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We consider the problem of segmenting cell nuclei instances from Hematoxylin and Eosin (H&E) stains with weak supervision. While most recent works focus on improving the segmentation quality, this is usually insufficient for instance segmentation of cell instances clumped together or with a small size. In this work, we propose a two-step post-processing procedure, Split and Expand, that directly improves the conversion of segmentation maps to instances. In the Split step, we split clumps of cells from the segmentation map into individual cell instances with the guidance of cell-center predictions through Gaussian Mixture Model clustering. In the Expand step, we find missing small cells using the cell-center predictions (which tend to capture small cells more consistently as they are trained using reliable point annotations), and utilize Layer-wise Relevance Propagation (LRP) explanation results to expand those cell-center predictions into cell instances. Our Split and Expand post-processing procedure is training-free and is executed at inference-time only. To further improve the performance of our method, a feature re-weighting loss based on LRP is proposed. We test our procedure on the MoNuSeg and TNBC datasets and show that our proposed method provides statistically significant improvements on object-level metrics. Our code will be made available.