The experimental characterisation of the swimming statistics of populations
of microorganisms or artificially propelled particles is essential for
understanding the physics of active systems and their exploitation. Here, we
construct a theoretical framework to extract information on the
three-dimensional motion of micro-swimmers from the Intermediate Scattering
Function (ISF) obtained from Differential Dynamic Microscopy (DDM). We derive
theoretical expressions for the ISF of helical and oscillatory breaststroke
swimmers, and test the theoretical framework by applying it to video sequences
generated from simulated swimmers with precisely-controlled dynamics. We then
discuss how our theory can be applied to the experimental study of helical
swimmers, such as active Janus colloids or suspensions of motile microalgae. In
particular, we show how fitting DDM data to a simple, non-helical ISF model can
be used to derive three-dimensional helical motility parameters, which can
therefore be obtained without specialised 3D microscopy equipment. Finally, we
discus how our results aid the study of active matter and describe applications
of biological and ecological importance