## 5 Metrics
### Metrics for Object-Wise Cell Detection  
Generally, in the case of cell detection, the ground truth is given in terms of bounding boxes of the cells contained in the images. Here, the IoU metric of Equation (1) can be adapted to evaluate the degree of overlap between the ground truth bounding boxes (\(GT\)) and the predicted bounding boxes (\(S\)). IoU presholding can then be used to decide if a detection is correct or not. For a given IoU threshold \(\alpha\), a true positive (\(TP\)), i.e., a correct positive prediction, is a detection for which \(\text{IoU}(GT,S)\geq\alpha\) and a false positive (\(FP\)), i.e., a wrong positive detection, is a detection for which \(\text{IoU}(GT,S)<\alpha\). A false negative (\(FN\)) is an actual instance that is not detected.  
Given these adapted concepts, the Recall, Precision, and F-score metrics defined in Equations (2)-(4) can be used to evaluate cell detection algorithms. These are also used to compute the **Average Precision at a given IoU threshold \(\alpha\)**, denoted as AP@\(\alpha\), defined as the Area Under the Precision-Recall Curve (AUC-PR) evaluated at the IoU threshold \(\alpha\), given as  
\[\text{AP@}\alpha=\int_{0}^{1}p(r)\,dr.\]  
According to the Common Objects in Context (COCO) [94] evaluation protocol ([[https://cocodataset.org/#detection-eval](https://cocodataset.org/#detection-eval)]([https://cocodataset.org/#detection-eval](https://cocodataset.org/#detection-eval)), accessed on 28 August 2022), single values for \(\alpha\) can be chosen for thresholding IoU (generally equal to 0.5 or 0.75). Moreover, a set of thresholds can be chosen and the **mean Average Precision** mAP over these IoU thresholds considered for cell detection evaluation. With the usual choice [26;74;75] of values for \(\alpha\) from 0.5 to 0.95 with a step size of 0.05, mAP is thus given by  
\[\text{mAP}=\[EMAIL_ADDRESS]+[EMAIL_ADDRESS]+\ldots+[EMAIL_ADDRESS]}{10}.\]  
In the CTC [22], the detection accuracy of the methods, denoted as **DET**, is adopted to estimate how accurately each given object has been identified ([[http://celltrackingnale.net/evaluation-methodology/](http://celltrackingnale.net/evaluation-methodology/)]([http://celltrackingnale.net/evaluation-methodology/](http://celltrackingnale.net/evaluation-methodology/)), accessed on 28 August 2022). It is based on the comparison of the nodes of the acyclic oriented graphs representing the objects in both the ground truth and the computed object detection result. Exploiting the Acyclic Oriented Graph Matching measure for detection (AOGM-D) [95], that gives the cost of transforming the set of nodes of the computed objects into the set of ground truth nodes, DET is defined as  
\[\text{DET}=1-\frac{\text{min}(\text{AOGM-D},\text{AOGM-D}_{0})}{\text{AOGM-D} _{0}},\]  
where AOGM-D\({}_{0}\) is the cost of creating the set of ground truth nodes from scratch. DET always falls in the [0,1] interval, with higher values corresponding to better detection performance. The DET metric is averaged with the SEG metric described in Section 5.1 to provide the overall performance for the CSB  
\[\text{OP}_{CSB}=\frac{1}{2}(\text{DET}+\text{SEG}).\]