Content uploaded by Mario Coutino

Author content

All content in this area was uploaded by Mario Coutino on Apr 13, 2017

Content may be subject to copyright.

GREEDY ALTERNATIVE FOR ROOM GEOMETRY ESTIMATION FROM

ACOUSTIC ECHOES: A SUBSPACE-BASED METHOD

Mario Coutino1, Martin Bo Møller2,3, Jesper Kjær Nielsen2,3, Richard Heusdens1

Delft University of Technology, The Netherlands

2Bang & Olufsen A/S, Denmark

3Aalborg University, Denmark

Introduction

Knowledge of the room shape can beneﬁt a large num-

ber of applications. For example, in the creation of

personal sound zones [1] one needs to know the room

impulse response (RIR) in diﬀerent locations, which

could be modeled if the geometry information of the

room is available.

Figure 1: Motivation for room geometry estimation.

Echoes generated by sound reﬂected from the room

walls carry information about the geometry of the en-

closure. By modeling room reﬂections using virtual

sources [2], it is possible to exploit the geometric dual-

ity of this representation to estimate the room bound-

aries.

Acoustic Echoes Sorting Problem

In instances where multiple microphones, randomly

placed in the room, are used to detect the acoustic

echos in the RIRs, ambiguities arise at the moment of

labeling the echoes according to the wall which pro-

duces them. This problem is illustrated in Fig. 2.

r1

r3

r2

r1r2r3

d1?/cd1?/c

ttt

d2?/cd2?/c d3?/c d3?/c

s

Figure 2: Ambiguity in the echoes labels due to diﬀerent order of

arrival of wall reﬂections

Data Model

The squared distance dm,n for the (m, n)-th

microphone-source pair can be written as

(xm−Xn)2+ (ym−Yn)2+ (zm−Zn)2=dm,n (1)

This can be expressed as an inner product as

RT

mSn=dm,n (2)

where the two vectors Rmand Snare given by

Rm= [rT

mrm−2xm−2ym−2zm1]T∈R5×1,(3)

Sn= [1 XnYnZnsT

nsn]T∈R5×1(4)

Collecting all the squared distances dm,n for the pairs

(m, n)leads to the distance matrix

D=RTS∈RM×N(5)

where R= [R1,...,RM]and S= [S1,...,SN]are

known microphone and unknown image source position

matrices, respectively.

Can we exploit the data model in (5) to avoid an NP-

hard problem and ﬁnd feasible echoes combinations?

Greedy Method

f(c)≤κc

M×NM

M× |C|

f(c1)/k˜

Dc1k2

2≤f(c2)/k˜

Dc2k2

2≤...≤f(c|C|)/k˜

Dc|C| k2

2

Region that contains true combinations

rank(Ec, ǫ)≤5

M×N

ǫ

Has unique elements?

D=

˜

D=

˜

DC=

Yes

Try columns sequentially

Yes

Generate ˜

Dfrom D

Correct combination

Wrong combination

Sorting (Ascending order)

Figure 3: Flow of the proposed greedy method for sorting acoustic

echoes.

How we can identify feasible combinations?

Orthogonal projection (Subspace ﬁltering) →f(c) = kΠN(R)˜

Dck2

2∀c∈[1, . . . , N M],ΠN(R)RT=0.

Rank Constraint for Euclidean Distance Matrices (EDMs) →rank(˜

Ec, )≤5

Results

Reconstruction results for 3D rectangular rooms. The proposed greedy method performs orders of magnitude

faster than the pure graph-based method [3], with comparable estimation accuracy.

100101102103104105106107

10−4

10−3

10−2

10−1

100

Indexes of sorted columns

||ΠR

⊥Dc|| / ||Dc||

σ=1mm

σ=5mm

σ=1cm

σ=3cm

σ=5cm

Figure 4: Columns of ˜

Dsorted by the value

of the projection for diﬀerent noise levels

0.001 0.005 0.01 0.03 0.05

10-2

10-1

100

101

Peaks position uncertainty (σ) [m]

RMSE [m]

(Modified) Graph−Based

SubSpace−based (Greedy)

Figure 5: Estimation error comparison for

M= 9 and N= 6.

0

5

10

15

20

25

30

Number of Microphones

Relative Computational Time

Subspace−based (Greedy)

Graph−based (Subspace Filtering)

Graph−based

567

Figure 6: Comparison of computation time

between the graph-based methods and greedy

approach.

Figure 7: Illustration of a 3D reconstruction of a rectangular room.

Contributions and Conclusion

•A Greedy approach for acoustic echoes labeling

using the complementary orthogonal projection of

the receivers’ location matrix is proposed.

•Perfect echo labeling through subspace ﬁltering in

the noise free case.

•In presence of noise, the combination of the rank

constraint for EDMs, the subspace ﬁltering, and a

sorting strategy allows to greedily label the acoustic

echoes.

•The proposed greedy method provides an accuracy

comparable with the current state-of-the-art method

based in graph theory, but at a reduced

computational cost.

•Eﬀects of uncertainties in the distance

measurements are shown through numerical

experiments.

References

[1] T. Betlehem, et al. "Personal Sound Zones: Delivering interface-free audio to multiple listeners." IEEE

Signal Processing Magazine 32.2 (2015): 81-91.

[2] J.B. Allen, and D.A. Berkley. "Image method for eﬃciently simulating small room acoustics." The Journal

of the Acoustical Society of America 65.4 (1979): 943-950.

[3] I. Jager, R. Heusdens, and N.D. Gaubitch. "Room geometry estimation from acoustic echoes using

graph-based echo labeling." 2016 IEEE International Conference on Acoustics, Speech and Signal

Processing (ICASSP). IEEE, 2016.