A novel sparsity-based variational model for nonrigid medical image registration

Nonrigid image registration aims at finding a displacement field densely aligning a moving image and a fixed one, which is a fundamental and challenging task in the field of medical image analysis. Variational models are widely used for nonrigid image registration, which consists of a data term and a regularization term. A common assumption of nonrigid medical image registration of the same modality is that the aligned pixels of the moving and fixed images have the same intensity values, which implies that the difference of registered moving and fixed images is highly sparse. However, existing data terms like sum of squared difference (SSD) and sum of absolute difference (SAD), (i.e., an ₂ norm and an ₁ norm, respectively) are not optimal choices for modeling sparsity. As to the regularization term, existing models usually assume the displacement field to be smooth, and hence the registration transformation is smooth or diffeomorphic, whereas for medical images, this assumption is not valid in many situations due to complex anatomy structures and possible lesions in images, especially along the edges of different organs and tissues. In this paper, we employ the ₀ norm as the data term to capture the sparsity of difference of registered images and regularize the displacement fields as functions of bounded generalized deformation (BGD), which allow both smooth and discontinuous displacement fields. To our knowledge, this is the first nonrigid image registration model that directly takes sparsity of the data term into account. We then introduce a log-sum type sparse approximation of the ₀-norm data term, which turns the NP-hard ₀ problem into a novel nonconvex and nonsmooth variational registration problem, called the log-BGD model. We prove the existence of its global solutions. We also derive a fast primal-dual iteratively reweighting (PDIR) algorithm for solving the model and prove its convergence. Numerical examples show the effectiveness of both the proposed log-BGD model and the derived algorithm.