- Technical complements on the proof of the main result, which were not included for space reasons.
- Videos of experiments.
- On the TV screen, the red dot depicts the source. The KEMAR head
is depicted by a blue circle centered on the origin of the frame, and
the attached blue bar is oriented towards the boresight direction.
All these ground truth data are got from an Optitrack real time motion
capture system.
- The audio-motor localization computes a Gaussian mixture
approximation of the head-to-source posterior pdf. The ellipsoids
drawn on the real time display are the 99%-probability
minimum-volume confidence sets associated to the hypotheses of this
posterior pdf. Their colors depict the posterior probability of the
associated hypothesis (red means high, blue means low, purple means
in-between).
- Open-loop uniform rotational movement of the head:

- Yet another open-loop uniform short rotational movement of the head:

- Open-loop uniform rectilinear translational movement of the head (up to little saturations of the neck velocity):

- Yet another open-loop uniform rectilinear translational movement of the head:

- Open-loop uniform nonholonomic circular movement of the head (its
relative angle w.r.t. the tangent vector of the trajectory is constant):

**Active motion as per ICASSP'2016 paper**

- Sketch of the measure of information held in the posterior
pdf
*vs*time index. Note that the applied velocities do not have the same magnitudes over time in the three compared cases, so that the rates of change of the relative plots may not be meaningful. However, note that the information measure obtained by the active motion strategy always decreases along time and reaches the lowest steady state value.

This page is structured into two parts.

The following case studies have been tested.

The head-to-source situation at time is characterized by the posterior state pdf approximated by the 2D Gaussian . This pdf can be mapped into the 1D Gaussian approximation of the predicted measurement pdf , by using the unscented transform. The aim is to maximize the variance so as to increase the entropy . This involves the composition of the several functions.

First the sigma-points
corresponding to
are computed from the
posterior mean
of the state vector at time
and the
Cholesky decomposition
of the posterior
covariance.

We consider that the random variable undergoes a rigid motion defined by the translations and the rotation , and that the dynamic noise is neglected. Then the sigma-points of the predicted state pdf can be obtained by applying the translation and rotation on each sigma-point :

Then each sigma-point of the predicted measurement can be obtained from the sigma-points defined in () by:

(3) |

and are the components of each sigma point . is the Woodworth-Schlosberg formula for interaural time difference approximation over a spherical head which is used to guide the exploration of the space by the head. function is used to retrieve the corresponding azimuth of each sigma-point. Finally the mean and the variance are computed using the unscented transform formulae such as

and are the classic weights of the unscented transform.

The variance can be defined as a function of the finite translation and rotation . However the maximium of this function is not analytically tractable. Its gradient around is then computed so as to point out the direction of its maximum, as follow:

The first order Taylor expansion of the functions
,
and
are composed around
with infinitesimal
translations and rotation
such as

is the gradient operator. is the Jacobian of taken in . Then the result of the composition, noted , is used to retrieve the mean and the variance with () and (). Finally, the first order Taylor expansion of is obtained, highlighting the gradient :

(9) |

Note: this part was generated using the
**LaTeX**2`HTML` translator Version 2008 (1.71)

Copyright © 1993, 1994, 1995, 1996,
Nikos Drakos,
Computer Based Learning Unit, University of Leeds.

Copyright © 1997, 1998, 1999,
Ross Moore,
Mathematics Department, Macquarie University, Sydney.

Last updated: Oct 2015 © LAAS/CNRS