## 4 Experiments
### Evaluation of Segmentation and Counting Performance
#### 4.3.2 Comparison with Other Methods  
In this part, some comparative experiments are applied for the yeast cell counting task. Some classical methods proposed in Section 2 and deep-learning-based methods proposed in Section 3 are compared, consisting Hough transformation [83], Otsu thresholding, Watershed, SegNet and U-Net-based segmentation approaches. Furthermore, we conduct some extra experiments using state-of-the-art approaches, containing Attention U-Net [66], Trans U-Net [67] and Swin U-Net [68].  
Due to the determination of \(k\) in clustering methods, such as \(k\)-means, is still an insoluble problem while counting; therefore, the clustering-based approaches cannot be applied here for dense tiny object counting. All comparative experiments have the same experimental setting, which can be referred to Section 4.1 for details. After image segmentation and object counting, the average evaluation indices are summarized in Table 4, and the example images of segmentation are shown in Figure 9.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Methods** & **A** & **D** & **J** & **P** & **C** & **H** \\ \hline PID-Net & 97.51 & 95.86 & 92.10 & 96.02 & 96.97 & 4.6272 \\ SegNet & 94.69 & 90.34 & 84.02 & 88.50 & 68.82 & 6.3604 \\ YeaZ (in [77]) & - & 94.00 & - & - & - & - \\ U-Net & 97.47 & 95.71 & 91.84 & 95.62 & 91.33 & 4.6666 \\ Attention U-Net & 96.62 & 93.36 & 88.96 & 92.67 & 83.44 & 5.1184 \\ Trans U-Net & 96.84 & 93.60 & 88.99 & 93.25 & 91.32 & 5.0715 \\ Swin U-Net & 96.47 & 92.99 & 88.32 & 92.43 & 91.95 & 5.3140 \\ Hough & 82.12 & 61.12 & 44.74 & 88.26 & 73.66 & 9.2486 \\ Otsu & 84.23 & 65.71 & 49.90 & 87.66 & 74.34 & 8.9165 \\ Watershed & 78.67 & 50.15 & 34.88 & 78.61 & 63.34 & 9.6873 \\ \hline \hline \end{tabular}
\end{table}
Table 4: The average segmentation evaluation indices of predicted images. A, D, J, P, C and H are abbreviations of the Accuracy, Dice, Jaccard, Precision, Counting Accuracy (in %) and Hausdorff Distance (in pixels/per image), respectively.  
Figure 8: The down-sampling block of PID-Net and PID-Net-M1.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Methods** & **A** & **D** & **J** & **P** & **C** & **H** \\ \hline PID-Net & 97.51 & 95.86 & 92.10 & 96.02 & 96.97 & 4.6272 \\ PID-Net-M1 & 96.90 & 94.71 & 90.05 & 94.71 & 67.84 & 5.0110 \\ PID-Net-M2 & 97.42 & 95.75 & 91.89 & 95.68 & 96.88 & 4.7204 \\ \hline \hline \end{tabular}
\end{table}
Table 3: The average segmentation evaluation indices of predicted images. A, D, J, P, C and H are abbreviations of the Accuracy, Dice, Jaccard, Precision, Counting Accuracy (in %) and Hausdorff Distance (in pixels/per image), respectively.  
From the evaluation indices summarized in Table 4, we can find that the PID-Net has the highest Accuracy, Dice, Jaccard, Precision and Counting Accuracy and the lowest Hausdorff distance, which means the proposed model performs best in the task of dense tiny object counting by comparing with other models. Even more, the Jaccard of PID-Net is higher than the YeaZ who proposed this yeast cell dataset. In general, the approaches based on deep learning perform better than the classical approaches.  
We find that the Counting Accuracy of several methods are very low abnormally, which may caused by the enormous difference between the GT and the predicted image. For instance, the single yeast cell image in Figure 9 performs unsatisfactory when the segmentation is not accurate. The segmentation results of SegNet and Attention-UNet have a large number of False Positive pixels, and the counting approach is based on the connected domain detection; hence, the value of \(\frac{|N_{pant}-N_{CT}|}{N_{GT}}\) is much higher than normal.  
From the best performance of the proposed PID-Net in the task of dense tiny object counting, we can infer that the down-sampling and skip connection part of PID-Net, which combines max-pooling and pixel interval down-sampling can obtain the feature maps of dense tiny objects and reconstruct the images better.