We present a novel convolutional neural network architecture designed for dense segmentation in a subset of the dimensions of the input data. The architecture takes an N-dimensional image as input, and produces a label for every pixel in M output dimensions, where 0< M < N. Large context is incorporated by an encoder-decoder structure, while funneling shortcut subnetworks provide precise localization. We demonstrate applicability of the architecture on two problems in retinal optical coherence tomography: segmentation of geographic atrophy and segmentation of retinal layers. Performance is compared against two baseline methods, that leave out either the encoder-decoder structure or the shortcut subnetworks. For segmentation of geographic atrophy, an average Dice score of 0.49+-0.21 was obtained, compared to 0.46+-0.22 and 0.28+-0.19 for the baseline methods, respectively. For the layer-segmentation task, the proposed architecture achieved a mean absolute error of 1.305+-0.547 pixels compared to 1.967+-0.841 and 2.166+-0.886 for the baseline methods.