We discuss the orthogonality problem of moving grids, where we minimize a cost function with a regularization term and improve the orthogonality of grids by making the angle between grids close to 90  .The initial grid used is obtained by a well-established method known as the grid deformation method. We will then replace the cost function in the orthogonality problem with sum of squared differences (SSD) to discuss the image registration problem. We will discuss the non-uniqueness of solutions, existence of optimal solutions and prove the existence of Lagrange multipliers of the image registration problem using the Direct Method in Calculus of Variation and then derive an optimality system based on the construction of a Lagrangian functional from which optimal transformations can be calculated.