A Generalized Framework for Analytic Regularization of Uniform Cubic B-spline Displacement Fields

TL;DR

Deformable image registration is inherently ill-posed and benefits from regularization of the displacement field. The authors develop a generalized analytic framework using uniform cubic B-spline parameterization to compute five regularizers (diffusion, curvature, linear elastic, third-order, and total displacement) with a combined penalty $S = \mu_1 S_1 + \mu_2 S_2 + \mu_3 S_3 + \mu_4 S_4 + \mu_5 S_5$, enabling exact, fast calculations via tile-based composite matrix operators. The method achieves accuracy close to numerical approaches (maximum MLS difference $7.4\%$) while delivering up to two orders of magnitude speedups, demonstrated on DIR-Lab 4D-CT data and implemented in Plastimatch. It leverages precomputed $64 \times 64$ matrices per term, supports parallel execution, and allows seamless combination of regularizers for domain-specific smoothness constraints. This framework makes robust, physically plausible deformable registration more computationally tractable for large-scale medical imaging tasks.

Abstract

Image registration is an inherently ill-posed problem that lacks the constraints needed for a unique mapping between voxels of the two images being registered. As such, one must regularize the registration to achieve physically meaningful transforms. The regularization penalty is usually a function of derivatives of the displacement-vector field, and can be calculated either analytically or numerically. The numerical approach, however, is computationally expensive depending on the image size, and therefore a computationally efficient analytical framework has been developed. Using cubic B-splines as the registration transform, we develop a generalized mathematical framework that supports five distinct regularizers: diffusion, curvature, linear elastic, third-order, and total displacement. We validate our approach by comparing each with its numerical counterpart in terms of accuracy. We also provide benchmarking results showing that the analytic solutions run significantly faster -- up to two orders of magnitude -- than finite differencing based numerical implementations.

Figures (5)

• Figure 1: Example of intra-subject registration demonstrating the need for regularization.
• Figure 2: Example of intra-subject registration showing a coronal slice of the (a) fixed, (b) moving and (c) warped images, with the corresponding selected landmark point overlaid.
• Figure 3: Mean landmark separation (a) and and minimum Jacobian determinant of the transform $\boldsymbol{T}$ (b) as a function of B-spline control-point spacing and the weight $\mu_n$, over the 10 thoracic cases using curvature (left), linear elastic (middle), and third-order (right) regularizer for the masked images.
• Figure 4: Coronal slice showing: (a) the fixed and the moving CT images overlaid; (b) displacement field generated without regularization; (c)--(d) displacement field generated by the curvature regularizer for different values of $\mu_2$. Displacement field shown in black indicates displacement less than 1 mm, green indicates displacement of around 2 mm, yellow indicates displacement of around 5 mm, and red indicates displacement greater than 10 mm,
• Figure 5: Execution times for OpenMP implementations as a function of number of tiles (a) and function of thread count for different numbers of tiles (b).