We propose a novel approach for image segmentation that combines Neural Ordinary Differential Equations (NODEs) and the Level Set method. Our approach parametrizes the evolution of an initial contour with a NODE that implicitly learns from data a forcing function describing the evolution. In cases where an initial contour is not available or to alleviate the need for careful choice or design of contour embedding functions, we propose using NODEs to directly evolve the embedding of an input image into a pixel-wise dense semantic label. We evaluate our methods on kidney segmentation (KiTS19) and on salient object detection (PASCAL-S, ECSSD and HKU-IS). In addition to improving initial contours provided by deep learning models while using a fraction of their number of parameters, our approach achieves F β scores that are higher than several state-of-the-art deep learning algorithms.
This paper proposes to apply the Neural ODE framework (Chen et al 2018) for image segmentation. The method relies on contour delineation through Level Sets. Since contour estimation requires to solve an ODE, this naturally allows to apply the work presented in (Chen et al 2018). The method is here applied in two segmentation tasks: kidney segmentation and salient object detection.
The concept underlying the paper is interesting, and leverages on very recent advances in the field. The idea of learning the dynamics required to evolve segmentation contours is original and certainly appealing. Unfortunately the content of this work seems quite preliminary in terms of presentation and experiments.
First, the methodology is only sketched, while the motivation underlying the modelling rationale is often missing. For example, it is not clear what is the difference between "image evolution" and "contour evolution" models, besides the implementation details, and what motivates the definition of these two different modelling approaches in parallel. 
Second, the experimental paradigm is controversial. The application on medical imaging is overly simplistic, as the authors do not consider the original 3D image stack, but rather the set of corresponding 2D slices modelled independently. The paper seems to ignore the large variety of body organ segmentation methods already available to the community, most of them working in 3D (e.g. [1-4]). The paper should necessarily compare with respect to these approaches and, even more importantly, with respect to standard level sets methods. 
From the practical perspective, the proposed method builds upon the results obtained with the UNet, and therefore is characterised by an additional computational burden. Given that the the training of neural ODE is not straightforward and computational expensive, the use of this model for achieving a tiny accuracy improvement seems overkill for this kind of application. Moreover, the segmentation accuracy is still computed slice-by-slice in the 2D images, and no information is available for the consistency of the reconstruction in 3D (regularity over the vertical axis).  
Finally, the results reported in Table 3 are not clear, why the metrics of the competing methods are approximated (e.g. ~ 0.7), while for the proposed methods are given up to the 3rd decimal term?
[1] 3D Kidney Segmentation from Abdominal Images Using Spatial-Appearance Models. Fahmi Khalifa, Ahmed Soliman, Adel Elmaghraby, Georgy Gimel'farb, and Ayman El-Baz 1. 
[2] Automatic Detection and Segmentation of Kidneys in 3D CT Images Using Random Forests. Rémi Cuingnet, Raphael Prevost, David Lesage, Laurent D. Cohen, Benoit Mory, Roberto Ardon. Medical Image Computing and Computer-Assisted Intervention – MICCAI 2012: 15th International Conference, Nice, France, October 1-5, 2012, Proceedings, Part III
[3] Multi-organ localization with cascaded global-to-local regression and shape prior. Medical image analysis. Gauriau, R., Cuingnet, R., Lesage, D., & Bloch, I. (2015), 23(1), 70-83.
[4] Joint Classification-Regression Forests for Spatially Structured Multi-object Segmentation. Ben Glocker, Olivier Pauly, Ender Konukoglu, Antonio Criminisi. ECCV 2012 pp 870-881