The morphometric parameters of the corneal endothelium - cell density (ECD), cell size variation (CV), and hexagonality (HEX) - provide clinically relevant information about the cornea. To estimate these parameters, the endothelium is commonly imaged with a non-contact specular microscope and cell segmentation is performed to these images. In previous work, we have developed several methods that, combined, can perform an automated estimation of the parameters: the inference of the cell edges, the detection of the region of interest (ROI), a post-processing method that combines both images (edges and ROI), and a refinement method that removes false edges. In this work, we first explore the possibility of using a CNN-based regressor to directly infer the parameters from the edge images, simplifying the framework. We use a dataset of 738 images coming from a study related to the implantation of a Baerveldt glaucoma device and a standard clinical care regarding DSAEK corneal transplantation, both from the Rotterdam Eye Hospital and both containing images of unhealthy endotheliums. This large dataset allows us to build a large training set that makes this approach feasible. We achieved a mean absolute percentage error (MAPE) of 4.32% for ECD, 7.07% for CV, and 11.74% for HEX. These results, while promising, do not outperform our previous work. In a second experiment, we explore the use of the CNN-based regressor to improve the post-processing method of our previous approach in order to adapt it to the specifics of each image. Our results showed no clear benefit and proved that our previous post-processing is already highly reliable and robust.