Realizing Unbiased Deformation: A Theoretical Consideration

Maps of local tissue compression or expansion are often recovered by comparing MRI scans using nonlinear registration techniques. The resulting changes can be analyzed using tensor-based morphometry (TBM) to make inferences about anatomical differences. Numerous deformation techniques have been developed, although there has not been much theoretical development examining the mathematical/statistical validity of each technique. In this paper, we propose a basic principle that any registration technique should satisfy: realizing unbiased test statistics under null distribution of the displacement. In other words, any registration technique should recover zero change in the test statistic when comparing two images differing only in noise. Based on this principle, we propose a fundamental framework for the construction and analysis of image deformation. Moreover, we argue that logarithmic transform is instrumental in the analysis of deformation maps. Combined with the proposed framework, this leads to a theoretical connection between image registration and other branches of applied mathematics including information theory and grid generation.