \caption{Summary of conditions in experiments 1--4.}
% \label{table:exp1-4}
% \begin{center}
% \scriptsize{
% \begin{tabular}{|l|c|c|c|} \hline
% & Synchronized & Source Loc. & AIR \\
% & AIR & Known & Known / BSI \\ \hline
% Exp. 1 & Yes & Yes & AIR Known \\
% Exp. 2 & Yes & No & AIR Known \\
% Exp. 3 & No & No & AIR Known \\
% Exp. 4 & No & No & BSI \\ \hline
% \end{tabular}}
% \end{center}
%\end{table}
%The Monte Carlo randomization occasionally produces in the impulse response which, due to similar path lengths from multiple reflectors, are too close to be distinguished as separate events. Such cases can be flagged as unreliable without need for the ground-truth reference as they result in nonunique solutions to the reflector localization in Experiment~\ref{en:sim_case1}. Successive random setups are created until 200 unique solutions are found and the same 200 setups are applied in Experiments~\ref{en:sim_case2}--\ref{en:sim_case4}.
\begin{figure}[!t]
\centering \epsfig{file=figures/reconst,width=0.95\columnwidth, trim = 10mm 5mm 10mm 0mm, clip}
\caption{Example inference result for a rectangular room measuring 3~$\times$~4~m. Ellipses and their corresponding reflector are drawn in the same colour.} \label{fig:sim_ex_result}
\end{figure}
%The results for Experiments 1--3 are given in Tables~\ref{table:sim_src_loc} -~\ref{table:sim_dis_ang}. These demonstrate that the proposed inference algorithm can reliably identify multiple reflectors both in the ideal case, where the source location is known and the AIRs are synchronized, and when the AIRs are unsynchronized and/or the source location is estimated using the method outlined in Section \ref{sec:measurement}.
%The results of Experiment 4 are shown in Figs.~\ref{fig:bsi_err}--\ref{fig:bsi_err3}. Reliable localization of all four reflecting walls can be achieved at input SNR values of 10 dB or greater. Errors begin to occur at SNR $<$ 10 dB although on average at least two walls can be identified at SNR$=-5$ dB ($\epsilon_{\textrm{l}} \geq 0.8$). Existing applications of BSI, such as dereverberation by channel equalization~\cite{N8aylor2010}, usually require significantly high SNRs in order to be effective as it is required that all taps be modelled well. The requirement for accuracy of the BSI is relaxed in the case of inference as only the times of the early reflections are needed. Further work into BSI in the context of inference is expected to exploit this relaxed requirement for the identified channels.
\subsection{Robustness Analysis}
In order to study the robustness of our method with respect to noise, we added additional white noise to...
\begin{figure}[!t]
\subfigure[Linear source arrangement]{
\epsfig{file=figures/5_linear_hough,width=0.95\columnwidth, trim = 10mm 5mm 5mm 5mm, clip}
\label{fig:subfig1}
}
\subfigure[Circular source arrangement]{
\epsfig{file=figures/5_circ_hough,width=0.95\columnwidth, trim = 10mm 5mm 5mm 5mm, clip}
\label{fig:subfig2}
}
\label{fig:subfigureExample}
\caption{Average distance, shown as the left $y$-axis, and angular reflector localization error, shown as the right $y$-axis, for a single reflector using five linearly \subref{fig:subfig1} and circularly \subref{fig:subfig2} arranged source positions, as a function of additive noise to the TOA estimates.}
\end{figure}
\subsection{Real Experiment}
The simulated experiments represent idealized environments in which the transfer function of the measurement apparatus is negligible and the floor and ceiling are perfectly absorbing. In the case of real-world measurements, geometric inference is a much more challenging problem. An experiment was devised in a small conference a room measuring $3.21 \times 3.58 \times 3.00$ m, with concrete walls and two flush-mounted wooden doors in the south and east walls. A microphone array consisting of four microphones spaced by 0.5 m in a `+' configuration and a fifth placed in the centre was positioned at (1.75,1.5) m from the south-west corner. A Genelec 8030A loudspeaker was moved in a circular trajectory around the array in 16 equiangle positions at a range of 1 m from the array centre, ensuring that the loudspeaker always faced towards the array. The microphone signals were sampled at 96 kHz. At each position, the acoustic impulse response between the source and microphone array was estimated using the MLS method~\cite{Vanderkooy1994}, followed by localization of the source and a single dominant reflector using the technique described in Section~\ref{sec:localization_single}. No effort was made to synchronize the recorded signals with the input stimulus. The line estimates were combined using the Hough transform and the parameters corresponding to the top four bins were used to estimate the bounding line reflectors.
The results are shown in Fig.~\ref{fig:real_ex_setup}. The error between the intended and estimated positions is due, in part, to the manual positioning of the loudspeaker. This is not problematic as the system makes no prior assumptions about the source location. The Hough data points, marked green, lie very close to the room boundaries and are well-fitted by the estimated line reflectors. Some erroneous data points are due to the source positions near multiples of 45$^\circ$ in which no single reflector is dominant; they are however treated as outliers and do not affect the estimated reflectors. Reflections from the walls were always dominant over those arising from the floor and ceiling as they are less reflective than the walls.
Averaged across all walls the distance and angular errors, summarized in Table~\ref{table:real_ref_loc}, are around one cm and less than a degree respectively.
The loudspeaker positions used in this experiment are similar to those used in a 2D wavefield synthesis array. It is known that the effect of reverberation impairs the quality of the synthesized acoustic scene, for which recent effort has been made that considers reflected (image) sources as virtual sources~\cite{}. Such compensation requires knowledge of the location of the reflecting boundaries that is achieved in this experiment, giving scope for a wavefield synthesis system that both senses and exploits the acoustic environment.
%\begin{figure}[!t]
%\centering \epsfig{file=figures/simulated_results,width=0.92\columnwidth}
%\caption{Blind system identification experiment: Alignment error (ranked by accuracy), shown as the left $y$-axis, and source localization accuracy, shown as the right $y$-axis, as a function of SNR.} \label{fig:bsi_err}
%\end{figure}
%
%\begin{figure}[!t]
%\centering \epsfig{file=figures/angular_distance_seperate,width=0.92\columnwidth}
%\caption{Blind system identification experiment: Distance, shown as the left $y$-axis, and angular reflector localization error (ranked by accuracy), shown as the right $y$-axis, as a function of SNR.} \label{fig:bsi_err2}
%\end{figure}
%
%\begin{figure}[!t]
%\centering \epsfig{file=figures/angular_distance_average,width=0.92\columnwidth}
%\caption{Blind system identification experiment: Average distance, shown as the left $y$-axis, and angular reflector localization error (for all walls combined), shown as the right $y$-axis, as a function of SNR.} \label{fig:bsi_err3}
%\end{figure}
\begin{figure}[!t]
\centering \epsfig{file=figures/real_world_xarray,width=0.9\columnwidth}
\caption{Room inference results using a microphone array, placed centrally in a small conference room, capturing a MLS sequence from 16 source positions in turn.} \label{fig:real_ex_setup}
\end{figure}
%
%Please note that along with the average reflector localization error ($e_{\mathrm{rl}}$) for all four walls, we also estimate the mean and variance of $e_{\mathrm{rl}}$ for the best, second best, second worst and worst localized reflector at each iteration of the simulation. In other words we order by accuracy from the best localized reflector to the worst localized reflector. Since it is not always possible to accurately estimate the location of all reflectors, an ordering by accuracy permits us to observe if our localization has failed in general, or if some reflectors can be accurately localized while others can not.
%\begin{table}[!t]
% \caption{Reflector Localization Results with Simulated Data and varying SNR}
% \label{table:sim_ref_loc}
% \begin{center}
% \scriptsize{
% \begin{tabular}{clcc}
% \hline\hline
% SNR. & Walls & $\mu(e_{\textrm{rl}})$ & $\sigma(e_{\textrm{rl}})$ \\ \hline
% 40 & \bf{All} & \bf{0.977} & \bf{0.098} \\
% 40 & Best & 1.000 & 0.004 \\
% 40 & 2nd best & 0.998 & 0.015 \\
% 40 & 2nd worst & 0.981 & 0.098 \\
% 40 & Worst & 0.929 & 0.217 \\ \hline
% 35 & \bf{All} & \bf{0.977} & \bf{0.098} \\
% 35 & Best & 1.000 & 0.004 \\
% 35 & 2nd best & 0.998 & 0.015 \\
% 35 & 2nd worst & 0.981 & 0.098 \\
% 35 & Worst & 0.929 & 0.217 \\ \hline
% 30 & \bf{All} & \bf{0.977} & \bf{0.098} \\
% 30 & Best & 1.000 & 0.004 \\
% 30 & 2nd best & 0.998 & 0.015 \\
% 30 & 2nd worst & 0.981 & 0.098 \\
% 30 & Worst & 0.929 & 0.217 \\ \hline
% 25 & \bf{All} & \bf{0.977} & \bf{0.098} \\
% 25 & Best & 1.000 & 0.004 \\
% 25 & 2nd best & 0.998 & 0.015 \\
% 25 & 2nd worst & 0.981 & 0.098 \\
% 25 & Worst & 0.929 & 0.217 \\ \hline
% 20 & \bf{All} & \bf{0.981} & \bf{0.091} \\
% 20 & Best & 1.000 & 0.000 \\
% 20 & 2nd best & 0.999 & 0.007 \\
% 20 & 2nd worst & 0.987 & 0.066 \\
% 20 & Worst & 0.940 & 0.196 \\ \hline
% 15 & \bf{All} & \bf{0.981} & \bf{0.091} \\
% 15 & Best & 1.000 & 0.000 \\
% 15 & 2nd best & 0.999 & 0.007 \\
% 15 & 2nd worst & 0.987 & 0.066 \\
% 15 & Worst & 0.940 & 0.196 \\ \hline
% 10 & \bf{All} & \bf{0.982} & \bf{0.091} \\
% 10 & Best & 1.000 & 0.000 \\
% 10 & 2nd best & 0.999 & 0.007 \\
% 10 & 2nd worst & 0.988 & 0.065 \\
% 10 & Worst & 0.940 & 0.195 \\ \hline
% 5 & \bf{All} & \bf{0.960} & \bf{0.150} \\
% 5 & Best & 1.000 & 0.003 \\
% 5 & 2nd best & 0.998 & 0.015 \\
% 5 & 2nd worst & 0.985 & 0.067 \\
% 5 & Worst & 0.858 & 0.321 \\ \hline
% 0 & \bf{All} & \bf{0.648} & \bf{0.171} \\
% 0 & Best & 0.989 & 0.031 \\
% 0 & 2nd best & 0.896 & 0.168 \\
% 0 & 2nd worst & 0.536 & 0.397 \\
% 0 & Worst & 0.170 & 0.360 \\ \hline
% \end{tabular}}
% \end{center}
%\end{table}
%\begin{table}[!t]
% \caption{Source Localization Results with Real-World Data}
% \label{table:real_src_loc}
% \begin{center}
% \scriptsize{
% \begin{tabular}{c|c}
% \hline\hline
% Exp. & $e_{\textrm{sl}}$ [cm] \\ \hline
% 5 & \\
% \end{tabular}}
% \end{center}
%\end{table}
%\begin{table}[!t]
% \caption{Reflector Localization: Alignment error results with Simulated Data}
% \label{table:sim_ref_loc}
% \begin{center}
% \scriptsize{
% \begin{tabular}{clcc}
% \hline\hline
% Exp. & Walls & $\mu(\epsilon_{\textrm{l}})$ & $\sigma(\epsilon_{\textrm{l}})$ \\ \hline
% 1 & \bf{All} & \bf{0.9953} & \bf{0.0631} \\
% 1 & Best & 1.0000 & 0.0000 \\
% 1 & 2nd best & 0.9998 & 0.0031 \\
% 1 & 2nd worst & 0.9931 & 0.0750 \\
% 1 & Worst & 0.9885 & 0.1011 \\ \hline
% 2 & \bf{All} & \bf{0.9996} & \bf{0.0040} \\
% 2 & Best & 1.000 & 0.0000 \\
% 2 & 2nd best & 1.000 & 0.0002 \\
% 2 & 2nd worst & 0.9998 & 0.0070 \\
% 2 & Worst & 0.9984 & 0.0079 \\ \hline
% 3 & \bf{All} & \bf{0.9947} & \bf{0.0647} \\
% 3 & Best & 1.000 & 0.0000 \\
% 3 & 2nd best & 1.000 & 0.0003 \\
% 3 & 2nd worst & 0.9920 & 0.0763 \\
% 3 & Worst & 0.9867 & 0.1041 \\
% \end{tabular}}
% \end{center}
%\end{table}
%
%\begin{table}[!t]
% \caption{Reflector Localization: Distance and Angular Error Results with Simulated Data}
% \label{table:sim_dis_ang}
% \begin{center}
% \scriptsize{
% \begin{tabular}{clcccc}
% \hline\hline
% Exp. & Walls & $\mu(\epsilon_{\textrm{d}})$ [cm] & $\sigma(\epsilon_{\textrm{d}})$ [cm] &$\mu(\epsilon_{\textrm{a}})$ [$^{\circ}$] & $\sigma(\epsilon_{\textrm{a}})$ [$^{\circ}$] \\ \hline
% 1 & \bf{All} & \bf{3.250} & \bf{15.780} &\bf{0.618} &\bf{2.815}\\
% 1 & Best & 0.300 & 0.250 & 0.039 & 0.055 \\
% 1 & 2nd best & 1.050 & 4.250 & 0.179 & 1.113\\
% 1 & 2nd worst & 3.240 & 10.860 & 0.759 & 3.029\\
% 1 & Worst & 8.430 & 28.690 & 1.494 & 4.482 \\ \hline
% 2 & \bf{All} & \bf{3.140} & \bf{13.180} &\bf{0.534} &\bf{2.084}\\
% 2 & Best & 0.500 & 0.650 & 0.055 & 0.079\\
% 2 & 2nd best & 1.080 & 1.150 & 0.137 & 0.202\\
% 2 & 2nd worst & 3.220 & 8.830 & 0.579 & 1.846\\
% 2 & Worst & 7.750 & 24.190 & 1.365 & 3.591\\ \hline
% 3 & \bf{All} & \bf{3.120} & \bf{17.280} &\bf{0.598} &\bf{3.098}\\
% 3 & Best & 0.300 & 2.160 & 0.045 & 0.314\\
% 3 & 2nd best & 0.870 & 4.870 & 0.147 & 0.711\\
% 3 & 2nd worst & 3.230 & 12.800 & 0.743 & 2.992\\
% 3 & Worst & 8.070 & 31.130 & 1.456 & 5.264\\
% \end{tabular}}
% \end{center}
%\end{table}
\begin{table}[!t]
\caption{Reflector Localization Results with Real-World Data}
\label{table:real_ref_loc}
\begin{center}
\scriptsize{
\begin{tabular}{c|cc}
\hline\hline
Wall & $\epsilon_{\textrm{d}}$ [cm] &$\epsilon_{\textrm{a}}$ [$^{\circ}$] \\ \hline
\bf{All} & \bf{1.100} & \bf{0.328} \\
North & 1.490 & 0.125 \\
East & 0.690 & 0.438 \\
South & 1.560 & 0.562 \\
West & 0.650 & 0.188
\end{tabular}}
\end{center}
\end{table}