Information-based source localization with distinct binaural cues
Patrick DANÈS(1)(1) LAAS-CNRS, Université de Toulouse, CNRS, UPS, Toulouse, France, patrick.danes@laas.fr

Abstract
Audio-motor binaural localization algorithms, which combine directional cues extracted from the sensed signals
with the motion of the sensor, are known to overcome shortcomings such as font-back ambiguity and source
range non-observability. They can be improved by closing the loop from their output to the control inputs of thesensor, i.e., the sensor motor commands. This paper presents an approach, coined “information-based feedback
control”, which drives in real time a binaural head so as to gather information on the location of a static source.
On the one hand, a “greedy” approach moves the head to its next best position. On the other hand, a multi-stepahead scheme determines its most effective path over a receding horizon of size N, by reasoning on average over
yet uncollected audio data. Both methods internally entail the prediction of binaural cues, e.g., ITDs, ILDs or a
combination of both. Some results can be given an elegant commonsense interpretation.
Keywords: Robot audition, Binaural hearing, Source localization, Audio-motor localization, Informationbased feedback control

1

INTRODUCTION

Since the early days of robot audition, sound source localization has been widely acknowledged as the early
fundamental stage required to endow robots with Cocktail Party ability in real environments, be this problem
tackled from the Computational Auditory Scene Analysis viewpoint or by means of standard signal processing
techniques [14]. First approaches relied on binaural sensors; in view of their moderate robustness to changing experimental conditions and of the difficulty to get and exploit accurate models of the scatterers holding
the microphones, microphone arrays were later envisaged [1]. However, the potential benefits of so-called “active” approaches, which seamlessly combine into audio-motor schemes the sensed audio stream with the motor
commands of the sensor, had long been identified [13]. This gave rise to a renewal of interest for binaural
setups, which avoid the engineering cost and burden of multichannel synchronous acquisition. The binaural
active audition paradigm also complies with psychology and neurology approaches which consider binaural audition in humans as an exploratory and purposive process, entailing the combination of hearing and motion, as
well as multiple feedback loops: e.g., from the cognitive level to low-level binaural processing stages (“shaping
the ears”), or from the cognitive level to planned motions (in order to reinforce/weaken hypotheses about themeaning of the scene) [3]. Feedbacks loops do also exist at the very low reflex level. As a matter of fact,
the unconscious interweave of binaural sensing and head motion can well be an information-based sensorimotor
feedback which moves the head so as to decrease the spatial uncertainty on the acoustic environment.
This paper outlines an engineering approach to information-based binaural localization with robots. The aim is
to augment an audio-motor localization scheme, which computes a belief on the relative position of a soundsource with respect to a moving binaural head, with a feedback control that computes in real time the headmotion that maximizes, on average, the spatial information on the source. Before entering into details, weoverview some neighboring problems addressed in the literature.
The early approaches to binaural audio-motor localization on artificial agents noticed the obvious impact of thehead trajectory on the uncertainty associated to the source position estimate. For instance, [11] propose a simulated comparison of various head displacements, which are combined with audio cues inside a particle filter.
More recently, still relying on a particle filter, a binaural robot endowed with vision actively moves so as tomaximize the decrease of entropies between the current and next beliefs on the hidden source position. In [2],the motions of a binaural head obtained by reinforcement learning on the basis of a frequency analysis of the2160

Figure 1. Overview.

Figure 2. Decomposition of active binaural localization into three stages.

sensed signals show that the expected source should be located in the median plane, or “auditory fovea”. The
Monte Carlo approach of [18] drives an agent towards a specific goal while lowering the spatial uncertainty ona sound source by optimizing a criterion which combines the distance to the goal and the entropy of the next
source position belief. All these strategies can be termed “greedy” or “myopic”, as their aim is to get the next
best head motion. Active motion has also been considered in [8] and [12]. In the context of robotics Simultaneous Localization and Mapping (SLAM) with other sensing modalities, greedy active motion strategies havealso been investigated, entailing Shannon entropy, mutual information, or Fisher information matrix, e.g., [4].
Besides, efforts have been made to cope with long-term criteria, that depict the gathered information after theapplication of a sequence of N motor commands. For instance, [20] designs a dynamic programming based
approximate solution to the minimization of the expectation, over the next unknown measurements, of a probabilistic occupation grid. Recently, a Monte Carlo Tree Search solution [15] minimizes the expectation, over themeasurements to appear on a receding horizon, of a criterion made up with approximations of the entropies ofthe source position beliefs over this horizon (themselves expressed as Gaussian sums by means of an extensionof the Gaussian sum filter proposed in [16]). Though this complete and convincing method assumes a countableadmissible set of control inputs, its major drawback is its computation time, so that it comes within planning
rather than feedback control.
The paper is organized as follows. Section 2 provides a mathematical statement of the considered informationbased binaural localization problem. It relies on a three-layer framework made up with short-term directionalsource localization, audio-motor localization, information-based feedback control. The exploration is guided
by virtual exploratory audio cues that can be related to the hidden head-to-source direction by means of ananalytical equation with constant-variance additive noise. Section 3 sketches the proposed mathematical solution.
It is shown that the complexity fundamentally differs depending on whether a greedy (one-step-ahead) or Nstep-ahead problem is addressed. Rather than entering into the mathematical details of the solution detailed
elsewhere, qualitative insights are given. Section 4 presents some results, in the form of sample trajectories
corresponding to the selected exploratory audio cues. Conclusions and prospects end the paper.

2

PROBLEM STATEMENT

2.1 Notations
Consider a binaural head of approximate radius a, endowed with two microphones R1 , R2 (Figure 1). Its associ−−−→
−
−
−
−
−
ated frame is F = (O, →
x ,→
y ,→
z ), with →x pointing downwards, →y = R2 R1 supporting the interaural axis, and
R

R

R

R

R

−−−→
kR2 R1 k→−zR pointing frontwards. An motionless omnidirectional sound source E emits continously. The unknown head−→−−to-source position is characterized by the (hidden) stateqvector x = (ey , ez )T ∈ R2 such that OE = ey →yR + ez →zR , or,equivalently, by the polar coordinates (r, θ ), with r =e2y + e2z and θ = −atan2(ey , ez ). As time evolves, these2161

variables are subscripted by the discrete index k.
The measurement vector zk gathers the binaural information collected by R1 , R2 over a short temporal snippet
ending at time k, typically the the left and right channel-time-frequency decompositions. The measurement
model between x and z, though unknown and very involved, is assumed static, i.e., zk depends solely on xk .
Between consecutive times k and k + 1, the head undergoes a rigid motion. Its control input vector, or “motor−−−−→ → −−−−→ −→ Tcommand”, is defined as uk = (TkT , φk )T ∈ R3 , with Tk = (u1k , u2k )T = (Ok Ok+1 .−yR k , Ok Ok+1 .zRk ) the 2D transla−
−−→)
\
−
→
tion and φ = u = (−
z→
,
z
the
finiterotation
which,
when
appliedin sequence, turn F into F .
Rk Rk+1k

3kk

around xRk

k+1

The prior dynamics pdf p(xk+1 |xk ) can be equivalently described by the stochastic state equation
xk+1 = RT (φk )xk − RT (φk )Tk + wk , wk ∼ N (0, Qk )


(1)



φ − sin φwhere R(φ ) = cos
stands for the nontrivial diagonal block of the 3 × 3 rotation matrix of angle φ
sin φ cos φ→−around x , and the additive dynamic noise w is assumed white zero-mean Gaussian with known covariance Q .
Rkk

Mutual independence of wk , the random initial state x0 and of all measurement noises is also assumed.
2.2 A three-layer framework to active binaural localization
Active binaural localization is addressed through the framework outlined on Figure 2. At each time k, thefollowing sequence is completed. Stage A processes the measurement zk so as to detect the activity of thesource and computes a HRTF-based “pseudo-likelihood” p(zk |θk ) of the head-to-source azimuth θk , along [17].
Stage B assimilates all these cues up to time k and combines them with the sensor motor commands into a
Gaussian sum Unscented Kalman filter, resulting in the state (head-to-source position) filtering pdf (or “belief”)
JkjJ ) expressed as a Gaussian mixture. Observe that if Q = 0 in (1), then the, Pk|kγkj N (xk ; x̂k|kp(xk |z1:k ) ≈ ∑ j=1
kcurrent filtering p(xk |z1:k ) and next prediction p(xk+1 |z1:k ) pdfs describe the same belief on the source location in
the world frame, but expressed from the respective viewpoints of Fk and Fk+1 . This audio-motor localization,
described in [16], enables front-back disambiguation as well as faithful source range recovery. This paper
addresses the feedback controller featured in Stage C, which processes the state filtering pdf from Stage B so
as to move the head in the most informative way.
2.3 Information-based optimum control strategy
The fundamental problem defining Stage C can now be stated.
Multi-step-ahead information-based
feedback control.
At time k, given the state filtering pdf
Jkjj p(xk |z1:k ) ≈ ∑ j=1
γkj N xk ; x̂k|k, Pk|k, find the sequence ū∗N = u?k : u∗k+N−1 of control inputs to be applied over a sliding
window of size N starting at k (i.e., from time k to k + N − 1) such that the expected value Ezk+1:k+N |z1:k h(xk+N |z1:k+N )of the entropy h(xk+N |z1:k+N ) of the state filtering pdf p(xk+N |z1:k+N ) at final time k + N is minimum.
This problem makes sense for two reasons. On the one hand, the entropy of a pdf is a convenient measure of
the information it conveys. On the other hand, as zk+1:k+N are not yet known at time k, the expectation of theentropy of the state filtering pdf at the end of the receding horizon over these future measurements (conditioned
known measurements up to k) is considered. Basically, N = 1 corresponds to the greedy strategy.
2.4 Simplifying assumptions
Binaural signals to be sensed in the future are hardly predictable, and so is the pdf p(zk+1:k+N |z1:k ) involved in
the criterion to be minimized at time k. This is why the following is assumed to hold.
Fundamental assumption. In Stages A and B of the three-stage framework to active binaural localization (Figure 2),z stands for the measurement variable (channel-time-frequency decomposition) depicted above. However, in theabove fundamental problem defining Stage C, zk+1 , . . . , zk+N stand for “exploratory audio cues” instead. A variable
z is said to be an exploratory audio cue iff it is linked to the hidden state x by a known closed-form measurement2162

model, the measurement noise of which is additive white zero-mean Gaussian (independent of all other noises) andhas a (co)variance unrelated to the hidden state value. In other words, the exploration (computation of ū∗N ) is driven
by kind of virtual measurements
zk+i = l(xk+i ) + vk+i , i = 1, . . . , N, vk+i ∼ N (0, Rk+i ) with Rk+i independent of hidden xk+i ,and v1:N white independent of x0 , w0:k+N , etc.

(2)Other less fundamental assumptions are also in effect.
• The state filtering pdf p(xk |z1:k ) is reduced to a single Gaussian, so that p(xk |z1:k ) ≈ N (xk ; x̂k|k , Pk|k ).
¯ k+i ), with l(.)¯ given analytically.
• Farfield exploratory audio cues are considered, i.e., l(xk+i ) = l(θ• Those cues are assumed scalar, i.e., zk+i ∈ R, Rk+i ∈ R.

3

SKETCH OF THE MATHEMATICAL SOLUTION

As aforementioned, the one-step-ahead (or “greedy”) problem is equivalent to N = 1. As it will be shown to besignificantly easier than the general N-step-ahed problem (N > 1), it deserves a specific treatment. Recall thatthe entropy h(N (x; x̄, P)) of the Gaussian pdf N (x; x̄, P) is an affine increasing function of det(P), so that, forthe purpose of its optimization it will be replaced (with a slight notation misuse) with det(P). The higher thevalue of det(P), the higher the volume of the 99%-probability confidence ellipsoid of N (x; x̄, P).
3.1 One-step-ahead problem
In view of the Unscented Kalman filter (UKF) equations [10], when starting from p(xk |z1:k ) ≈ N (xk ; x̂k|k , Pk|k )the next state filtering pdf comes as p(xk+1 |z1:k+1 ) ≈ N (xk+1 ; x̂k+1|k+1 , Pk+1|k+1 ), with Pk+1|k+1 independent of
zk+1 . Hence, the expectation Ezk+1 |z1:k h(xk+1 |z1:k+1 ) of the entropy h(xk+1 |z1:k+1 ) of p(xk+1 |z1:k+1 ) amounts toh(xk+1 |z1:k+1 ). Furthermore, if the dynamic noise covariance Qk is neglected, then the optimum control is [7, 5]ū?1 = (Tyk∗ , Tzk∗ , φk∗ ) = arg min J1 (ū1 ), with J1 (ū1 ) = h(xk+1 |z1:k+1 ) = K10 − h(zk+1 |z1:k ) and K10 independent of ū1 ,| {z }ū1 =uk =(Tyk ,Tzk ,φk )=:F1 (ū1 )(3)((Tyk ,Tzk ),φk )∈(T ×R)N
where
• T and R term the admissible translation and rotation domains (henceforth, respectively, a disk centeredon 0 of given radius for the 2D translation and a 1D line segment centered on 0 for the rotation);• h(zk+1 |z1:k ) stands for the entropy of the next measurement prediction pdf p(zk+1 |z1:k ); it depends onū1 = uk = (Tyk , Tzk , φk ) and is thus denoted by F1 (ū1 ).
If the Gaussian approximation p(zk+1 |z1:k ) ≈ N (zk+1 ; ẑk+1|k , Sk+1|k ) produced by the UKF is used for p(zk+1 |z1:k ),then the information-based one-step-ahead optimum control simplifies asū?1 = (Tyk∗ , Tzk∗ , φk∗ ) = arg

max((Tyk ,Tzk ),φk )∈T ×R

F1 (Tyk , Tzk , φk ), with F1 (Tyk , Tzk , φk ) = h(zk+1 |z1:k ) ≈ log det(Sk+1|k ). (4)Therein, the criterion F1 (ū1 ) comes as a combination of the image, through the function l(.) entailed in the
exploratory audio cues model (2), of sigma-points of the next state filtering pdf p(xk+1 |z1:k ). These sigmapoints are themselves functions of the decision variable ū1 . Hence, though the genuine F1 (ū1 ) has no closed
form, it can easily be approximated. So can its gradient, through Taylor expansion. Consequently, (3)-(4) canbe solved by means of the projected gradient algorithm.

2163

3.2 N-step-ahead problem
The corresponding information-based optimum control input sequence is given by [6, 5].
hiNū?N = u∗k : u∗k+N−1 = arg min JN (ūN ), with JN (ūN ) = KN0 − h(zk+1 |z1:k ) − ∑ Ezk+1:k+i−1 |z1:k h(zk+i |z1:k+i−1 ) (5)| {z } i=2ūN =uk :uk+N−1|{z}=:F1 (ū1 )ūN ∈(T ×R)N=:Fi (ūi ,zk+1:k+i−1 )and KN0 independent of ūN , where the entropy h(zk+i |z1:k+i−1 ) of the i-step-ahead measurement prediction pdf
p(zk+i |z1:k+i−1 ) depends on the sequence ūi of the i next control inputs to be determined and on the i − 1 future
exploratory cues zk+1:k+i−1 , so that it is denoted by Fi (ūi , zk+1:k+i−1 ). Its conditional expectation over these yet
unknown virtual measurements must be evaluated, what makes the N-step-ahead problem significantly harder.
A way out is to use the approximation [19]Z2(i−1)p(zk+1:k+i−1 |z1:k )Fi (ūi , zk+1:k+i−1 )dzk+1:k+i−1 ≈∑ w j Fi (ūi , Z j,i (ūi−1 )),

(6)

j=0

where {Z j,i } are the 2(i − 1) + 1 sigma-points, deterministically drawn by the Unscented Transform (UT),of the Gaussian approximation N (zk+1:k+i−1 ; ẑk+1:k+i−1|k , Sk+1:k+i−1|k ) of the joint measurement prediction pdf
p(zk+1:k+i−1 |z1:k ), and {w j } are the corresponding UT weights. N (zk+1:k+i−1 ; ẑk+1:k+i−1|k , Sk+1:k+i−1|k ) is readily
obtained by means of the UT and the exploratory audio cues model (2) on the basis of the Gaussian approximation N (xk+1:k+i−1 ; x̂k+1:k+i−1|k , Pk+1:k+i−1|k ) of the joint state prediction pdf p(xk+1:k+i−1 |z1:k ).
N (xk+1:k+i−1 ; x̂k+1:k+i−1|k , Pk+1:k+i−1|k ) itself can be obtained from N (xk ; x̂k|k , Pk|k ) ≈ p(xk |z1:k ) by repeated application of the prior dynamics (1) and use of the UT. Hence, x̂k+1:k+i−1|k , Pk+1:k+i−1|k , ẑk+1:k+i−1|k , Sk+1:k+i−1|kdepend on the sequence of decisions variables ūi−1 = uk : uk+i−2 . So do the sigma-points {Z j,i }, which aredenoted {Z j,i (ūi−1 )}. In summary, the information-based N-step-ahead optimum control simplifies as
N 2(i−1)ū?N = u∗k : u∗k+N−1 = arg max F1 (ū1 ) + ∑ūN ∈(T ×R)N

∑ w j Fi (ūi , Z j,i (ūi−1 )).

(7)

i=2 j=0

In spite of the apparent simplicity of (7), it must be kept in mind that for eachindex i in {2, . . . , N}, and foreach jth sigma-point Z j,i (ūi−1 ) associated to i, Fi (ūi , Z j,i (ūi−1 )) is the value Fi (ūi , zk+1:k+i−1 ) z
k+1:k+i−1 =Z j,i (ūi−1 )of the entropy Fi (ūi , zk+1:k+i−1 ) = h(zk+i |z1:k+i−1 ) of p(zk+i |z1:k+i−1 ), when evaluated at the sequence of virtual measurements zk+1:k+i−1 = Z j,i (ūi−1 ). For each sigma-point Z j,i , which implicitly depends
 on the sequenceūi−1 of decision variables under optimization, a Gaussian approximation p(zk+i |z1:k+i−1 ) z≈
k+1:k+i−1 =Z j,i (ūi−1 )N (zk+i ; ẑ(k+i|k+i−1), j , S(k+i|k+i−1), j ) can be produced by a UKF, leading to Fi (ūi , Z j,i (ūi−1 )) ≈ log det(S(k+i|k+i−1), j ),in the vein of (4). The obtained approximation of the genuine criterion of (5) is quite impossible to write byhand. Nevertheless it can be programmed, and even if Taylor expansions are hardly conceivable, automaticdifferentiation in the framework of dual numbers algebra [9] is still possible to get its gradient. This enables
again a solution by means of the projected gradient algorithm.
3.3 Qualitative insights
Commonsense characterization of one-step-ahead (greedy) control optimality can be established. Figure 3-(a)displays the binaural head at time k with its attached frame, the genuine position of the source E (yellow), as
well as the 99%-probability confidence ellipse (grey) corresponding to p(xk |z1:k ) ≈ N (xk ; x̂k|k , Pk|k ). The set ofradial lines rigidly linked to the head and intersecting at its center are the loci of the head-to-source positions
which correspond to evenly-spaced values of the exploratory measurement at time k. These drawn on the picture
typically correspond to the Woodworth-Schlosberg approximation of ITDs: the locus z = 0 is supported by −z→
Rk ;it splits the 2D plane into two halves onto which z keeps the same sign and evolves monotonically; around −z→
Rkis the auditory fovea, for which the sensitivity of z to the head-to-source position is maximum.

2164

Figure 3. Geometric interpretation of h(zk+1 |z1:k ) for various motions, when using ITD-like exploratory cues.
Let the motor command uk applied to the head between times k and k + 1 correspond to a clockwise rotation, and consider the situation at k + 1 (Figure 3-(b)). The blue 99%-probability confidence ellipse corresponds to p(xk+1 |z1:k ) ≈ N (xk+1 ; x̂k+1|k , Pk+1|k ). The green cone represents the 99%-probability confidence
set of the pdf p(xk+1 |zk+1 ), i.e., the spatial uncertainty sector on the head-to-source position extracted froma single hypothetical measurement zk+1 . As the measurement noise (co)variance Rk+1 is constant, the denser
are the iso-z radial lines (which are still rigidly linked to the head) around zk+1 , the smaller is the extent of
this confidence cone. The red 99%-probability confidence ellipse corresponds to the next state filtering pdf
p(xk+1 |z1:k+1 ) ≈ N (xk+1 ; x̂k+1|k+1 , Pk+1|k+1 ) obtained by fusing the two last pdfs. As aforementioned, the entropy of p(xk+1 |z1:k+1 ) is all the smaller as the volume of this ellipse is low. Heuristically, the entropy of thenext measurement prediction pdf p(zk+1 |z1:k ) is all the bigger (resp. smaller) as the number of iso-z radial lines
intersected by the blue 99%-probability confidence ellipse of p(xk+1 |z1:k ) is high (resp. low).
By comparing Figures 3-(b)-(c)-(d), the following heuristical rules of thumb can drawn so that uk is optimum
when using approximate Woodworth-Schlosberg ITDs as exploratory audio cues (virtual measurements): while
keeping into its admissible set T × R, uk must
• orient the auditory fovea, rather than the interaural axis, towards the confidence ellipsoid associated to theinitial belief p(xk |z1:k );• drive the head center on the line supporting the minor axis, rather than the major axis, of this ellipsoid;• make the head get closer to this ellipsoid;
so that after the head motion this ellipsoid intersects as many iso-z radial lines as possible.

4

RESULTS WITH DISTINCT BINAURAL CUES

An experimental setup comprising a KEMAR head-and-torso-simulator mounted on a nonholonomic mobile
platform and endowed with a motorized neck has been designed in such a way that its binaural head is omnidirectional. Simulations and live tests were conducted, which implement the Woodworth-Schlosberg approximation of ITDs for a spherical head of same average radius. The simulations can be considered realistic,
as they rely on the binaural simulator developed in the framework of the EU FP7 Two!Ears project (http://docs.twoears.eu/en/latest/binsim/) which can faithfully emulates the sensed binaural signals. As
aforementioned, the Woodworth-Schlosberg ITDs iso-z loci looks like the radial lines sketched on Figure 3-(a).
Due to space reasons, only simulated results with such virtual measurements are presented.
Nevertheless, exact farfield ITDs and ILDs for a spherical binaural head have also been implemented in simulation. To this aim, the developments onto Legendre polynomials of the farfield left and right HRTFs as well as
their derivatives have been carefully implemented so as to keep within predefined tolerances. The results withfarfield spherical ITDs look like these sketched below for their Woodworth-Schlosberg approximation. Farfield
ITDs constitute relevant exploratory cues in view of their limited sensitivity to the assumed source frequency
and because they constitute a convenient approximation of the genuine range-dependent ITDs for an impor-2165

Figure 4. One-step-ahead optimum head motion with Woodworth-Schlosberg ITD exploratory cues (Case 2)

Figure 5. One-step-ahead optimum head motion with Woodworth-Schlosberg ITD exploratory cues

Figure 6. One- (left) vs Five- (right) step-ahead optimum head motion with Woodworth-Schlosberg ITD cues
tant set of head-to-source positions. However, controlling the head motion on the basis of virtual farfield ILD
mesurements can give rise to bad results if precautions are not taken. Unsurprisingly, iso-z loci of genuine
spherical ILDs strongly depend on the assumed source frequency. Moreover, their sensitivity to the source
range is such that for a broad interval of source azimuths outside the auditory fovea, they may strongly differ
from iso-z radial lines of their farfield limits. Last, farfield ILDs do not vary monotonically on each side of thehead, so that they may lead to quite surprising and unefficient motions in some circumstances.
Figures 4 and 5 display the one-step-ahead optimum head motion. It can be seen that the obtained behavior of
the head complies with the heuristic rules of thumb sketched in Section 3.3. Notice that Figure 4 corresponds
to an advantageous initial belief such that the further fusion with measurements is very efficient.
Figure 6 compares the greedy strategy with the 5-step optimum head motion. It can be seen that the second
policy leads to a more “cautious” behavior of the head, in that it tends to first rotate and then follow a straight
path. The greedy policy can be transiently better in terms of performance (information criterion), but is lessefficient in the long term.

5

CONCLUSION AND PROSPECTS

The information-based feedback for binaural localization outlined in this paper aims at minimizing the expected
entropy of the head-to-source belief at the end of a receding N-element horizon. A tractable approximate solution is obtained through the unscented transform, automatic differentiation, and the projected gradient algorithm.
With no specific code optimization, it can run in real time on the binaural mobile robot for small N’s. It
has been extended to a multi-objective problem similar to [18]. Prospects consist in the incorporation of thisstochastic and optimization method with machine learning in order to consider multiple sources in unanechoic/dynamic environments.

ACKNOWLEDGEMENTS
The foundations of this work were developed in collaboration with A. Portello and G. Bustamante during their
PhD theses. They are gratefully acknowledged.

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