---
abstract: 'This paper addresses the problem of ad hoc microphone array calibration where only partial information about the distances between microphones is available. We construct a matrix consisting of the pairwise distances and propose to estimate the missing entries based on a novel Euclidean distance matrix completion algorithm by alternative low-rank matrix completion and projection onto the Euclidean distance space. This approach confines the recovered matrix to the EDM cone at each iteration of the matrix completion algorithm. The theoretical guarantees of the calibration performance are obtained considering the random and locally structured missing entries as well as the measurement noise on the known distances. [This study elucidates the links between the calibration error and the number of microphones along with the noise level and the ratio of missing distances. Thorough experiments on real data recordings and simulated setups are conducted to demonstrate these theoretical insights.]{} A significant improvement is achieved by the proposed Euclidean distance matrix completion algorithm over the state-of-the-art techniques for ad hoc microphone array calibration.'
address:
- 'Idiap Research Institute, Martigny, Switzerland'
- 'École Polytechnique Fédérale de Lausanne (EPFL), Switzerland'
- 'Emails: {mohammad.taghizadeh, phil.garner, herve.bourlard, afsaneh.asaei}@idiap.ch, reza.parhizkar@epfl.ch'
author:
- 'Mohammad J. Taghizadeh'
- Reza Parhizkar
- 'Philip N. Garner'
- Hervé Bourlard
- |
    \
    Afsaneh Asaei
bibliography:
- 'strings11.bib'
title: |
    Ad Hoc Microphone Array Calibration: Euclidean Distance\
    Matrix Completion Algorithm and Theoretical Guarantees
---

Ad-hoc microphone array calibration, Diffuse noise coherence model, Cadzow algorithm, EDM cone, Euclidean distance matrix completion.

Conclusions {#sec:conclusion}
===========

We proposed a Euclidean distance matrix completion algorithm for calibration of ad hoc microphone arrays from partially known pairwise distances. This approach exploits the low-rank property of the distance matrix and recovers the missing entries based on a matrix completion optimization scheme. To incorporate the properties of a Euclidean distance matrix, the estimated matrix at each iteration of the matrix completion is projected onto the EDM cone. Furthermore, we derived the theoretical bounds on the calibration error using matrix completion algorithm. The experimental evaluations conducted on real data recordings demonstrate that the proposed method outperforms the state-of-the-art techniques for ad hoc array calibration. This study confirmed that exploiting the combination of the rank condition of EDMs, similarity in the measured distances, and iterative projection on the EDM cone leads to the best position reconstruction results. The proposed algorithm and the theoretical guarantees are applicable to the general framework of ad hoc sensor networks calibration.

Acknowledgment {#acknowledgment .unnumbered}
==============

This work was supported by the Swiss National Science Foundation under the National Center of Competence in Research (NCCR) on “Interactive Multi-modal Information Management” (IM2). The authors would like to acknowledge the anonymous reviewers for their comments and remarks to improve the quality and clarity of the manuscript.

  Symbol        Meaning                                                             Symbol   Meaning
  ------------- ------------------------------------------------------------------- -------- ------------------------------------
  $N$           number of microphones                                               $\D$     complete noiseless distance matrix
  $a$           radius of the circular table on which microphones are distributed   $\M$     squared distance matrix
  $\varsigma$   normalized standard deviation of noise                              $\tM$    noisy squared distance matrix
  $\Pc_E$       projection into matrices with entries on index set $E$              $\hM$    estimated squared distance matrix
  $\cP_e$       projection to EDM cone                                              $\Z$     noise matrix
  $p$           probability of having random missing entries                        $\M^E$   observed matrix
  $d_{max}$     radius of the circle defining structured observed entries           $\X$     positions matrix
  $\Ms$         distance matrix with observed entries on index set $S$              $\hX$    estimated positions matrix

  : Summary of the notation.

\[tab:notations\]

![[Coherence of the signal of two microphones at $d_{ij}=20\;$cm and the fitted ${{\rm sinc}}$ function using real data recordings.]{}[]{data-label="fig:sinc2"}](1.pdf){width="\linewidth" height=".4\linewidth"}

![Matrix completion with projection onto the EDM cone. []{data-label="fig:grassman"}](2.pdf){width="0.6\linewidth"}

![[Calibration error (logarithmic scale) as defined in  versus the number of microphones. The standard deviation of noise on measured distances is $\varsigma\, d_{ij}$ where $\varsigma = 0.0167$. The error bars correspond to one standard deviation from the mean estimates.]{}[]{data-label="fig:num_mic"}](3.pdf){width="1\linewidth"}

![[Mean position error (logarithmic scale) as defined in  versus the number of microphones. The standard deviation of the noise on measured distances is $\varsigma\, d_{ij}$ where $\varsigma = 0.0167$. The error bars correspond to one standard deviation from the mean estimates.]{}[]{data-label="fig:num_mic_crb"}](4.pdf){width="1\linewidth"}

![[Calibration error (logarithmic scale) as quantified in  versus $\varsigma$. The error bars correspond to one standard deviation from the mean estimates.]{}[]{data-label="fig:snr_sim"}](5.pdf){width="1\linewidth"}

![[Mean position error (logarithmic scale) as defined in  versus $\varsigma$. The error bars correspond to one standard deviation from the mean estimates.]{}[]{data-label="fig:snr_crb_sim"}](6.pdf){width="1\linewidth"}

![[Mean position calibration error versus the ratio of missing pairwise distances for 30 sources and 30 microphones [(60 nodes in total)]{} considered in the self-calibration method [@crocco2012bilinear] and 60 microphones used for the proposed E-MC$^2$ algorithm. The standard deviation of noise in pairwise distance estimation is 0.02.]{}[]{data-label="fig:missing"}](7.pdf){width="1\linewidth"}

![[Effect of jitter on E-MC$^2$ algorithm quantified in terms of (a) mean position error as defined in  as well as (b) calibration error as defined in  versus $\varsigma$. The error bars correspond to one standard deviation from the mean estimates. The number of microphones is 45 and 60% of the pairwise distances are missing.]{}[]{data-label="fig:jitter"}](8.pdf){width="1\linewidth"}

[@c\*2[c]{}c&gt;c@]{} & &\
& & & & [[**Calibration (cm$^2$)**]{.nodecor}]{}\
(lr)[2-3]{}(l)[4-5]{} & 3.3 (0.72) & 175.8 & 3.18 (0.73) & 170.5\
(lr)[2-3]{}(lr)[4-5]{} & 2.1 (0.3) & 96.3 & 4.64 (0.65) & 258.8\
(lr)[2-3]{}(lr)[4-5]{} & 6.8 (0.96) & 265 & 7.05 (0.92) & 281.5\
(lr)[2-3]{}(lr)[4-5]{} & 6.9 (1.35) & 272 & 7.5 (1.55) & 305\
(lr)[2-3]{}(lr)[4-5]{} & 6.56 (0.91) & 225.1 & 6.8 (0.94) & 274\
(lr)[2-3]{}(lr)[4-5]{} & 1.58 (0.37) & 95.5 & 1.71 (0.41) & 105.83\

\[tab:distributed\]

![Calibration of the eleven-element microphone array while several pairwise distances are missing. The geometries are estimated using MDS-MAP, SDP, S-stress and the proposed proposed algorithm E-MC$^2$.[]{data-label="fig:11mic_old"}](90.pdf){width="1\linewidth"}

![Calibration of the eleven-element microphone array while several pairwise distances are missing. The geometries are estimated using MC, MC+Cadzow (MC$^2$), and the proposed algorithm E-MC$^2$.[]{data-label="fig:11mic_adv"}](91.pdf){width="1\linewidth"}

![Calibration of the nine-element microphone array. The geometries are estimated using MDS-MAP, S-stress, SDP and the proposed Euclidean distance matrix completion algorithm, E-MC$^2$.[]{data-label="fig:9mic_old"}](92.pdf){width="1\linewidth"}

![[Calibration of the nine-element microphone array. The geometries are estimated using MC, MC+Cadzow (MC$^2$) and the proposed algorithm E-MC$^2$.]{}[]{data-label="fig:9mic_adv"}](93.pdf){width="1\linewidth"}

[@c\*2[c]{} c&gt;c@]{} & &\
& & & & [[**12-mic**]{.nodecor}]{}\
(lr)[2-3]{}(l)[4-5]{} & 9 & 8.13 & 434.4 & 472\
(lr)[2-3]{}(lr)[4-5]{} & 9.09 & 8.63 & 141 & 135\
(lr)[2-3]{}(lr)[4-5]{} & 6.86 & 6.14 & 125 & 95\
(lr)[2-3]{}(lr)[4-5]{} & 10.6 & 9.75 & 133 & 115\
(lr)[2-3]{}(lr)[4-5]{} & 9.2 & 7.68 & 119 & 52\
(lr)[2-3]{}(lr)[4-5]{} & 6.5 & 5.85 & 49.6 & 46\

\[tab:all-theory-cal\]

[@c\*2[c]{} c&gt;c@]{} & &\
& & & & [[**12-mic**]{.nodecor}]{}\
(lr)[2-3]{}(l)[4-5]{} & 0.83 & 0.78 & 6.34 & 7.23\
(lr)[2-3]{}(lr)[4-5]{} & 0.86 & 0.81 & 2.88 & 2.35\
(lr)[2-3]{}(lr)[4-5]{} & 0.69 & 0.61 & 2.5 & 1.9\
(lr)[2-3]{}(lr)[4-5]{} & 1.1 & 0.97 & 2.6 & 2.1\
(lr)[2-3]{}(lr)[4-5]{} & 0.91 & 0.74 & 2.16 & 1.7\
(lr)[2-3]{}(lr)[4-5]{} & 0.64 & 0.97 & 1.06 & 1\

\[tab:all-theory-pos\]

![Scenario corresponding to the (I) lower bound and (II) upper bound of the probability $q$ of structured missing distances.[]{data-label="fig:bounding"}](94.pdf){width="1\linewidth"}
