Conditional Entropies as Over-Segmentation and Under-Segmentation Metrics for Multi-Part Image Segmentation

In this paper, we define two conditional entropy measures for performance evaluation of general image segmentation. Given a segmentation label map and a ground truth label map, our measures describe their compatibility in two ways. The first one is the conditional entropy of the segmentation given the ground truth, which indicates the oversegmentation rate. The second one is that of the ground truth given the segmentation, which indicates the under-segmentation rate. The two conditional entropies indicate the trade-off between smaller and larger granularities like false positive rate and false negative rate in ROC, and precision and recall in PR curve. Our measures are easy to implement, and involve no threshold or other parameter, have very intuitive explanation and many good theoretical properties, e.g., good bounds, monotonicity, continuity. Experiments show that our measures work well on Berkeley Image Segmentation Benchmark using three segmentation algorithms, Efficient Graph- Based segmentation, Mean Shift and Normalized Cut. We also give an asymmetric similarity measure based on the two entropies and compared it with Variation of Information. The comparison revealled that our method has advantages in many situations.We also checked the coarse-to-fine compatibility of segmentation results with changing parameters and ground truths from different annotators.