Minimal Structure and Motion Problems for TOA and TDOA

Measurements with Collinearity Constraints

Erik Ask, Simon Burgess and Kalle

Astr

Centre for Mathematical Sciences, Lund University, Lund, Sweden

Keywords:

Structure from Sound, TOA, TDOA, Minimal Solvers.

Abstract:

Structure from sound can be phrased as the problem of determining the position of a number of microphones

and a number of sound sources given only the recorded sounds. In this paper we study minimal structure from

sound problems in both TOA (time of arrival) and TDOA (time difference of arrival) settings with collinear

constraints on e.g. the microphone positions. Three such minimal cases are analyzed and solved with efﬁcient

and numerically stable techniques. An experimental validation of the solvers are performed on both simulated

and real data. In the paper we also show how such solvers can be utilized in a RANSAC framework to

perform robust matching of sound features and then used as initial estimates in a robust non-linear least-

squares optimization.

1 INTRODUCTION

Sound ranging or sound localization has been used

since world war I, to determine the sound source us-

ing a number of microphones at known locations and

measuring the time-difference of arrival of sounds.

The same mathematical model is today used both for

applications based on acoustics and radio and both

for signal strength or time-based information such as

time of arrival (TOA) or time differences of arrival

(TDOA), or a combination thereof. Although such

problems have been studied extensively in the litera-

ture in the form of localization of e.g. a sound source

using a calibrated detector array, the problem of cali-

bration of a sensor array using only measurement, i.e.

the initialization problem for sensor network calibra-

tion, has received much less attention. One technique

used for sensor network calibration is to manually

measure the inter-distance between pairs of micro-

phones and use multi-dimensional scaling to compute

microphone locations, (Birchﬁeld and Subramanya,

2005). Another option is to use GPS, (Niculescu and

Nath, 2001), or to use additional transmitters (radio

or audio), close to each receiver, (Elnahrawy et al.,

2004; Raykar et al., 2005; Sallai et al., 2004). Sensor

network calibration is treated in (Biswas and Thrun,

2004). In (Chen et al., 2002) it is shown how to esti-

mate additional microphones, once an initial estimate

of the position of some microphones are known. In

(Thrun, 2005) the far ﬁeld approximation is used to

initialize the calibration of sensor networks. Ini-

tialization of TOA networks has been studied in

(Stew

enius, 2005), where solutions to the minimal

case of three transmitters and three receivers in the

plane is given. The minimal case in 3D is determined

to be four receivers and six transmitters for TOA, but

this is not solved. Initialization of TDOA networks

is studied in (Pollefeys and Nister, 2008), where solu-

tions were give to two non-minimal cases of ten trans-

mitters and ﬁve receivers, whereas the minimal solu-

tion for far ﬁeld approximation in this paper are six

transmitters and four receivers. In (Wendeberg et al.,

2011) a TDOA setup is used for indoor navigation

based on non-linear optimization, but the method can

get stuck in local minima and is dependent on initial-

ization.

In this paper we will study the effects of restrict-

ing one set of synchronized sensors to a line (we will

assume receivers). For TOA measurements applica-

tions could be to determine all positions by travelling

along a line and measuring distances to ﬁxed posi-

tions. In TDOA it could be used to calibrate linear

sensor-arrays, easily setting up scenarios for indoor

navigation by placing sensors along a wall. A more

complicated setting could be if the line synchroniza-

tion could be emulated, by for instance using known

periodic signals from the transmitters, to again esti-

mate positions of both a receiver and known transmit-

ters by a linear motion. For example a moving car in

range of cellular antennas.

425

Ask E., Burgess S. and Åström K. (2013).

Minimal Structure and Motion Problems for TOA and TDOA Measurements with Collinearity Constraints.

In Proceedings of the 2nd International Conference on Pattern Recognition Applications and Methods, pages 425-429

DOI: 10.5220/0004202504250429

 SciTePress

2 PROBLEM FORMULATION

We will denote the position of transmitter i as z

and position of receiver j as m

. As is shown in

the next section one may without loss of general-

ity assume planar conﬁgurations, i.e. z

= (x

, y

) and

= (u

, v

). Throughout the paper D = [d

i j

] is a

matrix with time measurements between transmitters

and receivers. Since we are only interested in measur-

ing distances or relative distances it is indifferent what

type of sensor is placed collinearly, assuming syn-

chronization is on line, but for consistency we shall

assume the receivers are placed collinearly for our ar-

guments.

2.1 Linear Restriction

The main purpose of the linear restriction is to reduce

the number of unknowns, and hence the number of

necessary equations. This will reduce the size of the

minimal case and has the dual advantage of more sta-

ble numerical performance and a reduced requirement

in the number of transmitters and receivers needed.

The cost being the reduced usability in that either

transmitters or receivers need to be placed in linear

constellations. Since we only measure distances from

a common line, the results presented hold in all di-

mensions, with the dimension of the solution set being

of size 2 smaller than the dimension of the problem.

To see this consider any measurement d

i j

= ||z

−m

and let the ﬁrst dimension, denoted u be deﬁned by

the collinear receivers. Then z

has an orthogonal de-

composition z

⊥

with

= u

a unique point

on the u axis. Then u

can be uniquely determined

from any two known measurement points on the line

but only the length of

⊥

i j

. The ambiguity in direc-

tion gives us several equally feasible solutions. In 2D

there are 2 directions that with known distance ﬁxates

two points (0D). In 3D the perpendicular vector can

be rotated around the line 360 degrees tracing out a

circle (1D). We summarize the above in a theorem.

Theorem 1. The Structure and motion problem for

linear motion based on measurements to reference

points is equivalent in all dimensions ND up to the

dimension of the solution set.

Another important aspect is that the linear conﬁgura-

tion most likely is a degenerate case of any full mini-

mal solver. In (Stew

enius, 2005) It is shown that the

solver for the planar unrestricted case for TOA the lin-

ear setup causes the algorithm to become unstable and

it is impossible to solve for the non-line placed sen-

sors without adapting the method for handling null

spaces. There are no minimal polynomial solvers for

TDOA in any dimension, but theorem 1 essentially

states that the linear case results in similar considera-

tions on null spaces.

2.2 TOA

The TOA case occurs when time synchronization is

possible between transmitters i and receivers j. By

our assumptions this implies that all distances d

i j

are

known.

With k receivers are placed on a line and all n

transmitters are unrestricted we get kn measurements

and 2n+k−1 unknowns, where one receiver is placed

in origo and the remaining receivers placed on the ﬁrst

axis.The minimal case is then given by the smallest

possible integer solution to

kn = 2n +k − 1 (1)

easily conﬁrmed to be k = 3 and n = 2. Using what

we know from Theorem 1 we now have

Lemma 1. The minimal case for linear TOA in N ≥ 2

dimensions is 3 receivers and 2 transmitters, and has

a N − 2 dimensional solution set.

To derive the solution we have, since by assumption

= 0 for each measurement d

i j

that

i j

: d

i j

= x

− 2x

+ u

+ y

. (2)

This gives a total of 6 equations and 6 unknowns since

we can set u

= 0. Forming the two combinations

− E

) − (E

− E

) and (E

− E

) − (E

−

) gives

− d

+ d

= u

(2x

− 2x

)

− d

+ d

= u

(2x

− 2x

)

, (3)

and hence that

− d

+ d

− d

+ d

, (4)

giving us the possibility to exchange u

for a constant

times u

. A second order equation containing only u

can then be obtained by

− d

+ d

− d

+ d

− E

) − (E

− E

)

with u

substituted in (E

− E

). This polynomial

is trivial to solve and the remaining variables can be

obtained by back substitution in intermediate results.

2.3 TDOA

The motivation for TDOA is to avoid the restriction of

synchronization between transmitters and receivers.

In this setting it is not possible to directly transform

measured times to distances as it is unknown at what

ICPRAM2013-InternationalConferenceonPatternRecognitionApplicationsandMethods

426

point in time the signal was originally transmitted.

By instead imposing a restriction that all collinear re-

ceiver are synchronized we can instead look at the dif-

ference in time of arrival. First the relation in equa-

tion 2 is modiﬁed as to account for the ambiguity by

for each transmitter i introducing an unknown offset

i j

− u

)

+ (y

− v

)

+ o

. (5)

The following lemma gives the minimal cases under

these settings

Lemma 2 . The minimal case in N ≥ 2 dimensions

with no synchronization between transmitters and re-

ceivers, but synchronized receivers is either 2 trans-

mitters and 5 receivers or 3 transmitters and 4 re-

ceivers, and has a N − 2 dimensional solution set.

Proof: We place the receivers on the ﬁrst axis and

one receiver in origo. As all k receivers are assumed

synchronized we have no new parameters and we still

get k − 1 unknowns. For the n transmitters we now

have both the unknown spatial coordinates as well as

the offset giving us 3n unknowns. As before we get

kn equations. It is simple to verify that k = 5, n = 2

and k = 4, n = 3 are the minimal integer solutions

to k − 1 + 3n = kn. The independence of dimension

follows directly from theorem 1

In the subsequent discussions we will refer to the

above situations as (5, 2) and (4, 3) respectively. The

relation between measurements and positions as given

in equation 5 are not on polynomial form and hence

can not be solved directly by polynomial solvers. By

ﬁrst eliminating the square root one obtains

i j

− o

)

= (x

− u

)

+ (y

− v

)

which with v

= 0 can be written as

i j

− 2d

i j

+ o

= x

− 2x

+ u

+ y

. (6)

By subtracting any two such relations for any ﬁxed

i effectively eliminates both the o

, x

and y

terms.

We choose again to set u

= 0 and will use the corre-

sponding equations to subtract obtaining

i j

− d

− 2o

i j

− d

) + 2x

− u

= 0 . (7)

If we interpret this as a linear system in the monomial

, x

and 1 we get for each transmitter j the system







− d

+ d

− d

+ d

− d

+ d

− d

+ d











−2o

−1





=0,

(8)

for the (5, 2) case and an equivalent system with the

last line in the matrix removed for the (4, 3) case. By

basic linear algebra such systems have non-trivial so-

lutions exactly when the determinant of the matrix is

zero. For the over determined (5, 2) case this must

hold for all 3 × 3 sub matrices. In the (4, 3) case

we have a total of 3 square matrices and hence as

many determinants. The determinants form polyno-

mial equations in the unknowns u

, u

and u

. This

means we have reduced our problem from 12 equa-

tions in 12 unknowns to just solving 3 equations in

3 unknowns. For the (5, 2) case we get 2 rectangu-

lar matrices with a total of 8 sub determinants for the

unknowns u

, u

and u

. A subset would be sufﬁ-

cient, but as more equations will be generated later in

the solution algorithm in practice all are used. Again

the number of unknowns and equations are reduced.

From 10 unknowns and 10 equations to 4 unknowns

and 4 to 8 equations. Both reductions are important

for keeping the size of the problems manageable when

solving them.

3 SOLVING POLYNOMIAL

SYSTEMS

For the (3, 2) case solving the system is a matter of

solving a series of 1 variable 2nd degree polynomi-

als as described above. For the (4, 3) and (5, 2) cases

solvers based on (Byr

od et al., 2009) were imple-

mented. The technique is based on forming an ex-

panded set of equations, by multiplying the original

equations with a number of monomials, typically low

order monomials up to a certain degree. All expanded

equations are then expressed as a sparse coefﬁcient

matrix C times a monomial vector m, i.e. the equa-

tions are Cm = 0. Using numerical linear algebra it

is possible to calculate the action matrix M of the lin-

ear mapping T

: p 7→ pm

for some monomial m

The solutions to the original equations can then be

calculated from the eigenvectors and eigenvalues of

the action matrix M.

4 EXPERIMENTAL VALIDATION

4.1 Numerical Stability

Receivers were placed randomly in the interval [0, 1]

and transmitters randomly in the square ([0, 1], [0, 1]).

Figure 1, 2 and 3 shows histograms of the error resid-

uals of recovered positions for the (3, 2), (4, 3) and

(5, 2) cases respectively. The residual is the l

norm

between the true receiver positions and the recon-

structed positions given by the minimal solvers. All

MinimalStructureandMotionProblemsforTOAandTDOAMeasurementswithCollinearityConstraints

427

−20 −15 −10 −5 0 5

200

400

600

log10(res)

Figure 1: Residuals for the (3, 2) solver.

solvers have excellent numerical performance, in par-

ticular the (3, 2) solver, which is expected as it is

just a series of one variable 2nd degree solvers. In

a few instances the (5, 2) solver gives high residuals

or outright fail. This is related to a higher sensitive to

both proximity of degenerate cases and due to a larger

problem size being more prone to cancellation errors.

A total of 1000 experiments per solver were run.

−20 −15 −10 −5 0 5

100

200

300

log10(res)

Figure 2: Residuals for the (4, 3) solver.

−20 −15 −10 −5 0 5

100

200

300

log10(res)

Figure 3: Residuals for the (5, 2) solver.

4.2 Real Data

For the experiments with real data 8 microphones

(Shure SV100) were placed on a line along wall in an

ofﬁce and connected to an audio interface (M-Audio

Fast Track Ultra 8R), which is then connected to a

computer. Relatively distinct sounds were generated

by moving around in the room and clapping. The

8 synchronized channels were recorded at 44.1kHz.

Signal processing was performed by a crude inter-

est point detector on each of the eight signals. In-

terest points were deﬁned as edges between periods

Table 1: Reconstructed microphone array (top) compared

with ground truth (bottom) with origo omitted.

Microphone positions (m)

0.34 0.66 1.00 1.40 1.80 2.14 5.05

0.42 0.69 1.02 1.37 1.77 2.09 4.95

with low energy and periods with high energy. Each

interest point was then matched to the other seven

signals using normalized cross-correlation. Thus ap-

proximately 180 hypothetical matches were found in

the dataset. Among the several error sources in the

setup were reﬂections in hard surfaces (walls, books,

shelves, computer monitor), receivers not placed per-

fectly collinearly and non-exact estimate of the speed

of sound. A RANSAC procedure using 50 iterations

randomly selecting points and solver (5-2,4-3) saving

the best hypothesis. Scoring here are how many addi-

tional audio signals were consistent within 2dm other

than the 2 or 3 randomly selected. The ﬁnal result is

obtained using a bundle adjustment (non-linear least

squares) on the found inlier set. Table 1 shows ground

truth and reconstructed coordinates for the points on

the line. Given the error sources and the fact that the

microphones had a diameter of 3 cm the results are

very satisfactory. No ground truth were taken for the

sound sources But the spatial layout (not shown) is

reasonable in regards to the proportions of the ofﬁce.

5 CONCLUSIONS

In this paper we have studied, modelled and solved

three important minimal cases for structure from

sound assuming that e g the microphones are posi-

tioned on a line. For each of the case we present and

publish efﬁcient and numerically stable solvers. Such

solvers could be used in RANSAC schemes to weed

out the outliers in real data or be integrated in the low-

level audio or radio matching schemes. In the paper

we demonstrate the efﬁciency and numerical stabil-

ity on simulated data and demonstrate a small system

using low-level feature detection, matching, RANSAC

and bundling, to enable automatic microphone sensor

array calibration using only synchronized audio as in-

put.

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