## 3 Results  
The evaluation of the shape deformation of the skull before and after surgery was performed qualitatively and quantitatively. In the qualitative evaluations, the results of the color mapping of the local deformation are visually presented for the degrees of expansion and reduction between the preoperative skull and the postoperative skull, while in the quantitative evaluations, the proposed expansion and reduction indices of the regional cranial bones are presented. Figure 5 shows representative examples of the color-coded visualization of shape deformation (mm) in the skulls before and after surgery for severe and mild sagittal synostosis, severe and mild bilateral coronal synostosis, unilateral coronal synostosis, and unilateral lambdoid synostosis. For color-coded visualization, expansion is represented by cyan, blue, and purple, and reduction is represented by green, yellow, and red. With regard to severe sagittal synostosis, the entire parietal bones were expanded; specifically, the upper part of the parietal bone was significantly expanded, while the forehead area of the frontal bones and the entire occipital bone were reduced. In mild sagittal synostosis case, some parts of the parietal bones were slightly reduced, and the entire cranial bone was expanded and reduced but lower than in severe sagittal synostosis cases. For severe bilateral coronal synostosis, the forehead area of the frontal bones and posterior part of the parietal bones were expanded, with especially the occipital bone significantly expanded, while the upper part of frontal bones and the parietal bones overall were significantly reduced. In mild bilateral coronal synostosis case, the entire cranial bone showed less expansion and reduction than in severe bilateral coronal synostosis case. For unilateral coronal synostosis, the entire left frontal bone and the front part of the left parietal bone were significantly expanded, and the entire occipital bone was slightly expanded, while the entire right frontal and parietal bones were reduced. In unilateral lambdoid synostosis case, the entire frontal bones were slightly expanded, specifically, the posterior part of the left parietal bone and left part of the occipital bone were significantly expanded, while the anterior part of the parietal bones and the right part of the occipital bone were slightly reduced. Table 1 shows the average values of the expansion and reduction indices for all subjects with representative synostoses of craniosynostosis. In the sagittal synostosis case, the preoperative skull has a biparietal narrow and elongated form. However, after surgery, the quantification results showed that the frontal and occipital bones were reduced, and the parietal bones were expanded. In severe case, the expansion and reduction indices were greater than those in mild case; specifically, the reduction index in the occipital bone showed a considerable difference. In the bilateral coronal synostosis case, the clinically preoperative skull has a biparietal broad and short form. However, after surgery, the quantification results showed that the frontal bones were reduced, the parietal bones were similarly expanded and reduced, and the occipital bone was expanded. In severe case, the expansion and reduction indices were greater than in mild case, and the expansion index of the occipital bone especially showed the highest among all reduction and expansion indices. With regard to unilateral coronal synostosis, the preoperative skull has a forehead asymmetry and a short form. However, after surgery, the quantification results showed that left frontal and parietal bones and occipital bone were expanded while the right frontal and parietal bones were reduced. For unilateral lambdoid synostosis, the preoperative skull shows posterior skull asymmetry. However, after surgery, the quantification results showed that the left parietal bone was expanded, while the right parietal bone was reduced and the frontal and occipital bones were expanded.