L1 rotation averaging using the Weiszfeld algorithm

Richard I. Hartley, Khurrum Aftab, Jochen Trumpf

Abstract: We consider the problem of rotation averaging under the L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> norm. This problem is related to the classic Fermat-Weber problem for finding the geometric median of a set of points in IR <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> . 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 <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> mean. This results in an extremely simple and rapid averaging algorithm, without the need for line search. The choice of L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> mean (also called geometric median) is motivated by its greater robustness compared with rotation averaging under the L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> norm (the usual averaging process). We apply this problem to both single-rotation averaging (under which the algorithm provably finds the global L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> optimum) and multiple rotation averaging (for which no such proof exists). The algorithm is demonstrated to give markedly improved results, compared with L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> averaging. We achieve a median rotation error of 0.82 degrees on the 595 images of the Notre Dame image set.