Non-rigid refinement step

Only rigid model is not enough for multi-modal carotid image registration, since the neck of the patient is in different state of bending and twisting during MR and US imaging. Based on rigid initialization step, a non-rigid transformation is optimized to obtain high registration accuracy. Surfaces interpolated by 3D B-spline from 2D contours are used as features instead of centerline due to more local information in surfaces.

In non-rigid model, registration accuracy is highly dependent on the point set matching, because the model is more flexible than rigid model. To better handle the mismatching between ECA and ICA especially near bifurcation, we employ the same CICP algorithm with the correct label configuration from the first step.

A widely-used model, 3D thin-plate-spline (TPS) model [13], is employed as the non-rigid model in CICP algorithm. It fits a mapping function f(m_i) between moving point set {m_i} and corresponding fixed subset {fi′} by minimizing the energy function below:

where λ ≥ 0 is a regularization constant. Suppose each point p is represented in homogeneous coordinates, i.e. p = [1, p_x, p_y, p_z]^T, the mapping function is as follows:

where D is a 4 × 4 matrix for the affine transformation, and P is a 4 × N_m warping coefficient matrix for non-rigid deformation. The υ(m_i) is an N_m × 1 vector for each point m_i, where each element υ_j(m_i) = ‖m_i−m_j‖₂. The TPS model has a closed-form solution, which can be solved efficiently. Please refer to [18] for more details.