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We propose an efficient non-parametric diffeomorphic image registration algorithm based on Thirion's demons algorithm. In the first part of this paper, we show that Thirion's demons algorithm can be seen as an optimization procedure on the entire space of displacement fields. We provide strong theoretical roots to the different variants of Thirion's demons algorithm. This analysis predicts a theoretical advantage for the symmetric forces variant of the demons algorithm. We show on controlleddoi:10.1016/j.neuroimage.2008.10.040 pmid:19041946 fatcat:kyzcf5d5qzbw3bsgqyrg2pfkqe