Sparse Constrained Transformation Model Based on Radial Basis Function Expansion: Application to Cardiac and Brain Image Registration

Estimating robust transformations based on noisy landmark correspondences is challenging and cannot be ensured to be exact. In this paper, we propose a novel sparse transformation model based on corresponding landmarks. First, we construct a new transformation model that uses compact supported radial basis functions (CSRBFs) with multiple supports, with a least-squares cost function constrained by the l
₁
and l
₂
norms of the elastic and affine deformation coefficients used to estimate the CSRBF coefficients. This sparse model can be used to select CSRBFs with different supports and construct a robust deformation field. Then, the relationship between the CSRBF coefficients and the bending energy of the deformation field is analyzed in a reproducing kernel Hilbert space; this bending energy is introduced into the cost function as a regularization term. The cost function is optimized by using the fast iterative shrinkagethreshold algorithm to compute coefficients in the transformation model. To demonstrate the performance of our sparse transformation model, we combine it with robust point matching to simultaneously estimate the correspondence and transformation between landmarks. Experiments on synthetic data, brain images, and cardiac images show that the transformations estimated by our sparse transformation model are robust to noised landmark correspondences, preserving registration accuracy, minimizing the bending energy of the deformation field, and preserving the topology of the deformation field.