## 2 Background  
The aim of image registration is to compute a transformation to spatially align two images, where the source (moving) image is registered to the reference (fixed) image. The computed transformation is then applied to the moving image to create the moved image. In medical image registration, three basic transformation types are applied: rigid, affine and deformable. A rigid transformation includes translation and rotation and is represented by  
\[T_{rigid}(x)=Rx+t \tag{1}\]where \(R\) describes the rotation matrix, \(x\) a point and \(t\) the translation vector. For three-dimensional images, the rotation matrix and the translation vector require three parameters each. Thus, six parameters have to be calculated for a rigid transformation for three-dimensional images.  
The affine transformation is composed of translation, rotation, scaling and shearing. Matrix multiplication is used to merge the single transformations, as described by [33]. For three-dimensional images, the affine transformation is represented by a 4 \(\times\) 4 matrix that contains 12 parameters, including the translation parameters in the right column (parameters \(a_{14},a_{24},a_{34}\)), while the other nine parameters represent the combined values of rotation, scaling and shearing.  
\[T_{affine}=\begin{bmatrix}a_{11}&a_{12}&a_{13}&a_{14}\\ a_{21}&a_{22}&a_{23}&a_{24}\\ a_{31}&a_{32}&a_{33}&a_{34}\\ 0&0&0&1\end{bmatrix} \tag{2}\]  
Nonlinear (deformable) transformations are used to model local deformations in images that rigid and affine transformations cannot capture. These transformations are complex and do not preserve straightness or parallelism. The transformation is represented by a deformation (displacement) field \(\phi\):  
\[T_{deformable}(x)=x+\phi(x) \tag{3}\]  
Since deformable image registration is an ill-posed problem, regularization is necessary to ensure a smooth and plausible deformation field, e.g., by using penalty terms during the optimization of the transformation (iterative methods) or training of the registration network (deep learning-based) [34].  
Conventional registration methods such as SimpleElastix [35] or NiftyReg [31; 32] rely on solving a mathematical optimization model separately for each pair of images, which can be a slow and computationally intensive process. This involves iteratively maximizing a function that measures the similarity between the fixed and moved images. While some conventional methods can be accelerated using GPU implementations, this still requires a GPU for each registration process [36].  
In contrast, neural networks offer a faster and more efficient approach to image registration. Instead of solving an optimization problem for each new pair of images, a neural network takes two n-dimensional images as an input and outputs the transformation in a single forward pass through the neural network [12]. The weights of the neural network are optimized through training, resulting in a globally optimized function that can be used for the registration of any test image pair without requiring an iterative optimization process. This approach saves time both on the CPU and GPU compared to conventional methods [12].