TDOA Matrices: Algebraic Properties and their Application to Robust Denoising with Missing Data

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by  Jose Velasco

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TDOA Matrices: Algebraic Properties and their
Application to Robust Denoising with Missing Data
Jose Velasco, Daniel Pizarro, Javier Macias-Guarasa and Afsaneh Asaei,
Abstract—Measuring the Time delay of Arrival (TDOA) be- are referred to as the non-redundant set. This problem has
tween a set of sensors is the basic setup for many applications, been studied in the past, showing that one can optimally
such as localization or signal beamforming. This paper presents
obtain the non-redundant set when TDOA measurements are
the set of TDOA matrices, which are built from noise-free
contaminated with additive Gaussian noise. This is known as
TDOA measurements, not requiring knowledge of the sensor
6 array geometry. We prove that TDOA matrices are rank-two the Gauss-Markov estimator [14]. However, in more realistic
1 and have a special SVD decomposition that leads to a compact scenarios errors are not Gaussian and some of the TDOA
0 linear parametric representation. Properties of TDOA matrices measurements may contain outliers. In these cases the Gauss-
2 are applied in this paper to perform denoising, by finding
Markov estimator performs poorly.
the TDOA matrix closest to the matrix composed with noisy
y This paper presents the TDOA matrix, which is created
measurements. The paper shows that this problem admits a
a
by the arrangement of the full TDOA set inside a skew-
M closed-formsolutionforTDOAmeasurementscontaminatedwith
Gaussian noise which extends to the case of having missing symmetric matrix, and studies the algebraic properties of this
data. The paper also proposes a novel robust denoising method matrix, showing that it has rank 2 and a SVD decomposition
4
2 resistanttooutliers,missingdataandinspiredinrecentadvances with n − 1 degrees of freedom. Such matrices have been
inrobustlow-rankestimation.Experimentsinsyntheticandreal
previously defined in the literature [15], but their properties
datasetsshowsignificantimprovementsoftheproposeddenoising
] and applications have not been studied until now.
T algorithms in TDOA-based localization, both in terms of TDOA
accuracy estimation and localization error. These algebraic properties are used in this paper to per-
I
. form denoising under different scenarios, which include the
s
c presence of missing TDOA measurements and outliers. These
[ I. INTRODUCTION denoisingalgorithmsaretestedinthecontextofspeakerlocal-

TIME delay of arrival (TDOA) estimation is an essential ization with microphone arrays, using synthetic and publicly
2
v pre-processing step for multiple applications in the con- available real datasets. Our denoising algorithms are able to
5 text of sensor array processing, such as multi-channel source recover accurate TDOA values for high rates of missing data
8 localization [1], self-calibration [2] and beamforming [3]. In and outliers, significantly outperforming the Gauss-Markov
4
all cases, performance is directly related to the accuracy of estimator in those cases. All the proposed methods don’t
4
the estimated TDOAs [4]. require knowledge of the sensor positions, so that they can
0
. Estimating TDOA in noisy environments has been subject also be used for calibration [2].
1
of study during the last two decades [5]–[7], and is still an The main contributions of this work are threefold: i) Def-
0
6 active area of research, benefiting from current advances in inition of the algebraic properties of TDOA matrices. ii) A
1 signal processing and optimization strategies [8]–[11]. closed-form solution for TDOA denoising for Gaussian noise
: Typically, the TDOA between a single pair of sensors is and the presence of missing data. iii) Novel robust-denoising
v
i obtained by measuring the peak of the generalized cross- methods for handling additive correlated noise, outliers and
X
correlation(GCC)ofthereceivedsignalsoneachsensor[12], missing data.
r whichareassumedtobegeneratedfromasinglesource.Many
a
factors, such as the spectral content of the signal, multipath A. Notation
propagation, and noise contribute to errors in the estimation Real scalar values are represented by lowercase letters
of the TDOA. (e.g. δ). Vectors are by default arranged column-wise and are
Given a set of sensors, TDOA measurements can be ob- represented by lowercase bold letters (e.g. x). Matrices are
tained for every possible pair of sensors. This is commonly represented by uppercase bold letters (e.g. M). Lower-case
known as the full TDOA set or spherical set [13]. This paper letters are reserved to define vector and set sizes (e.g. vector
studieshowtoreducenoiseanderrorsfromthefullTDOAset. x=(x ,··· ,x )(cid:62) isofsizen),andx(cid:62) denotestransposeof
1 n
The intuition behind this denoising is to exploit redundancy vector x. Calligraphic fonts are reserved to represent generic
ofthefullTDOAset.Fornsensors,thefullsetofn(n−1)/2 sets(e.g.G)orfunctionsappliedtomatrices(e.g.P(X)).The
measurements can be represented by n − 1 values, which l norm (cid:107)·(cid:107) will be written by default as (cid:107)·(cid:107) for simplicity,
2 2
and (cid:107)·(cid:107) is the Frobenius norm, while |·| is reserved to
F
J.Velasco,D.Pizarro,andJ.Macias-GuarasaarewiththeDepartmentof
represent absolute values of scalars. The l norm of a matrix,
Electronics,EscuelaPolite´cnicaSuperior,UniversityofAlcala´,28805Alcala´ 0
deHenares,Spain. written(cid:107)·(cid:107)0,isdefinedasthenumberofnon-zeroelementsof
A.AsaeiiswithIdiapResearchInstitute,Martigny,Switzerland. thematrix.A◦BistheHadamardproductbetweenAandB,
E-mails: {jose.velasco, pizarro, macias}@depeca.uah.es,
defined as the entrywise multiplication of the corresponding
[email protected]).
ManuscriptreceivedMonthDay,2015;revisedMonthDay,2015. matrices. tr(·) is the trace function.
2
We also define the normalized unitary vector ˆ1 as Typically, when only two sensors are employed, the peak
√
ˆ1 = (1,...,1)(cid:62)/ n, and the null vector ˆ0 as of the generalized cross-correlation (GCC) function of the
ˆ0 = (0,...,0)(cid:62), both of them having size n. Finally, signals of two sensors is a good estimator for the TDOA,
1 = n ˆ1ˆ1(cid:62) is a n × n matrix with all elements equal to for reasonable noise and reverberation levels [12].
1, D is a n×n diagonal matrix where its main diagonal is When more than two sensors are used (n > 2), there are
x
the vector x, and I is the identity matrix. n(n−1)/2 different TDOA measurements from all possible
pairs of sensors, forming the full TDOA set or spherical
set [13]. However, all those TDOA measurements are redun-
B. Paper Structure
dant. In fact, usually one sensor is considered the reference
The rest of the paper is distributed as follows. Section II sensor, and only the subset of n − 1 TDOA measurements
describes the related work and Section III the problem state- which involve that sensor are considered. That non-redundant
ment. In section IV TDOA matrices are described along set is the set of measurement used by the majority of TDOA-
with a derivation of their properties. TDOA denoising in based positioning algorithms proposed in the literature [16],
Gaussian noise case is addressed in section V, also providing [18]–[22]. Nevertheless, an optimal (denoised) version of the
a closed-form solution. In sections VI and VII we propose non-redundant set can be estimated from the redundant set
novel algorithms for robust handling of noise and missing using a Bayesian Linear Unbiased Estimator (BLUE), also
data, respectively. Section VIII combine the proposals of the known as the Gauss-Markov estimator [14].
previous two sections into an unified algorithm. We also A closed-form solution for the BLUE estimator is provided
provide an extensive experimentation to validate the proposed in [23], also proving that it is equal to the standard least
algorithms using both synthetic (Section IX) and real data squares estimator, and that it reaches the Cramer-Rao lower
(Section X). Finally, conclusions are drawn in section XI. bound for positioning estimation. However, all the results in
that work are based on the assumption of additive Gaussian
noise,whichisunrealisticinmanypracticalapplications[24],
II. RELATEDWORK
anddoesn’tyieldgoodresultswhencorrelatednoiseispresent
TDOA estimation is an essential first step for multiple as a consequence, for instance, of multipath propagation.
applications related to 1) localization, 2) self-calibration and Additionally, the experimental results shown in their work are
3) beamforming (among others): only applied to synthetic data, thus not allowing to assess the
1) Localization: widely used in radar, sonar and acoustics, performance of their proposal in real scenarios (in section X
since no synchronization between the source and sensor is we show the limitations of their method when evaluated on
needed. The TDOA information is combined with knowledge real data).
of the sensors’ positions to generate a Maximum Likelihood A least-squares solution to TDOA denoising is proposed
spatial estimator made from hyperboloids intersected in some in [25]. It is based on projecting the non-redundant set
optimal sense. A linear closed-form solution of the former of TDOA measurements into a set of “feasible” bivectors
problem,validwhentheTDOAestimationerrorsaresmall,is (rank 2, antisymmetric tensors) that show the same geometric
given in [16]. properties of TDOA matrices. This denoising is also optimal
2)Self-calibration:sinceknowingthepositionofsensorsis for Gaussian noise but as in [24] the experimental analysis is
mandatory for localization techniques, some strategies have based on simulated data and it does not cope with missing
been also proposed in order to calibrate them using only data or the presence of outliers in the TDOA measurements.
TDOA measurements. In [2], [17], the TDOA problem is Periodicity in correlated signals, coherent noise and multi-
converted in a Time of Arrival (TOA) problem estimating the path due to reverberation are the major sources of non-
departuretimeofsignals.Then,self-calibrationtechniquesfor GaussianerrorinTDOAestimation.Differentapproacheshave
TOAcanbeemployed.Themaindrawbackofthisapproachis been proposed to deal with them. A basic method consists
that the conversion step from TDOA to TOA is very sensitive in making the GCC function more robust, de-emphasizing
to outliers and correlated noise. the frequency-dependent weighting. The Phase Transform
3) Beamforming: precise TDOA estimations is also critical (PHAT) [26] is one example of this procedure which has
for beamforming techniques and its applications. In [3], for received considerable attention as the basis of acoustic source
example, additional steps are proposed for selecting the ap- localizationsystemsduetoitsrobustnessinrealworldscenar-
propriate TDOA value among the correlation peaks, and also ios [27], [28]. Other approaches are based in blind estimation
dealing with TDOA outliers. These steps include a Viterbi of multi-path (room impulse response) [29] but they need a
decoding based algorithm which maximizes the continuity good initialization to perform well.
of the TDOA estimations in several frames. However, the Somepreviousworkshavealsoproposedmorecomplicated
TDOA selection criteria is just based on their distance to structures in order to represent TDOA redundancy, while not
surrounding TDOA values and their GCC-PHAT values, thus imposingstrongassumptionsonthenoisedistribution.In[30]
not attempting to benefit from the actual redundancy of the a representation based on graphs allows to disambiguate if a
TDOA measurements. peak in correlation was generated by the direct path or by
Hence, an accurate estimation of TDOA is essential for a reverberationapplyinganefficientsearchalgorithmamongall
good performance of any of the former applications based on possiblecombinations.However,theydonotexplicitlyattempt
these measurements. to provide improved TDOA estimations by exploiting their
3
redundancy.
Also different matrix representations have been used in
the bibliography regarding TDOA formulation. For example
[31] uses a representation slightly different to the TDOA
matrices we describe here, but such representation does not
have the algebraic properties that TDOA matrices have, and
their authors do not address an study in this sense.
So, to the best of our knowledge, there are no previous
reported work dealing with improving TDOA estimations by
exploiting their redundancy, while not imposing Gaussian
noiserestrictions,notrequiringthesensorpositions,andbeing
able to deal with the presence of outliers and missing mea- Fig.1:Exampleofthreedifferentgeometricalconfigurations(grey,green
andblue)of4sensorwithidenticalTDOAmatrix.
surements(errorsthatwillseverelyimpacttheperformanceof
applicationsbasedonTDOAmeasurements).Inthispaperwe
show that TDOA matrices are a powerful tool that combined with
with recent advances in robust low-rank estimation, are able ∆τ =(τ −τ ), (2)
ij i j
to generate novel solutions for these problems.
where τ is the time of arrival of the signal x(t) at the sensor
i
s .
III. PROBLEMSTATEMENT i
Hereafter, we assume only one source located at the We will also express M in terms of its
position r = (r ,r ,r )(cid:62), and n sensors synchro- columns as M = (m ,m ,··· ,m ), being
x y z 1 2 n
nized between them and placed in different positions m =(∆τ ,∆τ ,...,∆τ )(cid:62).
i 1i 2i ni
s =(s ,s ,s )(cid:62), i∈[1,n]. We denote as M (n) to the set of TDOA matrices of size
i ix iy iz T
Giventhissetup,wewillassumethatthesourceisemitting n×n.
an unknown signal x(t). Then, the signal received by the sen- Notice that there is a bijection between the full TDOA set
sori,x (t),iswithoutlossofgenerality,adelayedandattenu- and the corresponding TDOA matrix. Nevertheless expressing
i
atedversionofx(t)(directpropagation)inadditiontoasignal TDOA measurements as a matrix has important advantages,
g (t) which summarizes all the adverse effects, i.e. noise, that we will discover throughout this article.
i
interference, multipath, etc. Thus, x (t) = x(t−τ )+g (t), Note also that in the former definition, knowing the sensor
i i i
where τ = (cid:107)r−s (cid:107) /c is the time of arrival (TOA) of the array geometry is not required. For a given geometry, all the
i i 2
signal x(t) at the sensor s , being c the propagation speed. feasible TDOA matrices (those that are consistent with that
i
Assuming that TOA cannot be estimated directly, the time particular geometry) are a subset of M (n). Studying the
T
delay of arrival (TDOA) between the sensors i and j is properties of such subset is out of the scope of this paper
estimated by correlating the received signals x (t) and x (t) and the interested reader can refer to [8], [9], [32] for further
i j
(typicallyusingtheGeneralizedCross-CorrelationGCC[26]). details.
On the other hand, given a particular TDOA matrix, there
IV. TDOAMATRICES are infinite number of sensors geometries which match with
In this section we define TDOA matrices, and develop it. Left side of Fig. 1 shows that, given a set of TOAs
their main properties. In a nutshell, given any TDOA matrix (τ1,..., τn) compatible with the set of TDOA measurements,
M, we show that: i) M is rank 2 (Theorem 1), ii) M can the microphones can be situated in any place along the
be decomposed as M = (cid:0)xˆ1(cid:62)−ˆ1x(cid:62)(cid:1) with x = Mˆ1 circumference(sphereinthe3Dcase)withcenterinthesource
(Lemma 1) and iii) the previous decomposition is bijective (dottedlines),preservingitscorespondentTOA(andtherefore,
(Theorem 2). its TDOA). Right side of Fig. 1 shows that there are a infinite
These properties are the foundations of the denoising algo- number of TOA sets that comply with a given set of TDOA
rithms that we present in sections V and VI, and the missing measurements.
data recovery proposal described in section VII, plus their
combination described in section VIII. B. Rank of TDOA matrices
Theorem 1. Let M∈M (n), then M is rank 2 or 0 (trivial
T
A. Definition of TDOA matrices case).
Definition1. ATDOAmatrixM,isa(n×n)skew-symmetric
Proof: The matrix M can be expressed as:
matrixwheretheelement(i,j)isthetimedifferenceofarrival
(TDOA) between the signals arriving at sensor i and sensor M=T−T(cid:62), (3)
j:
 0 ∆τ ··· ∆τ  where T is a rank 1 matrix defined as:
12 1n
 
∆τ21 0 ··· ∆τ2n τ1 ··· τ1
M={∆τij}= ... ... ... ...  (1) T= ... ... ... , (4)
∆τ ∆τ ··· 0 τ ··· τ
n1 n2 n n
4
Applying the well known inequality: leads to a parametric representation of M that has important
properties that we will exploit later for TDOA denoising.
rank(A+B)≤rank(A)+rank(B), (5)
Corollary 2.1. Given any M ∈ M (n), the corresponding
T
we can deduce that rank(M)≤2. vector x∈Rn, perpendicular to 1, can be calculated as:
Moreover, since the rank of any skew-symmetric matrix
must be even, rank 1 is not feasible. So we can conclude that, x=Mˆ1. (7)
excepting the case that M is the zero matrix (trivial case), the
Corollary 2.2. M∈M (n) can also be expressed as:
rank of M is 2. This completes the proof. T
Note that the formerly referred as ’trivial case’ only occurs 1
M= √ (D 1−1D ) , ˆ1⊥xˆ, (8)
when the time of arrival is the same for all the sensors, i.e n x x
sensors are geometrically placed on a sphere with center in √ √
since xˆ1(cid:62) =D 1/ n and ˆ1x(cid:62) =1D / n,
the source. x x
Rank deficiency of TDOA matrices means that their rows 1) Singular Value Decomposition: Because M ∈ M (n)
T
and columns are linearly dependent. That is consistent with is a skew-symmetric matrix of rank 2, it has the following
the fact that, in the noise-free case, the full TDOA set can be singular value decomposition (SVD) [33, Supplementary ma-
generated from the non-redundant set using linear equations terial]:
[23]. In fact, in a TDOA matrix, the column j is the TDOA (cid:18) (cid:19)
σ 0
non-redundantsetwhenthesensorjisthereferenceforTDOA M=(uˆ ,−uˆ ) (uˆ ,uˆ )(cid:62) =σ(uˆ ,−uˆ )(uˆ ,uˆ )(cid:62),
2 1 0 σ 1 2 2 1 1 2
measurements. Hereafter, and without loss of generality, we
(9)
will consider the first sensor as the reference for the non
where uˆ and uˆ are orthonormal vectors and σ ≥ 0. Note
1 2
redundant-set.
that the SVD decomposition of M is not unique. Given any
orthogonal2×2matrixR,thevectors(vˆ ,vˆ )=(uˆ ,uˆ ) R
1 2 1 2
C. Bijective mapping for TDOA matrices also represent a valid SVD decomposition:
WeshownextaparticularrepresentationofTDOAmatrices M=σ(vˆ ,−vˆ )(vˆ ,vˆ )(cid:62) =σ(uˆ ,−uˆ )(uˆ ,uˆ )(cid:62). (10)
with n−1 parameters that describes their spectral properties 2 1 1 2 2 1 1 2
and also forms the base for our denoising algorithms. Among all possible SVD decompositions of M note that
theorem 2 guarantees that always exists one where uˆ = ˆ1.
Theorem 2. Let H ⊂Rn be a n−1 dimensional hyperplane 1
In such case SVD decomposition can be computed from (6)
of Rn such that span(ˆ1) (cid:42) H, then there always exits an
as stated in the following theorem:
isomorphism between M (n) and H of form:
T
Lemma 1. Given M∈M (n), it admits the following SVD
φ :H −→ M (n) T
H T decomposition:
z(cid:55)−→zˆ1(cid:62)−ˆ1z(cid:62)
M=σ(cid:0)uˆ,−ˆ1(cid:1)(cid:0)ˆ1,uˆ(cid:1)(cid:62) with uˆ = Mˆ1 σ =(cid:107)Mˆ1(cid:107)
Proof:Theorem2statesthatabijectivelinearmapexists (cid:107)Mˆ1(cid:107)
between MT(n) an a n−1 dimensional subset of Rn. Since (11)
the matrix T in (4) can be rewritten as T = zˆ1(cid:62), where
Note that, according the previous lemma, not all rank 2
z =(τ ,··· ,τ )(cid:62), we can define the following linear map:
1 n
skew-symmetric matrices are TDOA-Matrices. In fact, if vˆ
1
φ:Rn −→ MT(n) and vˆ2 in (10) are not coplanar with ˆ1 (i.e. (ˆ1(cid:62)vˆ1)vˆ1 +
z(cid:55)−→zˆ1(cid:62)−ˆ1z(cid:62) (ˆ1(cid:62)vˆ2)vˆ2 (cid:54)=ˆ1), then the resulting matrix is rank 2 and skew-
symmetric but not a TDOA matrix.
that is clearly surjective but not injective. That is, any vector
z(cid:48) = z+αˆ1 represents the same TDOA matrix. Indeed, the
V. TDOADENOISING
kernel of φ is the linear subspace of Rn generated by ˆ1.
In this section we propose a denoising strategy to deal
Since we are looking for an isomorphism, we will restrict
with Gaussian noise in the estimated TDOA measurements,
the domain of φ to ensure injectivity. It yields φ , where
H
H ⊂ Rn is a hyperplane of Rn not containing ker(φ). Note deriving a closed form solution for the proposed optimization
that φ is bijective as H ⊂ Rn keeps the surjectivity of φ problem. This solution is also compared with the Gauss-
H
Markov Estimator.
and, since ker(φ )={0}, the function is also injective.
H
Hereafter, we will only consider the particular case of φ
H
for the hyperplane H = ker(φ)⊥. It yields the following A. Denoising Strategy
expression for any M ∈MT(n): We assume now that each TDOA measurement is contami-
M=(cid:0)xˆ1(cid:62)−ˆ1x(cid:62)(cid:1) , x⊥ˆ1 ,x∈Rn (6) nated with uncorrelated Gaussian noise nij =−nji, such that
∆τ˜ = ∆τ +n . Therefore, the measured TDOA matrix
ij ij ij
Choosing x perpendicular to ˆ1 is very convenient, since M˜ ={∆τ˜ }isalsoaskew-symmetricmatrix,sumofanoise-
ij
both it simplifies φ−1 as shown in the corollary 2.1, and it free M ∈ M (n) and a skew-symmetric matrix containing
H T
alsocanbeusedforcalculatethesingularvaluedecomposition noise N={n }:
ij
(SVD) of any M ∈ M (n), as discussed in IV-C1. This M˜ =M+N. (12)
T
5
Because of the noise, M˜ ∈/ M (n) and thus Theorem 1 proposed solution (x∗ and λ∗) satisfies the equations of a
T
is no longer satisfied. Consequently, the rank of M˜ may be critical point, it is the global minimum
higher than two. Nevertheless, we will show that we can Since M˜ is skew-symmetric (M˜ −M˜(cid:62))=2M˜. Therefore,
take advantage of the structure of TDOA matrices in order (18) becomes:
to denoise the measured data.
2M˜ ˆ1−λˆ1
For denoising, we propose finding the closest x∗ = =M˜ ˆ1 (19a)
M∗ ∈ M (n), to the measured matrix M˜, in the sense 2
T 2ˆ1(cid:62)M˜ ˆ1
of the Frobenius norm. This approach yields the following λ= =2ˆ1(cid:62)M˜ ˆ1=0. (19b)
optimization problem: 2ˆ1(cid:62)ˆ1
M∗ = argmin (cid:13)(cid:13)M˜ −M(cid:13)(cid:13)2 . (13) In(19b)weusethefactthatˆ1(cid:62)Aˆ1=0forAbeingaskew-
M∈MT(n) (cid:13) (cid:13)F symmetric matrix. Also, it is interesting to note from (19a)
that x∗ follows the same expression as the one stated in (6)
B. Closed-Form Solution for x in the noise-free case.
A compact expression for M∗ can be easily derived from
Theorem 3. Problem (13) has the following closed form
solution: M∗ =(M˜ 1+1M˜)/n (19a) via (6):
Proof:From(6),thedenoisingproblem(13)isequivalent M∗ =(cid:16)ˆ1,M˜ ˆ1(cid:17)(cid:16)−M˜ ˆ1,ˆ1(cid:17)(cid:62) =M˜ ˆ1ˆ1(cid:62)+ˆ1ˆ1(cid:62)M˜ =
to the following constrained convex optimization problem:
=(M˜ 1+1M˜)/n. (20)
minimize (cid:13)(cid:13)M˜ −(cid:0)xˆ1(cid:62)−ˆ1x(cid:62)(cid:1)(cid:13)(cid:13)2
x (cid:13) (cid:13)F (14) This completes the proof.
subjectto ˆ1(cid:62)x=0. Sincelemma1relates(6)withSVD,theproposeddenoising
approach can be considered as a version of Eckart-Young-
Using the definition of Frobenius norm (cid:107)A(cid:107)2 = tr(AA(cid:62)),
F Mirsky theorem [34], [35] constrained to TDOA matrices.
and trace properties tr(AB)=tr(BA) and tr(A)=tr(A(cid:62))
we rewrite the cost as:
C. Equivalence with the Gauss-Markov Estimator
(cid:13)(cid:13)M˜ −(cid:0)xˆ1(cid:62)−ˆ1x(cid:62)(cid:1)(cid:13)(cid:13)2 = By operating in (20), each element (i,j) of the denoised
(cid:13) (cid:13)F matrix M∗ is obtained as follows:
=tr(cid:18)(cid:104)M˜ −(cid:0)xˆ1(cid:62)−ˆ1x(cid:62)(cid:1)(cid:105)(cid:104)M˜ −(cid:0)xˆ1(cid:62)−ˆ1x(cid:62)(cid:1)(cid:105)(cid:62)(cid:19) (cid:40)1 (cid:32)(cid:88)n (cid:33)(cid:41)
M∗ ={∆τ∗}= ∆τ +∆τ . (21)
(cid:16) (cid:17) ij n ik kj
=2 x(cid:62)x−x(cid:62)ˆ1ˆ1(cid:62)x−ˆ1(cid:62)M˜(cid:62)x+ˆ1(cid:62)M˜ x + k=1
(cid:16) (cid:17) (cid:16) (cid:17) The closed-form in (21) is identical to the one reported in
+tr M˜ M˜(cid:62) =f x;M˜ . (15) [23, eq.(14)] as the Gauss-Markov estimator of the TDOA
measurements, so that all the properties there can be extrapo-
Tosolvetheconstrainedproblem(14)weusethemethodof
lated to this work. This is not surprising as the least-squares
Lagrange multipliers, resulting in the following unconstrained
costof(13)isoptimalforGaussiannoise.Thesamedenoising
equivalent:
result was found in [25] by projecting TDOA measurements
x∗ = argmin [Λ(x;λ)], (16)
into the set of “feasible” bivectors in a least-squares sense.
x,λ
Under the assumption of Gaussian noise, we can conclude
where λ is the Lagrange multiplier and
that [23], [25] and our denoising result in (21) are completely
(cid:16) (cid:17)
Λ(x;λ)=f x;M˜ +λˆ1(cid:62)x. (17) equivalent.
We find extrema in (17) by taking first derivatives with
VI. ROBUSTTDOADENOISING
respect to both x and λ and solving the following system:
In some application scenarios, the assumption of uncorre-
∇Λ(x;λ)=ˆ0 ⇒ lated white noise made in section V is fully unrealistic. In
(cid:40)4x(cid:62)(cid:0)I−ˆ1ˆ1(cid:62)(cid:1)+2ˆ1(cid:62)(cid:16)M˜ −M˜(cid:62)(cid:17)+λˆ1(cid:62) =ˆ0(cid:62) cases where the noise is correlated with the signal, measure-
mentsarepronetocontainoutliersintheTDOAmeasurements
ˆ1(cid:62)x=0
duetospuriouspeaksinthecorrelation.Forsuchcases,amore
complete model for the measured matrix is:
(cid:16) (cid:17)
M˜ −M˜(cid:62) ˆ1−λˆ1
x∗ = M˜ =M+N+S, (22)
2
ˆ1(cid:62)(cid:16)M˜ −M˜(cid:62)(cid:17)ˆ1 where M∈MT(n), N is a skew-symmetric matrix contain-
λ∗ = . (18) ing Gaussian noise, much like in (12), and the new matrix
2ˆ1(cid:62)ˆ1 S models the addition of all the outliers. Since the number
Given that the objective function is strictly convex, the of outliers is usually small as compared with the number of
solution to the system is unique, and therefore because the measurements, we will assume S to be sparse and unknown.
6
In order to denoise M˜, we propose solving the following From now on, we will refer to this algorithm as Robust
optimization problem, finding both matrices M and S: DeN.
(cid:13) (cid:13)2
minimize (cid:13)M˜ −M−S(cid:13)
(cid:13) (cid:13)
M,S F VII. MISSINGDATARECOVERY
(23)
subjectto M∈M (n)
T
A. Recovery Strategy
(cid:107)S(cid:107) <2k,
0
In real scenarios, there may be situations where some of
where k is the maximum number of outliers supposed to be
the elements of M˜ might not be available (for instance, due
present in the TDOA measurements.
to communications failure) or even when they are available,
Robust denoising in (23) is a non-convex optimization
there are reasons to avoid using them (for example, due
problem with constraints that are not even differentiable. This
to a priori knowledge of unreliable measurements, or when
kind of optimization problems have been explored in Robust
calculating the whole redundant set is computationally too
PCA (RPCA) [36] or robust low-rank factorizations such
demanding) [32]. In such cases, we want to be able to avoid
in GoDec [37]. Despite TDOA matrices are low-rank, these
some measurements, thus performing estimations when part
algorithms are not well suited here as they do not include all
of the values in M˜ are missing.
the algebraic constraints in TDOA matrices.
Inordertosolve(23),weproposeaniterativealgorithm,in- In this section, we address the matrix completion problem
spiredinGoDec.Itconsistsofanalternationmethodinwhich ([38],[39])forTDOAmatrices.Weassumethatinameasured
M and S are obtained in turns, with close-form solutions for TDOA matrix M˜, some of its elements are unknown, and
these two steps (we use a subindex t to denote the iteration the rest are contaminated with additive Gaussian noise. We
count): take advantage of the redundancy present in TDOA matrices
to estimate a complete denoised TDOA matrix including the
 (cid:13) (cid:13)2
Mt =Ma∈rgMmTi(nn) (cid:13)(cid:13)M˜ −M−St−1(cid:13)(cid:13)F (24) miTsshiengmeantrtrixiesc.ompletion problem is stated as follows:
(cid:13) (cid:13)2
 St =a(cid:107)Srg(cid:107)0m<2ink (cid:13)(cid:13)M˜ −Mt−S(cid:13)(cid:13)F M∗ = argmin (cid:13)(cid:13)(cid:13)L ◦ (cid:16)M˜ −M(cid:17)(cid:13)(cid:13)(cid:13)2 , (26)
M∈MT(n) F
The first sub-problem of (24) is the same as our denoising
problem in (13), therefore M can be updated via (20). Then, where L is a symmetric binary matrix whose element (i,j) is
t
S is updated via entry-wise hard thresholding of M˜ −M . 1 if the TDOA between the sensor i and j is known, being 0
t t
Thus: otherwise. For convenience and without loss of generality, the
elements on the main diagonal of L will be set to 1.
 (cid:16) (cid:17) (cid:16) (cid:17) Solving (26) is equivalent to finding the full TDOA matrix
Mt = M˜ −St−1 ˆ1ˆ1(cid:62)+ˆ1ˆ1(cid:62) M˜ −St−1 whoseelementsbestfittheavailableelementsofM˜.Notethat,
(25)
 S =P (cid:16)M˜ −M (cid:17) L ◦ (M˜ −M)=(M˜L−L ◦ M), where M˜L =(L ◦ M˜) is
t 2k t the result of setting the unknown elements of M˜ to zero.
whereP (X)isanfunctionwhichgeneratesamatrixwiththe
l
samesizeofX,preservingthelelementsofXwiththelargest
absolute value, and making the rest of elements zero. Note B. Closed-Form Solution
that, since X is skew symmetric in our application, the result
Theorem 4. The problem (26) has the following closed form
provided by P2k(·) is also skew symmetric. The convergence solution: M∗ = (cid:0)D +L¯(cid:1)−1M˜ 1 + 1M˜ (cid:0)D +L¯(cid:1)−1
to a local minimum of this algorithm is guaranteed in similar where D = (cid:0)I ◦ LLβ(cid:62)(cid:1) is a n ×Ln diagonLal maβtrix with
circumstances as GoDec [37], as the solutions to both sub- β √
βββ =(cid:0)n−β¯ ,··· ,n−β¯ (cid:1)(cid:62) = nLˆ1 as its main diagonal.
problems in (25) are solved globally. 1 n
β¯ is the number of missing measurements with the sensor i.
So, the proposed robust denoising algorithm is shown in i
Alg. 1. Proof: Using Corollary 2.2, problem (26) is rewritten as
ARlegqouriirteh:mM˜1,Rk,ob(cid:15)ust denoising. minixmize (cid:13)(cid:13)(cid:13)(cid:13)M˜L− L ◦ (Dx√1n−1Dx)(cid:13)(cid:13)(cid:13)(cid:13)2 (27)
F
Ensure: M∈MT(n), (cid:107)S(cid:107)0 <2k, subjectto ˆ1(cid:62)x=0.
1: M0 =M˜ ; S0 =0 ; t=0
2: while (cid:107)M˜ −Mt−St(cid:107)2F/(cid:107)M˜(cid:107)2F >(cid:15) do Since 1 is the identity element of the hadamard product and
3: t=t+1 D is a diagonal matrix, we can rewrite (27) as:
x
4: Mt =(M˜ −St−1)ˆ1ˆ1(cid:62)+ˆ1ˆ1(cid:62)(M˜ −St−1)
56:: endSwt =hilPe2k(M˜ −Mt) minixmize (cid:13)(cid:13)(cid:13)(cid:13)M˜L− (DxL√−nLDx)(cid:13)(cid:13)(cid:13)(cid:13)2 (28)
F
7: return Mt, St subjectto ˆ1(cid:62)x=0.
7
Operating in a similar manner to (15) we get:
(cid:13)(cid:13)(cid:13)(cid:13)M˜L− (DxL√−nLDx)(cid:13)(cid:13)(cid:13)(cid:13)2 = n2 tr(cid:0)DxLL(cid:62)Dx(cid:1)− x∗ =n(cid:0)Dβ +L¯(cid:1)−1M˜Lˆ1. (38)
F Finally,thesolutionofproblem(26)canbecalculatedfrom
2 2 (cid:16)(cid:104) (cid:105) (cid:17)
− tr(D LD L)+ √ tr M˜ −M˜(cid:62) D L + (38) using (6):
n x x n L L x
(cid:16) (cid:17)
+tr M˜ M˜(cid:62) . (29) M∗ = (cid:0)D +L¯(cid:1)−1M˜ 1 + 1M˜ (cid:0)D +L¯(cid:1)−1 (39)
L L β L L β
Usingtheidentityx∗(A ◦ B)y=tr(cid:0)Dx∗ADyB(cid:62)(cid:1)weget: This completes the proof.
From now on, we will refer to this algorithm as MC. It
(cid:13)(cid:13)(cid:13)(cid:13)M˜L− (DxL√−nLDx)(cid:13)(cid:13)(cid:13)(cid:13)2F = n2x(cid:62) (cid:0)I ◦ LL(cid:62)(cid:1) x iims npootretawnotritnhfyortmoactoiomnmaebnotutthtahtetrheecomveartraibxil(itDyβof+mL¯is)sicnogndtaaitnas:
− 2x(cid:62) (cid:0)L ◦ L(cid:62)(cid:1) x+2ˆ1(cid:62) (cid:16)(cid:104)M˜ −M˜(cid:62)(cid:105) ◦ L(cid:62)(cid:17) x+ if it is full-rank, then the solution of (26) is unique and if
n L L (D +L¯) is rank-deficient, missing data is not recoverable
β
(cid:16) (cid:17) (cid:16) (cid:17)
+tr M˜ M˜(cid:62) =g x;M˜, L (30) uniquely without any further assumption.
L L
Furthermore, in the absence of missing data,
and finally: n(D +L¯)−1 = I, hence the matrix completion solution in
β
(39) becomes the solution of the denoising problem stated in
(cid:13)(cid:13)(cid:13)(cid:13)M˜L− (DxL√−nLDx)(cid:13)(cid:13)(cid:13)(cid:13)2 = (20).
F
2 (cid:16) (cid:16) (cid:17) (cid:17)
x(cid:62)D x−x(cid:62)Lx+nˆ1(cid:62) M˜ −M˜(cid:62) x +
n β L L VIII. ROBUSTTDOADENOISINGWITHMISSINGDATA
(cid:16) (cid:17)
+tr M˜ M˜(cid:62) =g(x;M˜, L). (31)
L L InthissectionweaimtocombinetheresultsofsectionsVI
and VII, addressing the more general case in which both out-
Itisimportanttonotethatequations(15)and(31)areiden-
ticalwhenthereisnomissingdatainM˜ (i.eL=1=nˆ1ˆ1(cid:62) liers and missing data are considered. Therefore, the problem
is a combination of (23) and (26) defined as:
and D =nI).
β
We use the method of Lagrange multipliers to express (27)
(cid:13) (cid:16) (cid:17)(cid:13)2
as the following unconstrained optimization problem: minimize (cid:13)L ◦ M˜ −M−S (cid:13)
(cid:13) (cid:13)
M,S F
Λ(x;λ)=g(cid:16)x;M˜, L(cid:17)+λˆ1(cid:62)x. (32) subjectto M∈MT(n) (40)
(cid:107)S(cid:107) <2k
0
By taking derivatives we obtain the following system:
S=L ◦ S.
∇Λ(x;λ)=ˆ0 ⇒
In the same way as in section VI, (40) can be solved
(cid:40) (cid:16) (cid:17)
2x(cid:62)(D −L)+ˆ1(cid:62) M˜ −M˜(cid:62) +λˆ1(cid:62) =ˆ0 by alternatively solving the following two subproblems until
n β L L (33)
convergence:
ˆ1(cid:62)x=0.
Since ˆ1(cid:62)x = 0 implies that 1x = ˆ0, we substitute in (33)
(cid:13) (cid:16) (cid:17)(cid:13)2
obtaining: M = argmin (cid:13)L ◦ M˜ −M−S (cid:13) (41a)
t (cid:13) t−1 (cid:13)
n2 (cid:0)Dβ +L¯(cid:1)x=(cid:16)M˜L−M˜(cid:62)L(cid:17)ˆ1−λˆ1, (34) St =Mar∈gMmTin(n) (cid:13)(cid:13)(cid:13)L ◦ (cid:16)M˜ −Mt(cid:17)−S(cid:13)(cid:13)(cid:13)2 . F (41b)
where (cid:0)L¯ =1−L(cid:1) (logical not operator over all elements of (cid:107)S(cid:107)0<2k F
L). Note that: (D +L¯) is symmetric and, furthermore, ˆ1 is The subproblem (41a) is equivalent to the missing
β
one of its eigenvectors. data problem solved in section VII but considering
(cid:0)D +L¯(cid:1)ˆ1= βββ√+βββ¯¯¯ =nˆ1. (35) Mit˜hLas=a(cLlo◦seMd˜fo−rmSts−o1lu).tiTonh:erefore, according to theorem 4,
β n
Therefore, if the two terms of (34) are multiplied on the right M∗ =(D +L¯)−1(L ◦ M˜ −S )1+
by ˆ1(cid:62) we get: t β t−1
+1(L ◦ M˜ −S )(D +L¯)−1. (42)
(cid:16) (cid:17) t−1 β
2ˆ1(cid:62)x=ˆ1(cid:62) M˜ −M˜(cid:62) ˆ1−λ. (36)
L L
Since (41b) is of the same form as the second subproblem in
Then applying to (36) the fact that (24), it can also be solved by entry-wise hard thresholding of
L ◦ (M˜ −M ).
ˆ1(cid:62)x=0 and ˆ1(cid:62)(M˜ −M˜(cid:62))ˆ1=0, (37) t
L L The pseudocode shown in Alg. 2 summarizes the proposed
we can conclude that λ=0. Thus, the solution of (27) is: algorithm for the general case.
8
Algorithm 2 Robust denoising with missing data. (a)Robust DeN (b)Gauss-Markov (c)Onlynon-redundantset
Require: M˜, L, k, (cid:15)
80 80 80
Ensure: M∈M (n), (cid:107)S(cid:107) <2k,
T 0 60 60 60
1: Dβ =I ◦ LL(cid:62) 40 40 40
2: Q=(Dβ +L¯)−1 20 20 20
5634:::: Mwh0itMl=e=t(cid:107)Mt=M˜˜+Q;−1S(LM0 =◦t−M0˜S;−tt(cid:107)=2FS/t0−(cid:107)1M˜)1(cid:107)2F+<1((cid:15)Ld◦o M˜ −St−1)Q 00#1o2f3o4u5t6lie7r8s91010100SD1o0f-1noi1s0e2-2[0ms1]0-300#1o2f3o4u5t6li3e70r8s910 100SD1o0f4-10noi1s0e-2[ms1]0-300#15o20f3o4u5t6lie7r8s910 10600SD1o0f-1noi1s0e-2[ms1]0-3
7: St =P2k(M˜ −Mt) Fig.2:Robustdenoisinginsyntheticdata:SNRindB,higherisbetter.
8: end while
9: return Mt, St
B. Evaluation of Robust TDOA Denoising
Note that in line 2 the matrix Q = (Dβ +L¯)−1 can be In this first experiment, we evaluated the performance of
precalculated in order to get an efficient implementation of theRobust DeNalgorithmproposedinsectionVI,imposing
the algorithm. that some TDOA values were outliers. To simulate this, we
From now on, we will refer to this algorithm as Robust randomly chose some measurements (between 0 and 10) and
DeN+MC. replaced them with a zero-mean Gaussian distributed noise,
with a standard deviation of 0.1 ms. It is worth mentioning
IX. EXPERIMENTSWITHSYNTHETICDATA thattheoutliervaluescalculatedthatwayarenotrelatedatall
Inthissectioncomputersimulationswillbeusedtocompare to the real TDOAs, thus being true outliers. The parameter k
theproposedalgorithmswithsomeofthealternativesexisting of the proposed algorithm, which fixes the maximum number
in the state of the art. of identifiable outliers, was set to 8.
For evaluating the Robust DeN and Robust DeN+MC 1) SNR Improvements Evaluation: Fig. 2a shows the SNR
algorithms, two different metrics will be used: values for the Robust DeN algorithm when modifying the
• The Signal-to-Noise-Ratio SNR [dB] of the noisestandarddeviationandthenumberofoutliers,compared
non-redundant set referenced to the first sensor with that obtained by the Gauss-Markov estimator (Fig. 2b),
(10log((cid:80)n (cid:107)∆τ (cid:107)2/(cid:80)n (cid:107)∆τ∗ −∆τ (cid:107)2)). This and also when only the non redundant set is used, i.e. not
i=1 i1 i=1 i1 i1
is an application independent metric (where ∆τ∗ is using the redundancy of TDOA measurements (Fig. 2c).
i1
the estimation of ∆τ ), that will allow assessing the As predicted in section V, when no outliers are present, the
i1
proposal improvements in the TDOA measurements per performance of the Robust DeN algorithm is the same as
se. Gauss-Markov(seerow0inFigs.2aand2b),henceitreaches
• The localization error, measured as the average distance the Cramer-Rao Bound [23], while being much better than
betweenthesourcegroundtruthpositionandtheposition using no redundancy. Nevertheless, the proposed algorithm
estimatedusinganygivenlocalizationalgorithmbasedon clearly outperforms the other two approaches when outliers
TDOA estimations (such as [16] in our case). This is an are present in the measurements (rows 1 through 10 in the
application dependent metric, that will allow assessing graphics of Fig. 2).
the actual benefits of the proposal in an example of 2) Source Localization Improvements Evaluation: The op-
a real task. Note that our proposal is not restricted timized non redundant set provided by the algorithms applied
to localization and can be used in other applications in Section IX-B1 were used in a localization algorithm using
that could benefit from denoised TDOAs (such as self- [16]. The average localization errors (in mm) are shown in
calibration or beamforming). Fig.3.Again,theRobust DeNalgorithmperformsasGauss-
Markovwhentherearenooutliers,butisclearlysuperiorwhen
outliers are present.
A. Experimental setup
It is also worth mentioning that the behaviour of the robust
For all the synthetic data experiments, a set of 10 sen-
denoising keeps the improvements at roughly the same level
sors (which implies 45 different sensor pairs) and 1 source
for increasing number of outliers present, thus validating
were randomly located. Therefore, 45 different TDOA mea-
the ability of the algorithm to pinpoint and eliminate their
surements were generated per experiment, and independent
presence.
Gaussian noise was added to them, using the same variance
for all the measurements.
C. Evaluation of Missing Data Recovery
The sensor locations were uniformly distributed in a cube
of 1 meter side, and the source positions were uniformly In this second experiment, we evaluated the capability of
distributed in a 2 meter side cube. The propagation speed the MC algorithm proposed in section VII to recover missing
of the signal was set to 343.313 m/s. In all the experiments values. For our purposes, the missing TDOA measurements
whereit’srequired,(cid:15)issetto10−10.Toincreasethestatistical were also chosen randomly but, in contrast to the previous
significanceoftheresults,theyareprovidedasaveragesof20 experiment, the matrix positions of the missing measurements
independent runs. were known.
9
(a)Robust DeN (b)Gauss-Markov (c)Onlynon-redundantset former corresponds to a standard deviation of 10−3 ms., and
the latter to 0.2 ms. ii) Two conditions related to the presence
2000 2000 2000
of outliers, imposing the existence of 2 or 6 outliers. iii) A
1500 1500 1500
variable number of missing TDOA measurements, defined as
1000 1000 1000
500 500 500 a percentage of missing TDOA values in the TDOA matrix.
0 0 0 In all cases, the number of outliers is fixed among the full
10#9o8f7o6u5tl4ie3r2s1020010-S3D1o0f-2no4i01s00e-1[ms1]0100#9o86f070o6u5tl4ie3r2s10 81000-S3D1o0f-2noi11s00e0-1[0ms1]0100#9o8f71o26u050tl4ie3r2s10 1104-S300D1o0f-2noi1s0e-1[ms1]00 TsthoDamtOethAiosfsiestthceaonnddsiisstctheaenrndt,ewdsiotmhmetehaesmurreeeaamsluceranestmes,emwnthasyeraebreeitdiosiusntcloaietrrdpseo.dsN,siiob.tleee.
Fig.3:Robustdenoisinginsyntheticdata:Localizationerrorinmm(using to anticipate where the outliers are.
[16]),lowerisbetter. Fig. 6 and Fig. 7 show, respectively, the SNR values, and
thelocalizationerrorfordifferentalgorithms,andfordifferent
(a)MatrixCompletion(MC) (b)Onlynon-redundantset evaluation scenarios.
As it can be seen in Fig. 6a, 6c, 7a, and 7c, when there
80 80
are a low number of outliers (2 in this case), the best results
60 60
areobtainedforlowerk values.However,whenthenumberof
40 40
outliersincrease,(Fig.6b,6d,7band7d,lowkvaluesperform
20 20
worse. So, we can conclude that k must be a number as low
0
%ofmissingmea0suremen5t0s 100 100SD10of-1nois1e0[m-2s] 10-3 %ofmissingme00asuremen5t0s 100 100 SD1o0f-1noise10[m-2s] 10-3 a6sdN,poe7vsbseirbtahlneed,lebs7usdt,iht(iwigshiwtehrortmhthaontroethooebusnetlurivmeerbst)ehrewobhfeehanacvtutihoaeluropoueftrliFceeirgns.t.a6gbe,
0 10 20 30 40 50 60 of missing data increases. It can be clearly seen that the
Fig.4:Missingdatarecoveryinsyntheticdata:SNRindB,higherisbetter. lines corresponding to different values of k are crossing
among them. This seems to indicate that as the missing data
(a)MatrixCompletion(MC) (b)Onlynon-redundantset percentage increases, the number of outliers that we are able
to detect decreases.
2000 2000 Anyway, the results obtained by the Robust DeN+MC
1500 1500 algorithm outperforms the Gauss-Markov estimator, asymp-
1000 1000 toticallyapproachingitwhenthenoiseisveryhigh.Notealso
500 500 that for high values of noise, the noise and the outliers are
1000 1000 practically indistinguishable.
%ofmissingmeasuremen5t0s 0 10-3SD10of-2nois1e0[m-1s] 100 %ofmissingmeasuremen5t0s 0 10-3SD10of-2nois1e0[m-1s] 100 X. EXPERIMENTSWITHREALDATA
200 400 600 800 1000 1200 1400 1600
The aim of this section is to evaluate whether the im-
Fig.5:Missingdatarecoveryinsyntheticdata:Localizationerrorinmm provements obtained in section IX using synthetic data are
(using[16]),lowerisbetter. actually found in real environments. To do this, the proposed
algorithms have been evaluated using audio recordings from
the AV16.3 database [40], an audio-visual corpus recorded in
Fig.4andFig.5show,respectively,theSNRvalues,andthe
localization error for the MC algorithm, when modifying the the Smart Meeting Room of the IDIAP Research Institute, in
Switzerland. The same metrics and localization algorithm of
noisestandarddeviationandthepercentageofmissingTDOA
the previous section has been employed.
values in the TDOA matrix, as compared with using only the
Additionally, the proposed Robust DeN algorithm is
non-redundant set (missing values were set to zero).
compared with other recent state-of-the-art methods in sec-
From the figures, it can be clearly seen that the proposed
tionX-D.Inthatcase,wehaveemployedthesameframework
algorithm can take advantage of the knowledge about which
used in [8], wherein many methods were already compared.
measurementsweremissing,achievingevenbetterresultsthan
In order to make the comparison as fair as possible, in this
whenthepositionsoftheoutlierswereunknown.Forexample,
part the localization algorithm will be the same as in [8].
removing 50% of the full TDOA set of measurements implies
23 missing values of 45 measurements , much more than the
maximum of 10 outliers evaluated in Fig. 2, while keeping A. Experimental Setup
good performance.
The IDIAP Meeting Room (shown in Fig. 8) is a
8.2m×3.6m×2.4m rectangular space containing a centrally
D. Evaluation of Robust TDOA Denoising with Missing Data located 4.8m×1.2m rectangular table, on top of which two
In this third experiment, we evaluated the capability of the circular microphone arrays of 10cm radius are located, each
Robust DeN+MC algorithm proposed in section VIII to face of them composed by 8 microphones. The centers of the two
both outliers and recover missing values. arrays are separated by 80cm and the origin of coordinates is
Toprovideawiderangeofevaluationscenarios,wedefined: locatedinthemiddlepointbetweenthetwoarrays.Adetailed
i) Two conditions related to noise, namely low and high. The description of the meeting room can be found in [41].
10
70 70 23 22
60 60 22 20
21
SNR[dB]345000 SNR[dB]345000 SNR[dB]112890 SNR[dB]111468 G(MNaaout rmsixsi −sCsMoinmagrpk mloeveti aosnu (rMemC)ents)
17
20 20 16 12 Robust DeN+MC (k=3)
Robust DeN+MC (k=5)
100 10 20 30 40 50 100 10 20 30 40 50 150 10 20 30 40 50 100 10 20 30 40 50 Robust DeN+MC (k=7)
%ofmissingmeasurements %ofmissingmeasurements %ofmissingmeasurements %ofmissingmeasurements
(a)SNRwithlownoiseand (b)SNRwithlownoiseand (c)SNRwithhighnoise (d)SNRwithhighnoise
2outliers 6outliers and2outliers and6outliers
Fig.6:Algorithmevaluationinsyntheticdata:SNRindB.
500 1000 800 1100
400 800 700 1000
900
Error[mm]230000 Error[mm]460000 Error[mm]456000000 Error[mm]678000000 G(MNaaout rmsixsi −sCsMoinmagrpk mloeveti aosnu (rMemC)ents)
500
100 200 300 400 Robust DeN+MC (k=3)
Robust DeN+MC (k=5)
00 10 20 30 40 50 00 10 20 30 40 50 2000 10 20 30 40 50 3000 10 20 30 40 50 Robust DeN+MC (k=7)
%ofmissingmeasurements %ofmissingmeasurements %ofmissingmeasurements %ofmissingmeasurements
(a)Localizationerrorwith (b)Localizationerrorwith (c)Localizationerrorwith (d)Localizationerrorwith
lownoiseand2outliers lownoiseand6outliers highnoiseand2outliers highnoiseand6outliers
Fig.7:Algorithmevaluationinsyntheticdata:Localizationerrorinmm([16]isusedforsourcepositionestimationfromtheoptimizedTDOAvalues).
3,60 m TABLEI:Robustdenoisingperformanceinrealdata
SNR(dB) AverageLocalizationerror(mm)
Robust DeN 27.46 354
Microphone .
Arrays Gauss-Markov 23.19 515
m Onlynon-redundantset 17.83 858
0
2
8,
Table B. Evaluation of Robust TDOA Denoising
In this experiment, all the microphone pairs have been
considered, hence 120 TDOA values have been computed for
(a)IDIAProomlayoutshowingthe (b)Positionsevaluatedin
each frame.
centeredtable,andthemicrophones therealdataexperiments.
arrangedintwocirculararrays. Onlytherelevantroom IntableIweshowanexampleoftheresultsfortheRobust
sectionisshown. DeNalgorithmwithk =10.Asitalsohappenedwithsynthetic
Fig.8:IDIAPSmartMeetingRoom:experimentaldetails. data, in this case the proposed algorithm outperforms the
Gauss-Markov estimator, yielding great improvements in both
SNRandlocalizationprecision.Fig.9showsthattheselection
The audio recordings are synchronously sampled at of k is not very critical in this dataset as improvements over
16 KHz, and the complete database along with the corre- Gauss-Markov are obtained for a wide range of k.
sponding annotation files containing the recordings ground These results are the baseline for the experiments with
truth (3D coordinates of the speaker’s mouth) is fully accessi- missing data described in the next subsection.
ble on-line at [42]. It is composed by several sequences from
which we are using sequence 01, with a single male speaker
generating digit strings in 16 positions (which can be seen C. Evaluation of Robust TDOA Denoising with Missing Data
as small circles in Fig. 8b), distributed along the room. The
In the second experiment with real data, we randomly
sequence duration accounts for 208 seconds in total, with 823
remove a set of TDOA measurements. Fig. 9 shows the ob-
ground truth frames.
tained results. The dotted lines correspond to the performance
The TDOA measurements ∆τ˜ij, from which the measured (SNR and localization error) achieved by the Gauss-Markov
TDOA matrix M˜ is built, where estimated using the highest estimator when there are no missing measurements. The solid
peak of the GCC-PHAT function[26]. lines with circular marks are the results obtained by the MC
As in a real scenario outliers are common but difficult to algorithm described in section VII.
anticipate or enforce, the sweep over noise levels and the On the other hand, the solid lines with
number of outliers that we performed with synthetic data are squared/triangular/diamond marks correspond with the
not feasible. Therefore, in our experiments with real data, results of the Robust DeN+MC algorithm presented in
we will only provide the SNR values and localization errors section VIII. The different colors/shapes indicate different
obtained after using each algorithm. values of the hyperparameter k.