3-D deformable image registration: a topology preservation scheme based on hierarchical deformation models and interval analysis optimization

Abstract

This paper deals with topology preservation in three-dimensional (3-D) deformable image registration. This work is a nontrivial extension of, which addresses the case of two-dimensional (2-D) topology preserving mappings. In both cases, the deformation map is modeled as a hierarchical displacement field, decomposed on a multiresolution B-spline basis. Topology preservation is enforced by controlling the Jacobian of the transformation. Finding the optimal displacement parameters amounts to solving a constrained optimization problem: The residual energy between the target image and the deformed source image is minimized under constraints on the Jacobian. Unlike the 2-D case, in which simple linear constraints are derived, the 3-D B-spline-based deformable mapping yields a difficult (until now, unsolved) optimization problem. In this paper, we tackle the problem by resorting to interval analysis optimization techniques. Care is taken to keep the computational burden as low as possible. Results on multipatient 3-D MRI registration illustrate the ability of the method to preserve topology on the continuous image domain.

Diffeomorphic demons: Efficient non-parametric image registration

“…Since the composition and inversion of B-spline transformations cannot be expressed on a B-spline basis, the advantage of using a parametric approach is not clear in this case. A complex optimization scheme on constrained B-splines has also been proposed in (Noblet et al, 2005) for this problem.…”

Review on Kernel based Target Tracking for Autonomous Driving

“…Telle and Ramdani [108] have utilized IA for camera calibration and 3D reconstruction, in which pixel coordinates were seen as two unknown but bounded variables (interval number), interval constraint propagation method was applied to propagate this uncertainties. IA can also be applied in auto-calibration [109], deformable image registration [110] and design of 2D image filter [111]. Considering the application of IA in the tracking framework (e.g., interval kernel function) can guarantee the truth-value of the tracking results and improve the robustness of tracking algorithm in autonomous driving.…”

Brain Functional Localization: A Survey of Image Registration Techniques

“…Some of the most important high-dimensional parametric models include: 1) FFD models with regular grid of control points, typically based on B-spline basis functions [51], [203], [214]- [222] or radial basis functions [57], [60], [77], [223]- [225]; 2) elastic-body spline warping models [226] such as thin plate spline (TPS) warping models [227]- [229] mainly used as interpolators for point set matching; 3) local affine transformations [166], [205], [206], [230]- [232]; 4) wavelet-mediated deformations [233]. A parametric particle method based on vector field formulation has also been developed [234].…”

“…Smoothness constraints based on the derivatives of the deformation field formulated as the bending energy have been used for a regular grid of control points in the FFD model [203]. Jacobian of transformation has also been incorporated as constraints for topology preservation via interval analysis optimization in a nonrigid registration algorithm [222].…”