For two-dimensional elasticity problems radial basis functions with high continuity are explored in the classical Galerkin elasticity representation. In the polynomial part of the approximations a few Trefftz functions are included. Wendland radial basis functions are chosen for the Galerkin functions in the elasticity representation. This will lead to compactly supported displacement functions with sufficient continuity. In this study no integrals are needed and submatrices in the discretization show sparse patterns and some symmetry. In a sequence of numerical test examples accurate stresses and displacements have been obtained. As a special application of this meshless algorithm, we considered medical image registration tasks in which information from CT (computed tomography) and MRI (Magnetic Resonance Imaging) or ultra sound images are combined. Medical images for the same region in a human body (liver, lungs, blood vessels, for example) can look a little different after using different scanning techniques. Organ shapes have to be mapped from one image to another image. From the numerical test examples it appears that the derived displacement functions and meshless discretization approach are suitable for elastic medical image registration tasks.