Abstract: Slice interpolation in medical images is one of the important concepts in medical image processing in which many attempts have been made to improve it. In general, it is divided into two groups that are intensity-based interpolation and object-based interpolation. This article is an overview of object-based interpolation, based on image registration. Due to the various types of image registration and curvature registration as the latest one, interpolation is done on the basis of this type of registration. According to this, there would be an optimization problem which is solved by the use of Euler-Lagrange. In this regard, an optimization problem converted into a partial differential equation that is solved by using finite difference method. Then, by applying the solution an optimal displacement vector for moving the image is obtained.