## 2 Materials and Methods
**The landmark-based comparison of hard tissues** in the second part of the study consisted of 30 individuals who had undergone corrective surgery of the upper and lower jaw as in the previous case. Two medical imaging data sets were available for each patient, obtained by cone beam computed tomography (CBCT). The sample consisted of 11 females and 19 males. The mean age at the start of treatment was 23.30 years--25.30 years for  
Figure 1: Preoperative and postoperative en face and profile measurements. women and 22.14 years for men. A Mann-Whitney test showed no statistically significant differences in age or time interval between men and women at the 5% significance level. For all participants, facial geometry was recorded using CBCT protocol. For each CBCT dataset, a set of 12 landmarks (or dx/sin, zy dx/sin, go dx/sin, al dx/sin, na, inc, pg, gn) was manually collected using the functionalities available in the Anatomage Invivo software (Figure 2). The entire dataset was first registered using partial Procrustes analysis, an alternative to generalized Procrustes analysis that standardizes the variation in position and rotation while preserving the variation in size. Once registered, the dataset was analyzed for differences between the preoperative and postoperative groups using one-way PERMANOVA. In addition, the centroid size was extracted from the coordinates and compared between the preoperative and postoperative sets using both a paired Wilcox test and an unpaired Mann-Whitney test. In the second step, the postoperative point configuration was fitted to the preoperative configuration using Ordinary Procrustes Analysis (OPA), which preserves the size variation. OPA involves the superimposition of two sets of points (as opposed to generalized Procrustes analysis, which typically involves three or more sets of points). The Procrustes distance was then calculated between the pairs of registered coordinates and was analyzed for differences based on the biological sex and age of the patient. In addition, Procrustes residuals were computed pairwise, tested for sex and age differences, and plotted on a mean point configuration to illustrate the magnitude and variance of changes associated with maxillofacial surgery. **Surface and landmark-based comparison of soft tissues** in the third part of the study was used on 30 patients (the same as in the second part of this study) from the Department of Stomatology, Second Medical Faculty, with a mean age of 23.57 years, resulting in a total of 60 facial scans. Each patient was scanned using the Vectra M3 3D non-invasive, ambient light-based scanner (Canfield Scientific Inc., Fairfield, NJ, USA). All individuals were scanned from the front and were instructed to maintain a neutral facial expression throughout the process. Raw data were processed using Mirror PhotoTools software, version 7.4 (Canfield Scientific Inc.), which generated a surface model represented as a mesh of several hundred thousand triangles. All images were taken with the patients in their natural upright head position to ensure that the Frankfort plane was parallel to the floor. A set of 26 features was manually collected for each facial model (Figure 3). For each surgical patient, a pair of preoperative and postoperative 3D facial models were directly compared. Initially, the postoperative 3D model was designated as the primary (reference) model, while the preoperative model was designated as the secondary (comparison) model. Iterative closest point (ICP) registration was then performed to align the secondary model  
Figure 2: The tested configuration of 12 landmarks (green color points: or dx/sin, zy dx/sin, go dx/sin, al dx/sin, na, inc, pg, gn) manually collected by the Anatomage Invivo software. to the primary model, preserving the size differences. The closest point distances were then calculated for each vertex of the primary model. Statistical descriptors, including root mean square (RMS) and 75 th percentile (PERC75), were extracted and used as measures of deviation.