Pulmonary fissure detection in computed tomography (CT) is a critical component for automatic lobar segmentation. The majority of fissure detection methods use feature descriptors that are hand-crafted, low-level, and have local spatial extent. The design of such feature detectors is typically targeted toward normal fissure anatomy, yielding low sensitivity to weak, and abnormal fissures that are common in clinical data sets. Furthermore, local features commonly suffer from low specificity, as the complex textures in the lung can be indistinguishable from the fissure when the global context is not considered. We propose a supervised discriminative learning framework for simultaneous feature extraction and classification. The proposed framework, called FissureNet, is a coarse-to-fine cascade of two convolutional neural networks. The coarse-to-fine strategy alleviates the challenges associated with training a network to segment a thin structure that represents a small fraction of the image voxels. FissureNet was evaluated on a cohort of 3706 subjects with inspiration and expiration 3DCT scans from the COPDGene clinical trial and a cohort of 20 subjects with 4DCT scans from a lung cancer clinical trial. On both data sets, FissureNet showed superior performance compared with a deep learning approach using the U-Net architecture and a Hessian-based fissure detection method in terms of area under the precision-recall curve (PR-AUC). The overall PR-AUC for FissureNet, U-Net, and Hessian on the COPDGene (lung cancer) data set was 0.980 (0.966), 0.963 (0.937), and 0.158 (0.182), respectively. On a subset of 30 COPDGene scans, FissureNet was compared with a recently proposed advanced fissure detection method called derivative of sticks (DoS) and showed superior performance with a PR-AUC of 0.991 compared with 0.668 for DoS.