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
Knowledge of the room shape can beneﬁt a large num-
ber of applications. For example, in the creation of
personal sound zones  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 , it is possible to exploit the geometric dual-
ity of this representation to estimate the room bound-
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.
d2?/cd2?/c d3?/c d3?/c
Figure 2: Ambiguity in the echoes labels due to diﬀerent order of
arrival of wall reﬂections
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
where the two vectors Rmand Snare given by
Sn= [1 XnYnZnsT
Collecting all the squared distances dm,n for the pairs
(m, n)leads to the distance matrix
where R= [R1,...,RM]and S= [S1,...,SN]are
known microphone and unknown image source position
Can we exploit the data model in (5) to avoid an NP-
hard problem and ﬁnd feasible echoes combinations?
Region that contains true combinations
Has unique elements?
Try columns sequentially
Sorting (Ascending order)
Figure 3: Flow of the proposed greedy method for sorting acoustic
How we can identify feasible combinations?
Orthogonal projection (Subspace ﬁltering) →f(c) = kΠN(R)˜
2∀c∈[1, . . . , N M],ΠN(R)RT=0.
Rank Constraint for Euclidean Distance Matrices (EDMs) →rank(˜
Reconstruction results for 3D rectangular rooms. The proposed greedy method performs orders of magnitude
faster than the pure graph-based method , with comparable estimation accuracy.
Indexes of sorted columns
⊥Dc|| / ||Dc||
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
Peaks position uncertainty (σ) [m]
Figure 5: Estimation error comparison for
M= 9 and N= 6.
Number of Microphones
Relative Computational Time
Graph−based (Subspace Filtering)
Figure 6: Comparison of computation time
between the graph-based methods and greedy
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
•The proposed greedy method provides an accuracy
comparable with the current state-of-the-art method
based in graph theory, but at a reduced
•Eﬀects of uncertainties in the distance
measurements are shown through numerical
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 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.
 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.