Efficient Gaussian Process-Based Modelling and Prediction of Image Time Series

Abstract

In this work we propose a novel Gaussian process-based spatio-temporal model of time series of images. By assuming separability of spatial and temporal processes we provide a very efficient and robust formulation for the marginal likelihood computation and the posterior prediction. The model adaptively accounts for local spatial correlations of the data, and the covariance structure is effectively parameterised by the Kronecker product of covariance matrices of very small size, each encoding only a single direction in space. We provide a simple and flexible framework for within- and between-subject modelling and prediction. In particular, we introduce the Hoffman-Ribak method for efficient inference on posterior processes and its uncertainty. The proposed framework is applied in the context of longitudinal modelling in Alzheimer's disease. We firstly demonstrate the advantage of our non-parametric method for modelling of within-subject structural changes. The results show that non-parametric methods demonstrably outperform conventional parametric methods. Then the framework is extended to optimize complex parametrized covariate kernels. Using Bayesian model comparison via marginal likelihood the framework enables to compare different hypotheses about individual change processes of images.

Fig. 1

Group-wise average absolute differences between extrapolated images and real ones. The GP model was trained on scans from 3 time points corresponding to baseline, 6 months and 1 year. Errors were generally found to be proportional to the extrapolation time.

Fig. 2

Predicted hippocampal progression for a sample MCIc patient. The model was estimated from 4 image time points (baseline to 2 years) in a bounding region including the hippocampus. The longitudinal sample distribution (gray dots) and mean prediction (red line) are estimated according to the marginal GP posterior of Section 3 by using the Hoffman-Ribak method.

Fig. 3

(A) Mean absolute error (MAE) of prediction maps in an independent testing sample of 105 subjects show increasingly better predictions using more predictor sets and Gaussian process models with ARD. (B) Predicted over true growth rates using model 4 in an example voxel showing correlation of r = 0.52.