We consider the problem of collaborative bearing estimation using a method with historic roots in set theoretic estimation techniques. We refer to this method as the Convex Combination Ellipsoid (CCE) method and show that it provides a less conservative covariance estimate than the well known Covariance Intersection (CI) method. The CCE method does not introduce additional uncertainty that was not already present in the prior estimates. Using our proposed approach for collaborative bearing estimation, the nonlinearity of the bearing measurement is captured as an uncertainty ellipsoid thereby avoiding the need for linearization or approximation via sampling procedures. Simulations are undertaken to evaluate the relative performance of the collaborative bearing estimation solution using the proposed (CCE) and typical (CI) methods.

This paper proposes a new method for estimating calibration parameters of plenoptic cameras by minimizing the nonlinear plenoptic reprojection error. Novel plenoptic feature types are proposed as data for the calibration method. These plenoptic disc features are in a natural one-to-one correspondence with physical points in front of the camera. We exploit the intrinsic geometry of plenoptic cameras in a novel projection model that relates the plenoptic disc features to physical points. The resulting calibration quality, as quantified by mean reprojection error and 3D reconstruction error, outperforms recently published results.

In many computer vision applications, a desired model of some type is computed by minimizing a cost function based on several measurements. Typically, one may compute the model that minimizes the L2 cost, that is the sum of squares of measurement errors with respect to the model. However, the Lq solution which minimizes the sum of the qth power of errors usually gives more robust results in the presence of outliers for some values of q, for example, q = 1. The Weiszfeld algorithm is a classic algorithm for finding the geometric L1 mean of a set of points in Euclidean space. It is provably optimal and requires neither differentiation, nor line search. The Weiszfeld algorithm has also been generalized to find the L1 mean of a set of points on a Riemannian manifold of non-negative curvature. This paper shows that the Weiszfeld approach may be extended to a wide variety of problems to find an Lq mean for 1 ≤ q <; 2, while maintaining simplicity and provable convergence. We apply this problem to both single-rotation averaging (under which the algorithm provably finds the global Lq optimum) and multiple rotation averaging (for which no such proof exists). Experimental results of Lq optimization for rotations show the improved reliability and robustness compared to L2 optimization.

We study the theory of projective reconstruction for multiple projections from an arbitrary dimensional projective space into lower-dimensional spaces. This problem is important due to its applications in the analysis of dynamical scenes. The current theory, due to Hartley and Schaffalitzky, is based on the Grassmann tensor, generalizing the ideas of fundamental matrix, trifocal tensor and quadrifocal tensor used in the well-studied case of 3D to 2D projections. We present a theory whose point of departure is the projective equations rather than the Grassmann tensor. This is a better fit for the analysis of approaches such as bundle adjustment and projective factorization which seek to directly solve the projective equations. In a first step, we prove that there is a unique Grassmann tensor corresponding to each set of image points, a question that remained open in the work of Hartley and Schaffalitzky. Then, we prove that projective equivalence follows from the set of projective equations given certain conditions on the estimated camera-point setup or the estimated projective depths. Finally, we demonstrate how wrong solutions to the projective factorization problem can happen, and classify such degenerate solutions based on the zero patterns in the estimated depth matrix.

We consider the problem of rotation averaging under the L1norm. This problem is related to the classic Fermat-Weber problem for finding the geometric median of a set of points in IRn. We apply the classical Weiszfeld algorithm to this problem, adapting it iteratively in tangent spaces of SO(3) to obtain a provably convergent algorithm for finding the L1mean. This results in an extremely simple and rapid averaging algorithm, without the need for line search. The choice of L1mean (also called geometric median) is motivated by its greater robustness compared with rotation averaging under the L2norm (the usual averaging process). We apply this problem to both single-rotation averaging (under which the algorithm provably finds the global L1optimum) and multiple rotation averaging (for which no such proof exists). The algorithm is demonstrated to give markedly improved results, compared with L2averaging. We achieve a median rotation error of 0.82 degrees on the 595 images of the Notre Dame image set.

A fundamental problem in any communication system is: given a communication channel between a transmitter and a receiver, how many ¿independent¿ signals can be exchanged between them? Arbitrary communication channels that can be described by linear compact channel operators mapping between normed spaces are examined in this paper. The (well-known) notions of degrees of freedom (DOF) at level¿and essential dimension of such channels are developed in this general setting. We argue that the DOF at level¿and the essential dimension fundamentally limit the number of independent signals that can be exchanged between the transmitter and the receiver. We also generalize the concept of singular values of compact operators to be applicable to compact operators defined on arbitrary normed spaces which do not necessarily carry a Hilbert space structure. We show how these generalized singular values, which we call DOF singular values, can be used to calculate the DOF at level¿and the essential dimension of compact operators that describe communication channels. We describe physically realistic channels that require such general channel models.

This paper considers the problem of estimating pose from inertial and bearing-only vision measurements. We present a non-linear observer that evolves directly on the special Euclidean group SE(3) from inertial measurements and bearing measurements, such as provided by a visual system tracking known landmarks. Local asymptotic convergence of the observer is proved. The observer is computationally simple and its gains are easy to tune. Simulation results demonstrate robustness to measurement noise and initial conditions.

A robust VAR-based (vector autoregressive) model is introduced for motion prediction in 3D hand tracking. This dynamic VAR motion model is learned in an online manner. The kinematic structure of the hand is accounted for in the form of constraints when solving for the parameters of the VAR model. Also integrated into the motion prediction model are adaptive weights that are optimised according to the reliability of past predictions. Experiments on synthetic and real video sequences show a substantial improvement in tracking performance when the robust VAR motion model is used. In fact, utilising the robust VAR model allows the tracker to handle fast out-of-plane hand movement with severe self-occlusion.