A PDE‐regularized smoothing method for space–time data over manifolds with application to medical data

Abstract

We propose an innovative statistical-numerical method to model spatiotemporal data, observed over a generic two-dimensional Riemanian manifold.The proposed approach consists of a regression model completed with a regularizing term based on the heat equation. The model is discretized through a finite element scheme set on the manifold, and solved by resorting to a fixed point-based iterative algorithm. This choice leads to a procedure which is highly efficient when compared with a monolithic approach, and which allows us to deal with massive datasets. After a preliminary assessment on simulation study cases, we investigate the performance of the new estimation tool in practical contexts, by dealing with neuroimaging and hemodynamic data.finite elements, fixed point algorithm, hemodynamics, neuroimaging, regularized regression | INTRODUCTIONThis work proposes a statistical-numerical methodology to analyze spatio-temporal data measured on general twodimensional Riemanian manifold domains. These kinds of data are very common in diverse contexts, from Engineering to Applied Sciences. In an Engineering design process, for instance, it is standard to study time-as well as space-varying quantities of interest observed over the surface of a three-dimensional prototype in order to optimize the design pipeline (e.g., the aerodynamic forces exerted on the surface of an airfoil when dealing with the design of an airplane). In Environmental Science, it is of paramount importance to accurately model space-time data distributed over regions characterized by a complex orography, for example, in order to better understand the Earth processes, or to control pollution or global climate changes, or to optimize the exploitation of natural resources. In this paper, we focus on some applications which arise from Life Science. Figure 1 refers to one of the analyzed contexts. The panel on the left shows the mesh approximating the cortical surface of a brain, on which the hemodynamic signal induced by the neuronal activity on the cortical surface, shown in the right panel, has been observed at a certain time. Standard spatio-temporal techniques, that rely on the Euclidean distance, are not suited, in general, to handle data such as the ones in Figure 1. Due to folded geometry of the domain, such methods can yield highly inaccurate estimates, by incorrectly identifying as close, data locations that actually are far apart on the real geometry. Thus, values observed over two distinct gyri could be artificially linked each other, although, in practice, separated by a sulcus. As a consequence, in order to obtain accurate estimates on complex manifolds, it becomes mandatory to appropriately comply with the complex geometry of the domain.