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We consider the problem of rotation averaging under the L 1 norm. This problem is related to the classic FermatWeber problem for finding the geometric median of a set of points in IR n . 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 L 1 mean. This results in an extremely simple and rapid averaging algorithm, without the need for line search. The choice of L 1 mean (also called geometric median) is motivated by its greater robustness compared with rotation averaging under the L 2 norm (the usual averaging process).We apply this problem to both single-rotation averaging (under which the algorithm provably finds the global L 1 optimum) and multiple rotation averaging (for which no such proof exists). The algorithm is demonstrated to give markedly improved results, compared with L 2 averaging. We achieve a median rotation error of 0.82 degrees on the 595 images of the Notre Dame image set.

This paper proposes a design methodology for non-linear state observers for invariant kinematic systems posed on finite dimensional connected Lie groups, and studies the associated fundamental system structure. The concept of synchrony of two dynamical systems is specialised to systems on Lie groups. For invariant systems this leads to a general factorisation theorem of a nonlinear observer into a synchronous (internal model) term and an innovation term. The synchronous term is fully specified by the system model. We propose a design methodology for the innovation term based on gradient-like terms derived from invariant or non-invariant cost functions. The resulting nonlinear observers have strong (almost) global convergence properties and examples are used to demonstrate the relevance of the proposed approach.

Let f be a real-valued function on a Riemannian submanifold of a Euclidean space, and letf be a local extension of f . We show that the Riemannian Hessian of f can be conveniently obtained from the Euclidean gradient and Hessian off by means of two manifoldspecific objects: the orthogonal projector onto the tangent space and the Weingarten map. Expressions for the Weingarten map are provided on various specific submanifolds.

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