**2.1 Subjects**

Fifteen patients from the Department of Oral and Maxillofacial Surgery, Orthodontics and Prosthodontics at the Peking University School and Hospital of Stomatology were recruited for this study. The inclusion criterion was an apparent facial asymmetry with a mandibular deviation of at least 3 mm from the facial midline, which is perpendicular to the interpupillary line at the soft tissue nasion when the patient is seated in a natural head position. The exclusion criteria were a history of previous craniofacial trauma, orthognathic surgery, orthodontic treatment, or congenital anomalies. This study was approved by the bioethics committee of the Peking University School and Hospital of Stomatology (PKUSSIRB-20163113) and was conducted in accordance with the guidelines and regulations for research involving human subjects. All participants were fully informed of the experimental purpose and procedure and provided an informed consent form prior to participating in the study.

**2.2 Experimental equipment and software**

A Face Scan 3D sensor system (3D-Shape Corp, Germany, Erlangen) was used to collect three-dimensional facial data from each patient, which were obtained in only 0.2–0.8 s with high accuracy (0.1 mm). The scanning range was 270°–320°, the imaging principle was raster scanning using 5 million charge-coupled device (CCD) pixels, and the approximate number of point cloud was 10 000, and 20 000 triangular meshes are formed.

For data processing, we used the reverse engineering software Geomagic Studio 2013 (3D System, USA, Morrisville), which is used to process three-dimensional facial data and conduct SRP extraction. The WPA algorithm developed in this study was based on the Python programming language, which optimises the objective function of the PA algorithm. The PA objective function, F, is shown in formula ①, the weight factor, wi, is shown in formula ②, and the WPA objective function F' is shown in formula ③.

where wi (i = 1,2, ..., 32) is the weight factor for each facial landmark (assigned according to the degree of asymmetry of the landmarks), LMK_Org is the original model landmark set, LMK_Mir is the mirror model landmark set, LMK_Orgi and LMK_Miri (i = 1,2, ..., n) are the corresponding landmarks in the original and mirror landmark set, respectively, Q is the spatial change matrix (contains translation, rotation, and scale; the scale value is 1 in this study), and p is the number of landmarks.

**2.3 Data capturing and processing **

When acquiring the three-dimensional facial data, we calibrated the equipment prior to use to ensure accurate image acquisition. Patients were guided by a clinician to a natural head position at distance of 135 cm from the scanner and a sitting position with both eyes looking forward, keeping the Frankfort horizontal (FH) plane parallel to the floor. Data was obtained when the facial expression was naturally relaxed. The criteria for the face scan data were an effective display of facial contours, a high-resolution image, no obvious movement, and a closed mouth.

Geomagic Studio 2013 was used to process images, which included removing extra data, smoothing the shells, and filling small holes. The original three-dimensional facial model was adjusted to the natural head position so that the FH plane of the natural head position coincided with the XZ plane of the global coordinate system and the sagittal plane coincided with the YZ plane of the global coordinate system. Three experienced clinical professors completed the extraction of anatomical landmarks from each original facial model (Model_Org). Thirty-two anatomical landmarks were selected from the overall region, including the glabella, nasion, pogonion, and alare et al. An example of a selected landmark is illustrated in Fig 1. Each researcher performed the extraction three times and calculated the mean coordinate value of the original landmark (LMK_Org). Next, the centre of gravity of the original model was moved to the origin of the global coordinate system, and the data was saved in a .obj file.

**2.4 Determining SRP**

2.4.1 Initial alignment of the original and mirror model

For all 15 case models in this study, the original model (Model_Org) was initially superimposed onto its YZ-plane mirror model to obtain an optimal weight distribution of the 32 PA landmarks. Geomagic Studio 2013 software was used for the global ICP registration function. During the process, the original model was fixed, and the mirror model was floated. The mirror model (Model_Mir) was obtained following superimposition, and the corresponding initial mirror landmarks were then established (LMK_Mir).

2.4.2 Test group_1: Determining SRP using PA algorithm

The three-dimensional coordinates of all landmarks in the original and mirror images (LMK_Org and LMK_Mir; 32 pairs of landmarks in total) were derived and entered into the PA algorithm program, which was based on the Python language, without weight differences. The transformation matrix of the mirror model was then calculated and loaded onto the Model_Mir using Geomagic Studio 2013. Finally, the SRP of the facial data for each patient was constructed by taking the union of the original and mirror models (Model Uni_PA) in Geomagic Studio using the function ‘‘plane’’ and "symmetry", defined as ‘SRP_PA’.

2.4.3 Test group_2: Determining SRP using WPA algorithm

Similarly, the three-dimensional coordinates of all landmarks in the original and mirror images (LMK_Org and LMK_Mir; 32 pairs of landmarks in total) were derived and entered into the WPA algorithm program, which was based on the Python language. The weight factor for each landmark was automatically calculated based on the distance of paired landmarks. For example, a landmark pair with good symmetry would be relatively close together post initial registration and would thus be given more weight. Conversely, a landmark pair with poor symmetry would be relatively far apart and would thus be given less weight.

The weighted landmarks of LMK_Org and LMK_Mir were superimposed three-dimensionally based on the least-weighted squares, so that optimal superimposition was obtained for the 32 pairs of landmarks and the WPA transformation matrix of the mirror model (Model_Mir) was derived. The transformation matrix was loaded onto the Model_Mir using Geomagic Studio 2013. Finally, the SRP of facial data for each patient was constructed by taking the union of the original and mirror models (Model Uni_WPA), the same procedure as test group_1, defined as ‘SRP_WPA’.

2.4.4 Reference group: Determining the ground truth

Studies have shown that the alignment of the original and mirror models for SRP abstraction, based on areas defined by experts having good symmetry, exhibits sufficient adaptability for facial asymmetry cases, but the reliance on expert definitions reduced the degree of algorithm automation. The SRP of an algorithm based on professional expertise and empirical data was regarded as the ground truth in this study. Regions with good facial symmetry from the original and mirror models (Model_Org and Model_Mir) were manually selected by senior doctors using Geomagic Studio software, and regional registration was conducted with the two models (Model_Org fixed and Model_Mir floated). Finally, the SRP of the facial data for each patient was constructed by taking the union of the original and mirror models (Model Uni_Ref). These SRPs were defined as the ground truth (‘SRP_Ref’).

The SRPs constructed using the WPA, PA, and professional algorithms are shown in Fig 2.

**2.5 SRP measurement evaluation**

2.5.1 Angle error of planes

For each of the 15 three-dimensional mandibular deviation models, the angles between SRP_PA and SRP_Ref and between SRP_WPA and SRP_Ref were calculated and recorded as Err_Ang_PA and Err_Ang_WPA, respectively. The average and standard deviation of the angle error for each sample were also calculated.

2.5.2 Position error of the mirrored landmarks

The position error of the mirrored landmarks was defined as a new quantitative index to evaluate SRP, which may further validate the result of the weighted landmarks. The position error indicator was designed to obtain the weight distribution of the WPA algorithm landmarks and professional landmarks (implied empirical information) by calculating the distance between corresponding landmarks in the WPA and professional algorithms. If the two weights are consistent, then the mirror landmark overlap is suitable, and the position error is small. Conversely, if the weights are inconsistent, then the position error is large. The mean value of the position error reflects the consistency between the SRPs of the WPA and professional algorithms in accounting for the weight distribution of the global facial landmarks.

The mirror landmarks of each model (LMK_PA and LMK_WPA) were obtained from the mirror and original models using the SRP_PA and SRP_WPA. The mirror landmarks of the reference group (LMK_Ref) were similarly obtained. The global position error was defined as the average distance of the 32 landmarks pairs in LMK_PA and LMK_Ref and in LMK_WPA and LMK_Ref. During this process, the original model was fixed in the test and reference groups. The closer each mirror landmark constructed by the SRPs of the test groups was to the same landmark in the reference group (i.e. the smaller the position error), the closer the SRP to the reference plane. The global position error was calculated based on 32 paired landmarks (Err_LMK_WPA and Err_LMK_PA) (Fig 3).

Huang has shown that facial asymmetry is more obvious in the lower face than upper face[10]. For mandibular deviation patients, the degree of landmarks asymmetry in the lower part of the face is significantly higher than those in the middle and upper parts. Therefore, the weight distribution of features in different regions should differ and cannot be analysed with the global position error. Thus, we also evaluated the regional position error of the three facial partitions. The regional position error was calculated for landmarks in each facial third partitions: 4 landmarks in the upper third, 17 in the middle third, and 11 in the lower third, named Err_LMK_WPA_Up and Err_LMK _PA_Up, Err_LMK_WPA_Mid and Err_LMK _PA_Mid, and Err_LMK_WPA_Low and Err_LMK _PA_Low, respectively. The average and standard deviation of the global and regional position error were calculated for each sample.

2.5.3 FAI error

The FAI error was calculated based on the SRP constructed for the test groups of the 15 facial data and defined as the sum of the distance from the medical landmark to the SRP and the difference between bilateral landmarks and the SRP. FAI_PA, FAI_WPA, and FAI_Ref were obtained according to formula ④. Err_FAI_WPA and Err_FAI_PA were defined as the difference between the FAI values of the WPA and professional algorithm and the difference between the FAI values of the PA and professional algorithms, respectively. The average value and standard deviation of the FAI error of each sample were calculated.

*Md*irepresents the distance from the medical landmark to the SRP. *Rd*i and *Ld*i represent the differences between the right landmark and the SRP, and that between the left landmark and the SRP, respectively.

**2.6 Statistical analysis**

Statistical analysis was conducted using SPSS software (Version 21, SPSS Inc., Chicago, IL, USA). A K-S normality test was conducted for the angle error (of two groups), the global position error (of two groups), the regional position error (of six groups), and the FAI error (of two groups) to assess data distribution (15 calculated values per group).

The workflow of the experimental procedures and evaluation methods are shown in Fig. 4. We performed a paired* t*-test analysis of the position error of both the WPA and PA algorithm groups of 15 patients to evaluate the overlapping differences of the WPA and PA algorithms in terms of global and regional landmarks. A statistical significance was set at P < 0.05.

A one-way ANOVA analysis was performed on regional landmarks of position error to examine whether differences in the position error of different facial partitions were statistically significant. A homogeneity-of-variance test was also performed. Tukey’s honesty significance test was used for multiple comparisons. A paired *t*-test analysis was also conducted to compare angle and FAI errors.