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

  • Marco Lorenzi
  • Gabriel Ziegler
  • Daniel C. Alexander
  • Sebastien Ourselin
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9123)

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.

Notes

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].

References

  1. 1.
    Ashburner, J., Friston, K.: Unified segmentation. NeuroImage 26, 839–851 (2005)CrossRefGoogle Scholar
  2. 2.
    Ashburner, J., Ridgway, G.: Symmetric diffeomorphic modeling of longitudinal structural MRI. Frontiers Neurosci. 6(197) (02 2013)Google Scholar
  3. 3.
    Davis, B.C., Fletcher, P.T., Bullitt, E., Joshi, S.C.: Population shape regression from random design data. IJCV 90(2), 255–266 (2010)CrossRefGoogle Scholar
  4. 4.
    Flandin, G., Penny, W.D.: Bayesian fMRI data analysis with sparse spatial basis function priors. NeuroImage 34(3), 1108–1125 (2007)CrossRefGoogle Scholar
  5. 5.
    Friston, K.J., Holmes, A., Worsley, K.J.: Statistical parametric maps in functional imaging: a general linear approach. Hum. Brain Mapp. 2, 189–210 (1995)CrossRefGoogle Scholar
  6. 6.
    Gelfand, A., Fuentes, M., Guttorp, P., Diggle, P.: Handbook of Spatial Statistics. Chapman & Hall/CRC Handbooks of Modern Statistical Methods. Taylor & Francis, London (2010) MATHCrossRefGoogle Scholar
  7. 7.
    Harrison, L.M., Green, G.G.: A Bayesian spatiotemporal model for very large data sets. NeuroImage 50(3), 1126–1141 (2010)CrossRefGoogle Scholar
  8. 8.
    Hoffman, Y., Ribak, E.: Constrained realizations of Gaussian fields -a simple algorithm. Astrophys. J. Lett. 380, L5–L8 (1991)CrossRefGoogle Scholar
  9. 9.
    Lorenzi, M., Ayache, N., Frisoni, G.B., Pennec, X.: The Alzheimer’s disease neuroimaging initiative: mapping the effects of A\(\beta \) \(_\text{1 }-\text{42 }\) levels on the longitudinal changes in healthy aging: hierarchical modeling based on stationary velocity fields. In: Fichtinger, G., Martel, A., Peters, T. (eds.) MICCAI 2011, Part II. LNCS, vol. 6892, pp. 663–670. Springer, Heidelberg (2011) CrossRefGoogle Scholar
  10. 10.
    Niethammer, M., Huang, Y., Vialard, F.-X.: Geodesic regression for image time-series. In: Fichtinger, G., Martel, A., Peters, T. (eds.) MICCAI 2011, Part II. LNCS, vol. 6892, pp. 655–662. Springer, Heidelberg (2011) CrossRefGoogle Scholar
  11. 11.
    Rasmussen, C.E., Williams, C.K.I.: Gaussian Processes for Machine Learning. The MIT Press, Cambridge (2005)Google Scholar
  12. 12.
    Stegle, O., Lippert, C., Mooij, J.M., et al.: Efficient inference in matrix-variate gaussian models with iid observation noise. In: Shawe-Taylor, J., Zemel, S., Bartlett, P.L., Pereira, F.C.N., Weinberger, K.Q. (eds.) Advances in Neural Information Processing Systems 24, pp. 630–638. Second Life, Granada (2011)Google Scholar
  13. 13.
    Ziegler, G., Ridgway, G.R., Dahnke, R., Gaser, C.: Individualized Gaussian process-based prediction and detection of local and global gray matter abnormalities in elderly subjects. NeuroImage 97, 333–348 (2014)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Marco Lorenzi
    • 1
  • Gabriel Ziegler
    • 2
  • Daniel C. Alexander
    • 1
  • Sebastien Ourselin
    • 1
  1. 1.Centre for Medical Image Computing, CMICUCLLondonUK
  2. 2.Wellcome Trust Centre for NeuroimagingUCLLondonUK

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