We study pseudo labelling and its generalisation for semi-supervised segmentation of medical images. Pseudo labelling has achieved great empirical successes in semi-supervised learning, by utilising raw inferences on unlabelled data as pseudo labels for self-training. In our paper, we build a connection between pseudo labelling and the Expectation Maximization algorithm which partially explains its empirical successes. We thereby realise that the original pseudo labelling is an empirical estimation of its underlying full formulation. Following this insight, we demonstrate the full generalisation of pseudo labels under Bayes' principle, called Bayesian Pseudo Labels. We then provide a variational approach to learn to approximate Bayesian Pseudo Labels, by learning a threshold to select good quality pseudo labels. In the rest of the paper, we demonstrate the applications of Pseudo Labelling and its generalisation Bayesian Psuedo Labelling in semi-supervised segmentation of medical images on: 1) 3D binary segmentation of lung vessels from CT volumes; 2) 2D multi class segmentation of brain tumours from MRI volumes; 3) 3D binary segmentation of brain tumours from MRI volumes. We also show that pseudo labels can enhance the robustness of the learnt representations.

This paper concerns pseudo labelling in segmentation. Our contribution is fourfold. Firstly, we present a new formulation of pseudo-labelling as an Expectation-Maximization (EM) algorithm for clear statistical interpretation. Secondly, we propose a semi-supervised medical image segmentation method purely based on the original pseudo labelling, namely SegPL. We demonstrate SegPL is a competitive approach against state-of-the-art consistency regularisation based methods on semi-supervised segmentation on a 2D multi-class MRI brain tumour segmentation task and a 3D binary CT lung vessel segmentation task. The simplicity of SegPL allows less computational cost comparing to prior methods. Thirdly, we demonstrate that the effectiveness of SegPL may originate from its robustness against out-of-distribution noises and adversarial attacks. Lastly, under the EM framework, we introduce a probabilistic generalisation of SegPL via variational inference, which learns a dynamic threshold for pseudo labelling during the training. We show that SegPL with variational inference can perform uncertainty estimation on par with the gold-standard method Deep Ensemble.