# Automatic Tortuosity Estimation of Nerve Fibers and Retinal Vessels in Ophthalmic Images
## 4 Experimental Results
### Retinal Vessels Tortuosity Grading  
In order to evaluate the tortuosity measurement performance of the proposed method on the RET-TORT dataset, we carried out a two steps experiments: In the first experiment, we use the manual annotation of the RET-TORT in order to show the effectiveness of the tortuosity metrics without accounting for possible mistakes of the segmentation. For each tortuosity metric, we calculated the tortuosity for arteries and veins images, respectively. Then we get the Spearman Correlation score of each metric correlated with tortuosity grading ground truth. The higher the correlation score is, the more accurate the metric is for the vessels tortuosity measurement. From the manual segmentation column of Table 3, we found that the TC get the best score for arteries tortuosity while our Kexp is the best for veins. But some tortuosity metrics (e.g., DCI, ICM, ACR) have low correlations and lead some errors.  
In the second experiment, we further validated our proposed method with the automated segmentation result on the RET-TORT dataset. We calculated Spearman Correlation scores of the tortuosity metrics on the segmentation results and compare their performances. The automatic segmentation pipeline is illustrated as Figure 5, we first perform the local phase filters enhancement on the original vessel images (Figure 5d-f). Then we segment the vessels by the Infinite Perimeter Active Contour with Hybrid Region (IPACHR) model (Figure 5g-i). After that, we get the skeletonization results of the segmentation and keep the most important vessel centerline for further tortuosity measurement (Figure 5j-l). As shown in the automated segmentation column of Table 3, all the  
\begin{table}
\begin{tabular}{l c c c c c} \hline \hline  & \multicolumn{2}{c}{**Manual Segmentation**} & \multicolumn{2}{c}{**Automated Segmentation**} & \\ \cline{2-5}  & **Arteries** & **Veins** & **Arteries** & **Veins** & **Error (\%)** \\ \hline WAC & 0.919 & 0.814 & 0.877 & 0.768 & 8.8 \\ DCI & 0.787 & 0.589 & 0.734 & 0.621 & 8.4 \\ TC & **0.949** & 0.853 & 0.919 & 0.812 & 7.1 \\ CHD & 0.801 & 0.662 & 0.756 & 0.638 & 6.9 \\ AC & 0.922 & 0.837 & 0.893 & 0.801 & 6.5 \\ ICM & 0.684 & 0.575 & 0.661 & 0.542 & 5.6 \\ CUR & 0.813 & 0.701 & 0.784 & 0.677 & 5.3 \\ SCC & 0.850 & 0.770 & 0.827 & 0.745 & 4.8 \\ SQC & 0.925 & 0.826 & 0.901 & 0.812 & 3.8 \\ MAC & 0.820 & 0.814 & 0.801 & 0.795 & 3.8 \\ ACR & 0.792 & 0.656 & 0.812 & 0.629 & 3.4 \\ TD & 0.890 & 0.760 & 0.912 & 0.753 & 2.9 \\ AAC & 0.838 & 0.695 & 0.841 & 0.677 & 2.1 \\ Kexp & 0.945 & **0.868** & **0.928** & **0.857** & **1.9** \\ \hline \hline \end{tabular}
\begin{tabular}{l c c c c} \hline \hline  & \multicolumn{2}{c}{Error = \(|\text{avg}(\text{SC}_{\text{manual}})-\text{avg}(\text{SC}_{\text{automated}})| \times 100\%\).} & \\ \hline \hline \end{tabular}
\end{table}
Table 3: Spearman Correlation scores of tortuosity metrics for retinal vessels with manual and automated segmentation.  
tortuosity metrics are calculated based on the centerline, and the Spearman Correlation scores of each metric are computed correlated with tortuosity grading ground truth. Our Kexp is also computed by using our exponential curvature metric on the centerline. From the results, we found that our Kexp achieved the best Spearman scores on the automatic segmentation group. In addition, when comparing results between the manual segmentation and the an automated segmentation group, the Spearman scores of automated group is lower than mannual ones. Because the IPACHR segmentation model lead some segmentation errors. The automated segmentation results of our Kexp estimation process is also slightly lower than manual ones with 1.9% difference ratio. Therefore, our proposed tortuosity estimation method is more accurate and robust than other metrics.