We present a novel approach where we address image registration with the concept of a sparse kernel machine. We formulate the registration problem as a regularized minimization functional where a reproducing kernel Hilbert space is used as transformation model. The regularization comprises a sparsity inducing l1-type norm and a well known l2 norm. We prove a representer theorem for this type of functional to guarantee a finite dimensional solution. The presented method brings the advantage of flexibly defining the admissible transformations by choosing a positive definite kernel jointly with an efficient sparse representation of the solution. As such, we introduce a new type of kernel function, which enables discontinuities in the transformation and simultaneously has nice interpolation properties. In addition, location-dependent smoothness is achieved within the same framework to further improve registration results. Finally, we make use of an adaptive grid refinement scheme to optimize on multiple scales and for a finer control point grid at locations of high gradients. We evaluate our new method with a public thoracic 4DCT dataset.