Statistical Properties of Jacobian Maps and the Realization of Unbiased Large-Deformation Nonlinear Image Registration

TLDR

Maps of local tissue compression or expansion are computed by nonlinear image registration of MRI scans, and the resulting changes are analyzed via tensor‑based morphometry using Jacobian maps to infer anatomical differences. The study aims to rigorously analyze Jacobian maps and develop a new numerical method for unbiased nonlinear image registration. The authors use a logarithmic transformation of Jacobian values, analyze log‑Jacobian distributions with a Kullback‑Leibler framework, and construct unbiased registration by minimizing the symmetric KL distance between the identity map and the deformation, with implementation details applicable to other methods. Permutation tests confirm that symmetrizing image registration statistically reduces skewness in the log‑Jacobian map, as demonstrated in a 3‑D brain mapping of a semantic dementia patient.

Abstract

Maps of local tissue compression or expansion are often computed by comparing magnetic resonance imaging (MRI) scans using nonlinear image registration. The resulting changes are commonly analyzed using tensor-based morphometry to make inferences about anatomical differences, often based on the Jacobian map, which estimates local tissue gain or loss. Here, we provide rigorous mathematical analyses of the Jacobian maps, and use themto motivate a new numerical method to construct unbiased nonlinear image registration. First, we argue that logarithmic transformation is crucial for analyzing Jacobian values representing morphometric differences. We then examine the statistical distributions of log-Jacobian maps by defining the Kullback-Leibler (KL) distance on material density functions arising in continuum-mechanical models. With this framework, unbiased image registration can be constructed by quantifying the symmetric KL-distance between the identity map and the resulting deformation. Implementation details, addressing the proposed unbiased registration as well as the minimization of symmetric image matching functionals, are then discussed and shown to be applicable to other registration methods, such as inverse consistent registration. In the results section, we test the proposed framework, as well as present an illustrative application mapping detailed 3-D brain changes in sequential magnetic resonance imaging scans of a patient diagnosed with semantic dementia. Using permutation tests, we show that the symmetrization of image registration statistically reduces skewness in the log-Jacobian map.