In this line of research our goal is to develop algorithms to distributely average quantities that lie on non-Euclidean manifolds. Of our particular interest is the case of pose estimation. We assume that the nodes are localized (meaning that each mote knows its position relative to its neighbours). We assume also that each node can estimate the pose of a common object (e.g. a face).
Our goal is then to aggregate the estimates through a distributed averaging. In Sensor Networks, a popular class of algorithms for computing averages are consensus algorithms. These techniques are attractive because require communications only between neighbours and converge under mild network connectivity requirements. Unfortunately, these algorithms are designed for scalar, Euclidean quantities. On the other hand, poses are represented with a pair (R,T), where R is a rotation and T is a translation. Rotations, in particular, live on the manifold of the special orthogonal group SO(3). This means that the appropriate metric (d_m in the Figure) has to be used, instead of the Euclidean one (d_e). This induces also a new concept of average, called Karcher mean. We have developed algorithms that:

• Take into account the appropriate measure of distance and the corresponding concept of Karcher mean in SO(3).
• Extend consensus algorithms to average quantities in SO(3).
• Show convergence to the correct average.

We have also applied our algorithm to a face recognition application, where each node estimates the pose of the face using Eigenfaces/Tensorfaces and then the various estimates are aggregated with our consensus algorithm on SO(3).

Localization problem in a camera sensor network refers to uniquely determining the pose of every camera with respect to a fixed reference frame. More precisely, assume we have a network of n cameras deployed in 3-D. The relative pose of each camera with respect to a reference frame is defined by its rotation R and its translation T with respect to that frame. We say the network is localized if there is a set of relative transformations (R_ij,T_ij) between camera i and j such that when the reference frame for node 1 is fixed at (R₁,T₁), the other poses (R_i,T_i) are uniquely determined. When the field-of-view's of two cameras intersect, it is possible to estimate their relative transformation(R_ij,T_ij) using two-view epipolar geometry. However, there are a couple of problems in directly using these pairwise estimates.

1. The estimates may not be consistent if the whole network is taken into consideration. For instance, if we go trough a loop in the network, the composition of all relative transformations should give the identity.
2. The translations are obtained only to a scale. It is necessary to exploit the constraints from the network topology in order to find these additional scales.

Previous work either is not applicable in this setting (typically, only the case of range measurements has been considered) or does not correctly exploit the manifold structure of SE(3). In our work we developed distributed algorithm that, given the pairwise measurement, find a complete, consistent localization of the network. Moreover, our algorithm is optimal with respect to the use of the appropriate metric in SE(3).
The problem of camera calibration for a single camera has been widely studied in the computer vision literature. Such methods typically require the user to show a calibration rig to the camera. Obviously, such methods do not scale well for large camera sensor networks, because they would require manual calibration of each camera. On the other hand, self-calibration methods automatically calibrate the cameras by solving nonlinear equations such as Kruppa's equations. While these methods are very elegant, they suffer from the fact that the problem of solving Kruppa's equations is numerically ill-conditioned. Moreover, these methods assume that all the cameras have the same intrinsic parameters so they are not readily applicable to camera sensor networks, where each camera can have different intrinsic parameters. We show that the problem of automatically calibrating a large number of cameras in a sensor network can be solved in a distributed way by solving a set of linear equations. We show that this is possible under the mild assumption that only one of the cameras needs to be calibrated.
Once the cameras' intrinsic and extrinsic parameters are known, the next problem is to recover the structure of the 3D scene (triangulation problem). However, solving the multiple view triangulation problem becomes difficult in a sensor network setup. This is because, while each pair of motes could easily compute the 3D structure of the scene via linear triangulation, the estimates from different pairs of motes may not be the same, especially in real situations where image data are noisy. We propose a method based on distributed consensus algorithms for estimating the 3D structure of a scene in a distributed way. We show that all the motes compute the same 3D structure, even though they communicate with only a few neighbors in the network. This method requires that all the cameras observe the same scene and that the network graph over which the nodes communicate be connected.