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The need for labour intensive pixel-wise annotation is a major limitation of many fully supervised learning methods for segmenting bioimages that can contain numerous object instances with thin separations. In this paper, we introduce a deep convolutional neural network for microscopy image segmentation. Annotation issues are circumvented by letting the network being trainable on coarse labels combined with only a very small number of images with pixel-wise annotations. We call this new labelling strategy 'lazy' labels. Image segmentation is stratified into three connected tasks: rough inner region detection, object separation and pixel-wise segmentation. These tasks are learned in an end-to-end multi-task learning framework. The method is demonstrated on two microscopy datasets, where we show that the model gives accurate segmentation results even if exact boundary labels are missing for a majority of annotated data. It brings more flexibility and efficiency for training deep neural networks that are data hungry and is applicable to biomedical images with poor contrast at the object boundaries or with diverse textures and repeated patterns.