A pipeline of unsupervised image analysis methods for extraction of geometrical features from retinal fundus images has previously been developed. Features related to vessel caliber, tortuosity and bifurcations, have been identified as potential biomarkers for a variety of diseases, including diabetes and Alzheimer’s. The current computationally expensive pipeline takes 24 minutes to process a single image, which impedes implementation in a screening setting. In this work, we approximate the pipeline with a convolutional neural network (CNN) that enables processing of a single image in a few seconds. As an additional benefit, the trained CNN is sensitive to key structures in the retina and can be used as a pretrained network for related disease classification tasks. Our model is based on the ResNet-50 architecture and outputs four biomarkers that describe global properties of the vascular tree in retinal fundus images. Intraclass correlation coefficients between the predictions of the CNN and the results of the pipeline showed strong agreement (0.86 - 0.91) for three of four biomarkers and moderate agreement (0.42) for one biomarker. Class activation maps were created to illustrate the attention of the network. The maps show qualitatively that the activations of the network overlap with the biomarkers of interest, and that the network is able to distinguish venules from arterioles. Moreover, local high and low tortuous regions are clearly identified, confirming that a CNN is sensitive to key structures in the retina.