A Tournament of Transformation Models: B-Spline-based vs. Mesh-based Multi-Objective Deformable Image Registration

Abstract: The transformation model is an essential component of any deformable image registration approach. It provides a representation of physical deformations between images, thereby defining the range and realism of registrations that can be found. Two types of transformation models have emerged as popular choices: B-spline models and mesh models. Although both models have been investigated in detail, a direct comparison has not yet been made, since the models are optimized using very different optimization methods in practice. B-spline models are predominantly optimized using gradient-descent methods, while mesh models are typically optimized using finite-element method solvers or evolutionary algorithms. Multi-objective optimization methods, which aim to find a diverse set of high-quality trade-off registrations, are increasingly acknowledged to be important in deformable image registration. Since these methods search for a diverse set of registrations, they can provide a more complete picture of the capabilities of different transformation models, making them suitable for a comparison of models. In this work, we conduct the first direct comparison between B-spline and mesh transformation models, by optimizing both models with the same state-of-the-art multi-objective optimization method, the Multi-Objective Real-Valued Gene-pool Optimal Mixing Evolutionary Algorithm (MO-RV-GOMEA). The combination with B-spline transformation models, moreover, is novel. We experimentally compare both models on two different registration problems that are both based on pelvic CT scans of cervical cancer patients, featuring large deformations. Our results, on three cervical cancer patients, indicate that the choice of transformation model can have a profound impact on the diversity and quality of achieved registration outcomes.

• The paper demonstrates that comparing B-spline and mesh models via MO-RV-GOMEA improves understanding of deformable image registration outcomes.
• It shows that mesh models yield more localized and precise deformations compared to B-spline models, especially in complex anatomical regions.
• The study highlights that multi-objective optimization enhances registration diversity and quality, promising better clinical decision-making.

Detailed Summary of "A Tournament of Transformation Models: B-Spline-based vs. Mesh-based Multi-Objective Deformable Image Registration"

This paper investigates the two predominant transformation models used in deformable image registration (DIR) — B-spline models and mesh models — in the context of multi-objective optimization. It provides the first direct comparison by applying the Multi-Objective Real-Valued Gene-pool Optimal Mixing Evolutionary Algorithm (MO-RV-GOMEA) to both transformation types. This study is vital in understanding how the choice of transformation model affects the diversity and quality of registration outcomes.

Transformation Models and Optimization Strategy

B-Spline and Mesh Models

B-spline models represent deformations through a grid of control points, offering smooth deformable mappings but with limitations in large deformations. Mesh models, on the other hand, utilize tetrahedral meshes that can handle complex physical deformations by representing local geometry and ensuring fold-free transformations. The choice between them impacts the registration’s ability to model realistic deformations.

Figure 1: 2D illustration of how the B-spline and mesh transformation models decompose the image domain and allow for local objective re-evaluations.

Multi-Objective Optimization with MO-RV-GOMEA

Both transformation models were optimized using MO-RV-GOMEA, which excels in leveraging partial evaluations that enhance optimization efficiency and reduce computational costs. Here, the novel integration of MO-RV-GOMEA with the B-spline model was achieved by decomposing the image domain into localized regions, allowing simultaneous partial optimizations.

Figure 2: Optimization loops of the three registration approaches compared in this work.

Experimentation and Results

Experiment Setup

The experiments focused on CT scans from three cervical cancer patients, evaluating two registration problems per patient: a simple bladder isolation and a multi-organ registration including bones, sigmoid, and rectum. The quality and diversity of registrations were assessed based on image similarity and deformation magnitude objectives, using a post-hoc voxel-based deformation metric for consistency.

Comparative Findings

Results show that direct optimization using MO-RV-GOMEA leads to superior performance compared to traditional gradient-descent approaches, particularly in delivering diverse registration options. In all cases, the mesh-based model demonstrated more localized and precise deformations compared to the broader adjustments observed with the B-spline model. These differences are noticeable in the registration of complex anatomical areas where the mesh model adhered closely to real anatomical displacement.

Figure 3: Sagittal slices of the source and target images for Patient 1.

Figure 4: Comparison of registrations found by the three approaches for Patient 1.

Figure 5: Comparison of registrations for Patient 2.

Figure 6: Comparison of registrations for Patient 3.

Practical and Theoretical Implications

This research highlights the critical role of selecting appropriate transformation models in medical image registration, suggesting that mesh models may offer superior lineage for problems with complex deformations. The employment of multi-objective optimization techniques like MO-RV-GOMEA is shown to enhance outcome diversity and quality, especially valuable in clinical contexts where bespoke solutions may be required.

Conclusion

This paper successfully bridges a gap in direct comparative studies of DIR transformation models by leveraging MO-RV-GOMEA across different model types. Future work might explore resolutions and multi-modal scenarios, potentially extending these methods to openly validate alternative transformation models and objectives under various practical constraints. Such advancements promise to enhance clinical decision-making and therapeutic precision, tailoring image registration solutions to patient-specific anatomical intricacies.