Traditional geometric filter algorithms are usually designed without considering the geo-metric properties of bias states.To deal with this problem,a group and a group action are pro-posed,and a new equivariant filter method is derived.To evaluate the performance of this filter al-gorithm,a visual-inertial odometry system based on an equivariant filter and a visual observation network,named Deep-EqF-VIO,is developed.Firstly,a navigation state dynamics system is con-structed on the manifold,then an equivariant filter method is derived by the equivariant filter prin-ciple,and finally a visual-inertial odometry system is constructed by combining with a visual ob-servation network.In order to further improve the performance of the system,a pretraining meth-od incorporating with dense optical flow pseudo-labels is proposed.This method guides the visual observation network to extract more generalized geometric motion features from input images via the optical flow estimation task.Experimental results show that Deep-EqF-VIO achieves the best accuracy in most scenes compared to similar algorithms.After retraining the VIO system by using the proposed pre-training method,the VIO performance can be further improved,with the maxi-mum error reduction rate reaching 49.80％.This method has a certain degree of application poten-tial.
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