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

TL;DR

This paper introduces a fast, analytical framework for regularizing cubic B-spline displacement fields in image registration, supporting multiple regularizers and outperforming numerical methods in speed.

Contribution

A generalized analytical framework for regularizing cubic B-spline displacement fields supporting five regularizers, with validation and benchmarking against numerical methods.

Findings

Analytical solutions are up to 100 times faster than numerical methods.

The framework accurately matches numerical regularizers.

Supports five different regularization types.

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.