For a proper three-dimensional reconstruction of histology serial sections, adjustment of the slices is necessary for combining serial sections. Deformations occur due to the sectioning and acquisition pro-cess of the microscopic analysis of histology. By reconstructing the deformations with a transformation of the sections, a mathematical correction on the images can be applied. By using image registration, a transformation function is searched for to minimize differences between the histology slices. Due to the non-linearity of the distortions, prior knowledge is required in order to have a solvable problem. Additional information in the form of elasticity regularisation is considered. Implementing the elastic regularisation with a finite element method, provides a continuous transformation function With the continuous function, in a natural way, alignment can be monitored for folding transformations. In this work, the (bi-)linear and (bi-)quadratic elements for the finite element method are implemented and compared with the finite difference method. It is observed that for the different kinds of elements, the (bi-)linear elements yield best results with the validity of the transformation. Moreover, the computa- tional costs for the bi-linear elements are the cheapest. Compared with the finite difference method, the differences in accuracy are not noteworthy but the computational time of the finite element method is longer. Furthermore, to steer the matching in an accurate direction, improvements are proposed by applying local stiffness of the elements or adding soft constraints on the volume of the elements. This results in significant improvements in the transformation. For these two approaches it is observed that local stiffness is more restrictive than volume-preserving. Solving the optimization problem, a Gauss-Newton method to search for descent directions is applied. A matrix-based and a matrix-free approach of elasticity regularisation is considered in the linear system of finding a descent direction. While the matrix-free approach decreases the memory usage, the computational costs are significantly increased.