## 1 Introduction  
Since its introduction to dentistry in 1998, cone beam computed tomographic (CBCT) imaging has become an essential radiographic technique for orthodontic diagnosis and treatment planning, particularly in situations requiring the evaluation of complex anatomical relationships within the maxillofacial skeleton [1; 2; 3]. The adoption of CBCT has been driven by its advantages in providing three-dimensional (3D) imaging over traditional two-dimensional radiography [4; 5; 6; 7; 8].  
It is recognized that CBCT imaging in orthodontics and pediatric dentistry exposes children to three to five times more radiation than adults during the same scanning session. However, patients with severe malocclusions and dentofacial deformities often require combined orthodontic treatment and maxillofacial surgery [9; 10]. In these cases, CBCT integrated with virtual surgical planning allows surgeons to simulate different treatment options and predict the postoperative facial profile [11; 12].  
Previous research has used comparisons of 3D visualizations of cranial models and configurations of landmarks on the skull and face for surgical planning [13; 14]. Basic formsof comparison, such as the evaluation of metric features and the calculation of distances between landmarks on 3D models, help determine the relative agreement or similarity between external structures or identify significant differences [15]. More complex data processing, such as landmark-based analysis through geometric morphometrics or surface-based processing, facilitates the exploration and representation of spatial relationships between craniofacial components [16; 17].  
Geometric morphometrics processes discrete points (expressed in \(X\), \(Y\), \(Z\) Cartesian coordinates) using General Procrustes Analysis (GPA) for alignment. Once aligned, these coordinates are used to compute the Procrustes distance (PD), a metric that indicates the dissimilarity or similarity between two or more shapes [18; 19]. Conversely, comparing two or more 3D surfaces [20] often requires more computationally intensive registration algorithms such as Iterative Closest Point (ICP), with differences expressed as nearest point-to-point distances or cumulative measures such as the Hausdorff distance.  
The size and shape of the facial skeleton are fundamental determinants of facial appearance. Small asymmetries in skeletal morphology are noticeable, but the changes resulting from orthodontic therapy or maxillofacial surgery are usually much more significant. Soft tissues, in particular, can either mask or accentuate deformities, but they do not always correspond directly to the underlying skeleton.  
Given these circumstances, orthodontics and maxillofacial surgery rely on long-term imaging techniques to monitor both soft and hard head tissues, primarily the profile, skeleton, and dentition, for therapy planning and orthodontic and orthognathic treatment. These techniques can also be applied to anthropology and forensic dentistry [21; 22]. The 3D treatment plan for orthodontic patients, which includes both soft and hard tissue considerations along with orthognathic surgery, allows for an accurate prognosis of the outcome when the skeletal basis of the face is available in two or more three-dimensional image sets.  
The aim of our study was to compare 3D visualizations of skull and facial models before and after surgery, to assess the degree of agreement or similarity, and to identify any morphological differences [23; 24]. All of these methods use advanced sensor systems, traditional signal and image processing tools, and computational intelligence methods to support the needs of 3D CBCT models in dentistry [25; 26]. The new 3D technology proposed in our research is based on CBCT recordings taken before and after orthodontic therapy, offering advantages such as (i) the accurate long-term planning of future maxillofacial surgery over several years and (ii) the ability to visualize future positions of the soft and hard tissues of the face in 3D space.