Companion webpage to G. Bustamante et al.' ICASSP'2016 submission

This page is structured into two parts.

Videos of Live experiments

The following videos show the real time behavior of the algorithm.

The following case studies have been tested.

Technical complements on the proof of the main result

The head-to-source situation at time $ k$ is characterized by the posterior state pdf $ p(x_{k}\vert z_{1:k}) $ approximated by the 2D Gaussian $ {\ensuremath{\mathcal{N}}}(x_k;\hat{x}_{k\vert k},P_{k\vert k})$ . This pdf can be mapped into the 1D Gaussian approximation $ {\ensuremath{\mathcal{N}}}(z_{k+1};\hat{z}_{k+1\vert k},S_{k+1\vert k})$ of the predicted measurement pdf $ p(z_{k+1}\vert z_{1:k})$ , by using the unscented transform. The aim is to maximize the variance $ S_{k+1\vert k}$ so as to increase the entropy $ h(z_{k+1}\vert z_{1:k})$ . This involves the composition of the several functions.

First the sigma-points $ \left\{X_{i}^{-}\right\}$ corresponding to $ p(x_{k}\vert z_{1:k})
= {\ensuremath{\mathcal{N}}}(x_k;\hat{x}_{k\vert k},P_{k\vert k})$ are computed from the posterior mean $ \hat{x}_{k\vert k}$ of the state vector at time $ k$ and the Cholesky decomposition $ P_{k\vert k} = L_{k\vert k}L_{k\vert k}^T$ of the posterior covariance.

$\displaystyle \left\{X_{i}^{-}\right\}$ $\displaystyle =$ $\displaystyle Sigma\_points \left(\hat{x}_{k\vert k},L_{k\vert k} \right)$ (1)

We consider that the random variable $ x_k \sim p(x_{k}\vert z_{1:k})$ undergoes a rigid motion defined by the translations $ \{T_y,T_z\}$ and the rotation $ \{\phi\}$ , and that the dynamic noise is neglected. Then the sigma-points $ \left\{X_{i}^{+}\right\}$ of the predicted state pdf $ p(x_{k+1}\vert z_{1:k}) = {\ensuremath{\mathcal{N}}}(x_k;\hat{x}_{k+1\vert k},P_{k+1\vert k})$ can be obtained by applying the translation and rotation on each sigma-point $ \left\{X_{i}^-\right\}$ :
$\displaystyle \left\{X_{i}^{+}\right\}$ $\displaystyle =$ $\displaystyle \Phi_{X_{i}^-}(T_y,T_z,\phi).$ (2)

Then each sigma-point $ \left\{Z_{i}^+\right\}$ of the predicted measurement $ p(z_{k+1}\vert z_{1:k}) =
{\ensuremath{\mathcal{N}}}(z_k;\hat{z}_{k+1\vert k},S_{k+1\vert k})$ can be obtained from the sigma-points $ X_{i}^+$ defined in ([*]) by:
$\displaystyle \left\{Z_{i}^+\right\}$ $\displaystyle =$ $\displaystyle g\left( atan2\left(X_{i}^+(1),X_{i}^+(2) \right) \right).$ (3)

$ X_{i}^+(1)$ and $ X_{i}^+(2)$ are the components of each sigma point $ X_{i}^+$ . $ g(\cdot)$ 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. $ atan2(\cdot,\cdot)$ function is used to retrieve the corresponding azimuth of each sigma-point. Finally the mean $ \hat{z}_{k+1\vert k}$ and the variance $ S_{k+1\vert k}$ are computed using the unscented transform formulae such as
$\displaystyle \hat{z}_{k+1\vert k}$ $\displaystyle =$ $\displaystyle \sum_i w_m^{i}Z_{i}^+$ (4)
$\displaystyle S_{k+1\vert k}$ $\displaystyle =$ $\displaystyle \sum_i w_c^{i}\left(Z_{i}^+ - \hat{z}_{k+1\vert k}\right)^2.$ (5)

$ \left\{w_m^i\right\}$ and $ \left\{w_c^i\right\}$ are the classic weights of the unscented transform.

The variance $ S_{k+1\vert k}$ can be defined as a function of the finite translation and rotation $ S_{k+1\vert k} = {F}_{k}(T_y,T_z,\phi)$ . However the maximium of this function is not analytically tractable. Its gradient around $ \overrightarrow{D_0} = (0,0,0)$ is then computed so as to point out the direction of its maximum, as follow:

The first order Taylor expansion of the functions $ \Phi_{X_{i}^-}$ , $ atan2$ and $ g$ are composed around $ \overrightarrow{D_0}$ with infinitesimal translations and rotation $ \overrightarrow{du} = (dT_y, dT_z, d\phi)^T$ such as

$\displaystyle \Phi_{X_{i}^-}(\overrightarrow{D_0} + \overrightarrow{du})$ $\displaystyle =$ $\displaystyle \Phi_{X_i^-}(\overrightarrow{D_0}) +
{J{\Phi_{X_i^-}}}(\overrightarrow{D_0}) \cdot \overrightarrow{du}$ (6)
$\displaystyle atan2(u,v)$ $\displaystyle =$ $\displaystyle atan2(u_0,v_0) + {\overrightarrow{\nabla
}^Tatan2}(u_0,v_0) \cdot \begin{pmatrix}u -
u_0 v-v_0 \end{pmatrix}$ (7)
$\displaystyle g(w)$ $\displaystyle =$ $\displaystyle g(w_0) + g'(w_0)(w-w_0).$ (8)

$ \overrightarrow{\nabla}$ is the gradient operator. $ J{\Phi_{X_i^-}}(\overrightarrow{D_0})$ is the Jacobian of $ \Phi_{X_i^-}$ taken in $ \overrightarrow{D_0}$ . Then the result of the composition, noted $ Z_{i}(dT_y,dT_z,d\phi)$ , is used to retrieve the mean and the variance with ([*]) and ([*]). Finally, the first order Taylor expansion of $ F_k(dT_y,dT_z,d\phi)$ is obtained, highlighting the gradient $ \overrightarrow{\nabla}F_k$ :
$\displaystyle F_k\left({\overrightarrow{D_0} +
\overrightarrow{du}}\right) =
... \overrightarrow{\nabla }^TF_k (\overrightarrow{D_0}) \cdot \overrightarrow{du}$     (9)

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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.

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