Geometric and topological features of corneal nerve fibers in confocal microscopy images are important indicators for the diagnosis of common diseases such as diabetic neuropathy. Quantitative analysis of these important biomarkers requires an accurate segmentation of the nerve fiber network. Currently, most of the analysis are performed based on manual annotations of the nerve fiber segments, while a fully automatic corneal nerve fiber extraction and analysis framework is still needed. In this paper, we establish a fully convolutional network method to precisely enhance and segment corneal nerve fibers in microscopy images. Based on the segmentation results, automatic tortuosity measurement and branching detection modules are established to extract valuable geometric and topological biomarkers. The proposed segmentation method is validated on a dataset with 142 images. The experimental results show that our deep learning-based framework outperforms state-of-the-art segmentation approaches. The biomarker extraction methods are validated on two different datasets, demonstrating high effectiveness and reliability of the proposed methods.