A global matching technique was applied for the global displacement caused by positional deviation or respiratory motion. In global matching, we perform a rigid 3D affine transformation. The transformation can be represented as Tglobal:
Fig. 4.8 Illustration of the overall scheme of the 3D temporal subtraction method  where 9x, 9y, and 9z are the rotation around x-, y-, and z-axes, respectively, and tx, ty, and tz are the displacement on the x-, y-, and z-axes, respectively. To obtain the voxel pairs between the current image and the transformed previous one, a linear interpolation is adopted. In this global matching procedure, the global shift vector, which is obtained from the template location, with the maximum of the cross correlation value obtained for the similarity of the current and previous images, is shown in Eq.4.6:
Here Ic and Ip show the current and previous image, respectively. Ic and Ip are mean values of the current and previous images, respectively.
Local Matching Based on the 3D Voxel Matching Technique
In our voxel matching technique, an image warping technique is first applied to the current and the previous image to obtain shift vectors which represent the extent of deformation of the previous image relative to the current image . Based on these shift vectors on the current image, the previous image can be warped to produce a temporal subtraction image. However, the temporal subtraction image obtained by the subtraction of the warped previous image from the current image usually contains considerable subtraction artifacts. With the voxel matching technique, for a given location in the current image, we initially identify the corresponding location in the warped previous image. We then search a voxel in the previous image within a small search volume, which is called a kernel, in order to identify the matching voxel which has a value identical or nearly equal to the value of the given voxel in the current image. This search for the matching voxel is repeated for all of the voxels in the current image. The warped previous image is then replaced by the matched voxel, warped previous image in which the voxel values are generally identical to or nearly equal to the voxel values in the current image except for the voxel values in new lesions or changes in existing abnormalities as shown in Fig. 4.9. The voxel matching technique is one of the optimized registration tools for removal of the subtraction artifacts in a temporal subtraction technique by searching the best matched voxels. In order to assess the temporal changes in two images, the voxel matching technique can be applied to 2D images such as chest radiographs by use of a pixel-matching method instead of the voxel matching in 3D images.