Unconstrained ℓ1 — regularized minimization with interpolated transformations for heart motion compensation

Motion compensation constitutes a challenging issue in minimally invasive beating heart surgery. Since the zone to be repaired has a dynamic behaviour, precision and surgeon´s dexterity decrease. In order to solve this problem, various proposals have been presented using ℓ2-norm. However, as they present some limitations in terms of robustness and efficiency, motion compensation is still considered an open problem. In this work, a solution based on the class of ℓ1 Regularized Optimization is proposed. It has been selected due to its mathematical properties and practical benefits. In particular, deformation is characterized by cubic B-splines since they offer an excellent balance between computational cost and accuracy. Moreover, due to the non-differentiability of the functional, the logarithmic barrier function is used for generating a standard optimization problem. Results have provided a very good tradeoff between accuracy and efficiency, indicating the potential of the proposed approach and proving its stability even under complex deformations.