4.1 The WPA SRP was more closely aligned with the ground truth plane than the standard PA SRP
The weighted algorithm is an important innovation of this study. The degree of the landmarks symmetry could be evaluated quantitatively and used as landmark weight factors to construct an SRP. Our WPA algorithm is designed to assign a small weight for landmarks with poor symmetry, post initial global ICP superimposition of the original and mirror models, and a large weight for landmarks with good symmetry. The weight calculation method was based on the reciprocal of the distance between the paired landmarks, which represents an inverse relationship between the distance and the corresponding assigned weight. Based on superimposition using least-weighted squares, all original and mirror PA landmarks were assigned different weights. The solution to the PA landmark set system (the WPA objective function) was minimised, thus achieving an optimal overlap result of the original and mirror landmarks.
The results indicated that the average angle error of WPA group for all enrolled patients with mandibular deviation was < 2°, although there was no significant result when compared with the average angle error of the standard unweighted PA algorithm (of which the error was > 2°), the result of the WPA SRP was closer to the ground truth (Fig. 2), and the angle error displayed a downward trend.
Wu et al. showed that the angle difference between the two planes is easily perceived when it is > 6°. The angle error between the WPA SRP and the reference plane was < 2°, which indicates that the accuracy of the WPA SRP was almost equal to that of the reference plane and therefore had a better clinical suitability than the PA SRP. Furthermore, the stability level of the WPA algorithm, with a standard deviation of 0.81°, was significantly higher than that of the PA algorithm, which had a standard deviation of 1.08°.
Additionally, the FAI value calculated for the WPA algorithm was closer to the professional result than was the FAI value calculated for the PA algorithm. Furthermore, the WPA FAI for patients with mandibular deviation was closer to the ground truth plane than the PA FAI. These results confirmed that the WPA algorithm performed better than the PA algorithm in constructing facial SRPs for facial asymmetry (mandibular deviation).
4.2 A new SRP evaluation indicator: the position error of mirror landmarks
In previous SRPs studies of the face and skull, SRP evaluation indicators have primarily included the angle and FAI errors [24, 25]. These two indicators can assess the global proximity of SRP, but neither can quantitatively analyse facial landmark asymmetry. In this study, we proposed to use the position error indicator as a novel SRP evaluation tool.
The mirror landmarks differed between the test and reference SRPs for mirroring the original facial model, while the original model was the same between the test and reference groups. Table 1 indicates that the mean values of the global position errors of the WPA and PA algorithms were 3.64 mm and 4.54 mm, respectively, and that the difference between them was statistically significant. This indicates that the global overlapping degree of the WPA algorithm mirrored features and reference mirrored features was more accurate than that of the PA algorithm. The weight distribution of the WPA algorithm was also significantly more accurate than that of the PA algorithm; the weight factor of the WPA algorithm had a significant effect.
The mean value of the regional position error for the upper, middle, and lower partitions also reflected the degree of consistency between the weight distribution of the WPA SRP and the reference SRP for each facial partition. The mean position error of the WPA algorithm was smaller than that of the PA algorithm for all three facial partitions. This difference was significant, indicating that the WPA algorithm for each facial partition was close to the professional algorithm.
Additionally, the position error of the WPA algorithm for the upper and lower parts of the face was considerably smaller than that of the PA algorithm, while that for the middle part was close to that of the PA algorithm. This is because the WPA algorithm allocated a lower weight for lower facial landmarks to reduce their influence on the global overlapping degree, while the upper landmarks were assigned higher weights to increase the overlapping degree, thus accounting for professional experience in the weight distribution of the landmarks. Compared with the PA algorithm without weight distribution, the position error of the WPA in each region was optimised, and an ideal SRP construction result was obtained.
4.3 Limitations and further research to improve the three-dimensional facial SRP
Previous studies on the original-mirror alignment method are divided with regards to using the ICP and PA algorithms. Among them, ICP is an algorithm that does not refer to anatomical landmarks. Although the reliability and repeatability of the ICP algorithm have verified when used for constructing SRPs with data from patients with normal facial symmetry, facial asymmetry data affects algorithm’s performance making SRPs construction unfeasible for patients with severe asymmetry. Scholars have since improved the global ICP algorithm by manually selecting facial regions with good symmetry for original and mirror models; the clinical suitability of this modified ICP algorithm has improved to some extent[26, 27]. This algorithm is referred to as the regional ICP algorithm, and although it reduces the degree of automation by introducing human interference, it remains suitable for use in oral clinics. Therefore, the regional ICP algorithm was used as the ground truth in this study to evaluate the accuracy of our proposed algorithm.
One of the differences between the PA and ICP algorithms is that SRP extraction using the PA algorithm relies more on anatomical facial landmarks, which is consistent with clinical diagnosis and treatment. PA algorithm is applicable for symmetry patients, but the asymmetric PA landmarks will have a Pinocchio effect on the PA algorithm.
One source of improvement is to filter PA landmarks. Landmarks have been sorted through the recursive PA algorithm, deleting the obvious asymmetric landmarks (outliers) and using the remaining for PA operation to avoid their interference . However, for patients with complex facial deformities (in which most landmark symmetries are not ideal), this algorithm may eliminate too many landmarks and tends to be locally over-optimised.
Our study has proposed another way to improve the standard PA algorithm by adding a weighted system. We hypothesised that by analysing the distance between the corresponding original and mirror landmarks post initial alignment, the degree of symmetry could be evaluated quantitatively and used as landmark weight factors to construct an SRP with personalised feature weight assignments. Our WPA algorithm did not have a reduced degree of automation and could therefore simulate the expression of the reference value weight of anatomical landmarks assigned according to clinical experience. This is advantageous with regards to SRP construction. Our results also indicated that the WPA algorithm was suitable for patients with complex mandibular deviation. However, the WPA algorithm tested in this study had some limitations.
First, the quantitative indicator of landmarks asymmetry (the reciprocal of the distance between paired landmarks) was indirectly obtained. To set the key parameters for the landmark weight factors, global ICP algorithm was used to initiate the registration of the original and mirror models. One-way to address this is to use an intelligent landmark weighting strategy based on direct morphological feature analysis, artificial intelligence, and deep learning technology, to improve the accuracy and rationality of landmark weight distribution leading to better SRP constructions that simulate expert clinical diagnosis.
Second, anatomical landmarks in this study need to be selected manually. We expect that our WPA algorithm will further combine mathematical facial mask, automatically extracting a general face mesh, thus improving its clinical suitability. Cases of mandibular deviation between 5 mm and 23 mm were quantitatively analysed in this study; although sample cases should be further expanded to evaluate our method’s suitability for different types and degrees of facial deformities to provide guideline for clinical application. Therefore, testing our method on samples representing a wider range of facial deformities is warranted.