Topologically Constrained Template Estimation via Morse--Smale Complexes Controls Its Statistical Consistency

In most neuroimaging studies, one builds a brain template that serves as a reference for normalizing the measurements of each individual subject into a common space. Such a template should be representative of the population under study, thus avoiding bias in subsequent statistical analyses. The template is often computed by iteratively registering all images to the current template and then averaging the intensities of the registered images. Geometrically, the procedure can be summarized as the computation of the template as the “Fréchet mean” of the images projected in a quotient space. It has been argued recently that this type of algorithm could actually be asymptotically biased and therefore inconsistent. In other words, even with an infinite number of brain images in the database, the template estimate may not converge to the brain anatomy it is meant to estimate. Our paper investigates this phenomenon. We present a methodology that spatially quantifies the brain template's asymptotic bias. We identify the main variables that influence inconsistency. This leads us to investigate the topology of the template's intensity level sets, represented by its Morse--Smale (MS) complex. We propose a topologically constrained adaptation of the template computation that constructs a hierarchical template with bounded bias. We apply our method to the analysis of a brain template of 136 T1 weighted MR images from the Open Access Series of Imaging Studies (OASIS) database.