## 4 Discussion  
We presented an evolutionary multi-objective class-solution learning approach for DIR, which we applied to breast MR DIR problems of increasing difficulty. We used our approach to learn weights typically used in DIR software, but the approach was general and could straightforwardly be used to consider other parameters, as well. The multi-objective class-solution approach presented in this work kept the benefits of a multi-objective perspective [10] and was more efficient than multi-objective patient-specific parameter optimization, by allowing parallel computation of DIRs without any further tuning or optimization, thereby much more so facilitating the use of multi-objective DIR in clinical practice. Although the concept of a class solution implies the existence of a unique solution that solves all instances of a particular problem well, we considered a set of class solutions, i.e., a multi-objective class solution, to be more appropriate for the problem of DIR. The idea that a single trade-off can work for all DIR cases of a certain type comes from a single-objective optimization perspective, but even among registration problems of the same type, parameter settings that give the desired results can differ [10]. A multi-objective optimization approach instead has the ability to capture high-quality trade-offs among which the preferred outcome is likely present. Moreover, selecting this outcome is insightful and easy [28]. It was also observed that what the users thought was a high-quality solution coincided with a mean TRE (very) close to the minimum mean TRE obtained, indicating that the results obtained by our method can be considered reliable. The evaluation of the quality of the DIR outcomes, in the absence of a ground truth, relied on the annotated landmarks, which has known limitations (it may not be representative of the quality of the deformation throughout the entire volume). A realistic phantom-based study could potentially circumvent this, but such an elaborate study is out of the scope of this work. We further saw that when the solutions were visually inspected by observers, what the users thought was a high-quality solution coincided with a mean TRE (very) close to the minimum mean TRE obtained [28], indicating that the mean TRE was representative of the overall quality of the solutions. Since EAs are state-of-the-art for multi-objective black-box optimization and given the limited amount of data present for the DIR problems studied in this work, the choice of an EA as a machine learning technique, as opposed to more widely-used machine learning approaches such as, e.g., convolutional neural networks, seems reasonable. Using the EA to tackle this problem, even in the presence of very limited data, was possible since the problem concerned a limited number of parameters (i.e., a low-dimensional manifold) to be learned and the learning of point sets. The EA employed in this work was specifically chosen as it was designed for continuous, multi-objective problems, and it has been shown to perform well for those problems [25]. Given that the bulk of the computation was in the DIRs during meta-optimization and given the performance of this EA on different benchmark problems compared to other EAs, we do not expect that choosing a different EA as a meta-optimizer would yield significant differences in terms of computation time. The results of this approach, which may be considered an evolutionary multi-objective machine learning approach [31], indicate that deriving a multi-objective class solution for DIR is feasible, even with a limited amount of data, for DIR problem classes that are solvable and for which the variation within the class is not too large, as in the case of prone-prone breast MR image registration.