## 5 Discussion
### Comparison to Published Multistage Registration Methods  
In classical (iterative) methods, approaches to multistage registration (affine and deformable or rigid-affine-deformable) exist. For instance, the SimpleElastix [35] documentation describes a non-rigid registration. Here, the affine algorithm is applied first and the bspline (deformable) algorithm afterwards. In the iterative registration method NiftyReg [31; 32], the algorithm reg_aladin applies a rigid and affine registration by default. Thus, by connecting reg_aladin (rigid and affine) and the deformable registration with reg_f3d, a multistage rigid-affine-deformable registration is obtained. The ANTS (Advanced Neuroimaging Tools) [49] toolbox provides an elastic registration that uses affine and deformable transformations. In addition, a variant of the symmetric image normalization method (SyN) combines rigid, affine and deformable transformations (SyNRA). However, these classical methods are often time-consuming as each new image pair requires iterative registration. Thus, the registration of a single image pair takes several  
Figure 7: Example results of the registration of two volumes from the M\({}^{2}\)OLIE liver dataset. The images show central slices of the coronal plane. The fixed images, moving images and result (moved) images are overlaid by the liver segmentations (column 1–3). The resulting composites for image and segmentation show the fixed data in blue and the moving/moved data in red. The arrow in the result image indicates the area with medical implausible deformation. The deformation field is visualized in RGB channels. minutes, depending on the registration method and the hardware used (CPU or GPU). With neural networks, the registration process speeds up. In comparison, our proposed multistage network takes under a second to register an image pair on CPU or GPU. Several neural networks have been published with multistage approaches for medical image registration [13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28]. Refs. [13; 14; 15; 16; 17; 18; 19; 20; 21; 23; 28] apply neural networks with two stages. In the first stage/subnetwork, an affine transformation is calculated, whereas in the second stage/subnetwork, a deformable transformation is calculated. Refs. [22; 24; 25; 26] also utilize an affine network as the first subnetwork, but this is followed by a cascade of multiple deformable subnetworks. And Ref. [27] applies Generative Adversarial Networks (GANs) for affine-deformable transformation. The generator consists of a modality translator (for multimodal image registration), an affine-transformation regressor and a nonlinear-deformation regressor. However, only affine and deformable transformations were applied in these papers. To the best of our knowledge, no multistage network has yet been published that combines the three transformations rigid, affine and deformable. Therefore, our proposed multistage network represents a new approach to medical image registration and addresses the challenge of larger rigid deformations beyond the typical capture ranges of affine registration approaches [29; 34] that might occur in medical images. A fair comparison of the results of our proposed rigid-affine-deformable multistage network with results in papers from published multistage approaches is difficult to achieve. In the published papers, different datasets, image modalities and target structures were used than in our experiments. The differences in the performance of the published multistage methods and our proposed multistage network could therefore also result from the different datasets and modalities used. Moreover, different target structures such as femur [14], brain [13; 15; 17; 21; 24; 25; 26; 27; 28], liver or kidney (used in our experiments) contain different degrees of rigid, affine or deformable transformations, which may affect the performance of the multistage methods.