# Incompressible image registration using divergence-conforming B-splines

## TL;DR

This paper introduces a novel image registration method using divergence-conforming B-splines to ensure exact volumetric preservation, improving accuracy and flexibility over previous approaches.

## Contribution

The paper presents a new framework for incompressible image registration with divergence-free velocity fields using divergence-conforming B-splines, enabling exact volume preservation.

## Key findings

- Achieves exact divergence-free velocity fields at any point in the domain.
- Demonstrates improved accuracy and volume preservation compared to state-of-the-art methods.
- Provides theoretical insights into numerical incompressibility error.

## Abstract

Anatomically plausible image registration often requires volumetric preservation. Previous approaches to incompressible image registration have exploited relaxed constraints, ad hoc optimisation methods or practically intractable computational schemes. Divergence-free velocity fields have been used to achieve incompressibility in the continuous domain, although, after discretisation, no guarantees have been provided. In this paper, we introduce stationary velocity fields (SVFs) parameterised by divergence-conforming B-splines in the context of image registration. We demonstrate that sparse linear constraints on the parameters of such divergence-conforming B-Splines SVFs lead to being exactly divergence-free at any point of the continuous spatial domain. In contrast to previous approaches, our framework can easily take advantage of modern solvers for constrained optimisation, symmetric registration approaches, arbitrary image similarity and additional regularisation terms. We study the numerical incompressibility error for the transformation in the case of an Euler integration, which gives theoretical insights on the improved accuracy error over previous methods. We evaluate the proposed framework using synthetically deformed multimodal brain images, and the STACOM11 myocardial tracking challenge. Accuracy measurements demonstrate that our method compares favourably with state-of-the-art methods whilst achieving volume preservation.