## 1 Introduction  
Craniosynostosis is one of the most common causes of cranial malformations in infants. It is a condition in which one or more cranial sutures becomes prematurely fused, resulting in an abnormal morphology as well as limited brain growth and intracranial hypertension. Craniosynostosis is classified according to the type of premature fused cranial suture involved, such as the sagittal, coronal, metopic, or lambdoid types [1]. Figure 1 shows the four types of craniosynostosis used in our study, such as the sagittal, bilateral coronal and unilateral coronal, and unilateral lambdoid synotoses. The normal skull in Figure 1 shows that all sutures are open, presenting five cranial bones, the left and right frontal bones, the left and right parietal bones, and the occipital bone where the sagittal suture is located between the two parietal bones, the coronal suture is located between the frontal and parietal bones, and the lambdoid suture is located between the parietal and occipital bones. Figure 1 presents an example of sagittal synostosis, which accounts for 40-60% of all cases of craniosynostosis. In this type, premature fusion of the sagittal suture causes the skull shape to show biparial narrowing and elongation, because the lateral growth of the skull is restricted and its anteroposterior growth continues. As shown in Figure 1, bilateral coronal synostosis accounts for 5-10% of all cases of craniosynostosis,with premature fusion of both coronal sutures causing the shape of the skull to be broad, high, and short. Unilateral coronal synostosis, shown in Figure 1d, accounts for 20-25% of all cases of craniosynostosis, and premature fusion of one side the coronal suture flattens the forehead toward one side and causes it to protrude on the opposite side. Figure 1e presents an example of unilateral lambdoid synostosis, which accounts for 0-5% of all cases of craniosynostosis, with premature fusion of one side of the lambdoid suture flattening the back of the skull on one side and causing it to protrude on the opposite side. Analyzing the surgical effects of craniosynostosis requires a quantitative assessment of the morphological changes in the skull before and after the surgery, but it is still performed by a clinician's visual inspection of the overall head shape or by measuring the cephalic index (CI) [2,3] and cranial vault asymmetry index (CVAI) [4,5]. The CI is usually calculated as the ratio of the maximum width of the skull divided by the maximum length from the top view of the volume as rendered by a 3D CT, and is commonly used as a severity index  
Figure 1: Volume rendering images of normal skull (**first row**) and skulls of four types of craniosynostosis patients (**second to fifth rows**: sagittal, bilateral coronal, unilateral coronal, and unilambdoid synostoses). (**a**) Front view, (**b**) back view, (**c**) top view, (**d**) right view, (**e**) left view. (LFB: left frontal bone, RFB: right frontal bone, LPB: left parietal bone, RPB: right parietal bone, OB: occipital bone). for sagittal and bilateral coronal cases characterized by a symmetric shape. However, it is difficult to determine the maximum skull width and length consistently, since the location of the most prominent points seen from the top view depends on the skull shape of synostosis [6; 7]. The CVAI is calculated by dividing the difference between two diagonal cranial diameters, 30 degrees from the \(Y\)-axis, by the short cranial diagonal diameter and multiplying by 100 in the cross-sectional plane at level 3. The level 3 is one-third of the distance from the reference plane including nasion and both tragion to the vertex. The CVAI is used as a severity index for unilateral coronal and unilateral lambdoid cases characterized by an asymmetric shape. However, it is difficult to reflect the deformation, in which case the position of the prominent area is different from the position of the estimated diagonal diameter with 30 degrees from the \(Y\)-axis.