The most general manner to injectively align true and predicted segments
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Kirilov et al (2019) develop a metric, called Panoptic Quality (PQ), to evaluate image segmentation methods. The metric is based on a confusion table, and compares a predicted to a ground truth segmentation. The only non straightforward part in this comparison is to align the segments in the two segmentations. A metric only works well if that alignment is a partial bijection. Kirilov et al (2019) list 3 desirable properties for a definition of alignment: it should be simple, interpretable and effectively computable. There are many definitions guaranteeing a partial bijection and these 3 properties. We present the weakest: one that is both sufficient and necessary to guarantee that the alignment is a partial bijection. This new condition is effectively computable and natural. It simply says that the number of correctly predicted elements (in image segmentation, the pixels) should be larger than the number of missed, and larger than the number of spurious elements. This is strictly weaker than the proposal in Kirilov et al (2019). In formulas, instead of |TP|> |FN| + |FP|, the weaker condition requires that |TP|> |FN| and |TP| > |FP|. We evaluate the new alignment condition theoretically and empirically.
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