4.1 The WPA SRP was more closely aligned than the standard PA SRP with the ground truth plane
The weighted algorithm is an important innovation of this study, the degree of symmetry of the landmarks 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 after the initial global ICP superimposition of the original and mirror models and a large weight for landmarks with good symmetry. The weight calculation method is based on the reciprocal of the distance between the paired landmarks, which reflects the 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. As a result, the optimal overlap result of the original and mirror landmarks was achieved.
The results indicated that the average angle error of WPA group for the 15 patients with mandibular deviation was less than 2°, and although there was no statistically significant result when compared with the average angle error of the standard unweighted PA algorithm (of which the error was greater than 2°), the result of the WPA SRP was closer to the ground truth (Fig. 2), and the angle error displayed a downward trend.
Jia Wu et al. showed that the angle difference between the two planes is easily perceived when it is more than 6°. The angle error between the WPA SRP and the professional plane was less than 2°, which indicates that the accuracy of the WPA SRP was almost equal to that of the professional plane and therefore had 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 also closer to the ground truth plane than was 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 studies of SRPs of the face and skull, SRP evaluation indicators have mainly included the angle error and FAI error[24, 25]. These two indicators can assess the global proximity of the 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 professional SRPs for mirroring the original facial model, while the original model was the same between the test and professional groups. The results in Table 1 indicate 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 the two was statistically significant. This indicates that the global overlapping degree of the WPA algorithm mirrored features and professional mirrored features was more accuracy 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 professional 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 statistically 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 of the face was close to that of the PA algorithm. This occurred 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 research on the original-mirror alignment method is mainly divided with regard to the use of the ICP and PA algorithms. Among them, ICP is an algorithm that does not refer to anatomical landmarks. Although scholars have verified the reliability and repeatability of the ICP algorithm when used for constructing SRPs with data from patients with normal facial symmetry, facia asymmetry data affects the algorithm’s performance and makes the construction of SRPs unfeasible for patients with severe asymmetry. Scholars have since improved the global ICP algorithm by manually selecting facial regions with good symmetry for the original and mirror models; the clinical suitability of this modified ICP algorithm has improved to some extent[26, 27]. This algorithm is called the regional ICP algorithm. Although the regional ICP algorithm reduces the degree of automation by introducing human involvement, research has shown that the regional ICP algorithm is suitable for use in oral clinics. Therefore, the regional ICP algorithm, which has been screened by experts, was used as the ground truth in this study to evaluate the accuracy of our proposed algorithm.
The significant difference 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. Some scholars have confirmed that the PA algorithm has a good applicability for symmetry patients, but similarity to ICP algorithm, asymmetry PA landmarks will have a Pinocchio effect on the PA algorithm.
One source of improvement is to filter PA landmarks. Some scholars have sorted landmarks through the recursive PA algorithm, deleting the obvious asymmetric landmarks (outliers) and using the remaining landmarks for PA operation to avoid the interference of asymmetric landmarks. However, for patients with complex facial deformities (in which most of the landmark symmetries are not ideal), this algorithm may eliminate too many landmarks and tend to be locally over-optimised.
Our study proposes another way to improve the standard PA algorithm, which is to add a weighted system. We hypothesised that by analysing the distance between the corresponding original and mirror landmarks after the initial alignment, the degree of symmetry of the landmarks could be evaluated quantitatively and used as landmark weight factors to construct an SRP with personalised feature weight assignments. The WPA algorithm developed in this study did not have a reduced degree of automation caused by human intervention and could therefore simulate the expression of the reference value weight of anatomical landmarks assigned according to clinical experience. Our study showed that this was advantageous with regard to SRP construction, and our results 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 asymmetry of the landmarks (the reciprocal of the distance between paired landmarks) was indirectly obtained; to set the key parameters for the landmark weight factors, the global ICP algorithm was used to initialise the registration of the original and mirror models. A potential way to address this limitation is to use an intelligent landmark weighting strategy based on direct morphological feature analysis, artificial intelligence, and deep learning technology, which can further improve the accuracy and rationality of landmark weight distribution and lead to better SRP constructions that simulate expert clinical diagnosis.
Second, only cases of mandibular deviation between 5 mm and 23 mm were quantitatively analysed in this study, and sample cases need to be further expanded to analyse the statistical and measurement suitability of our method for different types and degrees of facial deformities in order to provide guidance for clinical application. Therefore, testing our method on samples representing a wider range of facial deformities is a necessary area of further research.