We present an approach to leaf level segmentation of images of Arabidopsis thaliana plants based upon detected edges. We introduce a novel approach to edge classification, which forms an important part of a method to both count the leaves and establish the leaf area of a growing plant from images obtained in a high-throughput phenotyping system. Our technique uses a relatively shallow convolutional neural network to classify image edges as background, plant edge, leaf-on-leaf edge or internal leaf noise. The edges themselves were found using the Canny edge detector and the classified edges can be used with simple image processing techniques to generate a region-based segmentation in which the leaves are distinct. This approach is strong at distinguishing occluding pairs of leaves where one leaf is largely hidden, a situation which has proved troublesome for plant image analysis systems in the past. In addition, we introduce the publicly available plant image dataset that was used for this work.