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.
G. Ziegler—Joint first author.
Data used in preparation of this article were obtained from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) database (http://adni.loni.usc.edu/). As such, the investigators within the ADNI contributed to the design and implementation of ADNI and/or provided data but did not participate in analysis or writing of this report.
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Notes
- 1.
For simplicity we focus on an even sampling across spatial directions, even though the generalization of the proposed model to the uneven case is straightforward.
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Acknowledgements
Marco Lorenzi is grateful to Prof. John Ashburner, for his help in finalizing this work, and to Dr. Richard Turner, for his precious suggestions on the train toward London. Sebastien Ourselin receives funding from the EPSRC (EP/H046410/1, EP/J020990/1, EP/K005278), the MRC (MR/J01107X/1), the EU-FP7 project VPH-DARE@IT (FP7-ICT-2011-9-601055), the NIHR Biomedical Research Unit (Dementia) at UCL and the National Institute for Health Research University College London Hospitals Biomedical Research Centre (NIHR BRC UCLH/UCL High Impact Initiative- BW.mn.BRC10269). Gabriel Ziegler is supported in part by the German Academic Exchange Service (DAAD). The Wellcome Trust Centre for Neuroimaging is supported by core funding from the Wellcome Trust [grant number 091593/Z/10/Z].
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Lorenzi, M., Ziegler, G., Alexander, D.C., Ourselin, S. (2015). Efficient Gaussian Process-Based Modelling and Prediction of Image Time Series. In: Ourselin, S., Alexander, D., Westin, CF., Cardoso, M. (eds) Information Processing in Medical Imaging. IPMI 2015. Lecture Notes in Computer Science(), vol 9123. Springer, Cham. https://doi.org/10.1007/978-3-319-19992-4_49
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