In this article, we present a segmentation algorithm for assessing retinal image quality with respect to the visibility of the macular region. An image is considered of acceptable quality if the macular region is clearly visible and entirely in the field of view. Additionally, for acceptable images, the method is able to locate the fovea with a maximal error of 0.34 mm. The algorithm is based on a lightweight fully-convolutional network, several thousand times smaller than state-of-the-art networks investigated so far in preliminary studies. We obtain near-human performance for assessing macula visibility and fovea localization. The presented method can easily be embedded in tabletop or handheld retinographs, decreasing the number of ungradable images, saving both patient and physician time. It is an important step towards automatic screening of retinal pathologies, including diabetic retinopathy, which is a major global healthcare issue.