Background: Corneal endothelium (CE) images provide valuable clinical information regarding the health state of the cornea. Computation of the clinical morphometric parameters requires the segmentation of endothelial cell images. Current techniques to image the endothelium in vivo deliver low quality images, which makes automatic segmentation a complicated task. Here, we present two convolutional neural networks (CNN) to segment CE images: a global fully convolutional approach based on U-net, and a local sliding-window network (SW-net). We propose to use probabilistic labels instead of binary, we evaluate a preprocessing method to enhance the contrast of images, and we introduce a postprocessing method based on Fourier analysis and watershed to convert the CNN output images into the final cell segmentation. Both methods are applied to 50 images acquired with an SP-1P Topcon specular microscope. Estimates are compared against a manual delineation made by a trained observer. Results: U-net (AUC = 0.9938) yields slightly sharper, clearer images than SW-net (AUC = 0.9921). After postprocessing, U-net obtains a DICE = 0.981 and a MHD = 0.22 (modified Hausdorff distance), whereas SW-net yields a DICE = 0.978 and a MHD = 0.30. U-net generates a wrong cell segmentation in only 0.48% of the cells, versus 0.92% for the SW-net. U-net achieves statistically significant better precision and accuracy than both, Topcon and SW-net, for the estimates of three clinical parameters: cell density (ECD), polymegethism (CV), and pleomorphism (HEX). The mean relative error in U-net for the parameters is 0.4% in ECD, 2.8% in CV, and 1.3% in HEX. The computation time to segment an image and estimate the parameters is barely a few seconds. Conclusions: Both methods presented here provide a statistically significant improvement over the state of the art. U-net has reached the smallest error rate. We suggest a segmentation refinement based on our previous work to further improve the performance.

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Resolution in deep convolutional neural networks (CNNs) is typically bounded by the receptive field size through filter sizes, and subsampling layers or strided convolutions on feature maps. The optimal resolution may vary significantly depending on the dataset. Modern CNNs hard-code their resolution hyper-parameters in the network architecture which makes tuning such hyper-parameters cumbersome. We propose to do away with hard-coded resolution hyper-parameters and aim to learn the appropriate resolution from data. We use scale-space theory to obtain a self-similar parametrization of filters and make use of the N-Jet: a truncated Taylor series to approximate a filter by a learned combination of Gaussian derivative filters. The parameter σ of the Gaussian basis controls both the amount of detail the filter encodes and the spatial extent of the filter. Since σ is a continuous parameter, we can optimize it with respect to the loss. The proposed N-Jet layer achieves comparable performance when used in stateof-the art architectures, while learning the correct resolution in each layer automatically. We evaluate our N-Jet layer on both classification and segmentation, and we show that learning σ is especially beneficial when dealing with inputs at multiple sizes.