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A Gaussian Process Model for Unsupervised Analysis of High Dimensional Shape Data

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Machine Learning in Medical Imaging (MLMI 2021)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 12966))

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Abstract

Applications of medical image analysis are often faced with the challenge of modelling high-dimensional data with relatively few samples. In many settings, normal or healthy samples are prevalent while pathological samples are rarer, highly diverse, and/or difficult to model. In such cases, a robust model of the normal population in the high-dimensional space can be useful for characterizing pathologies. In this context, there is utility in hybrid models, such as probabilistic PCA, which learns a low-dimensional model, commensurates with the available data, and combines it with a generic, isotropic noise model for the remaining dimensions. However, the isotropic noise model ignores the inherent correlations that are evident in so many high-dimensional data sets associated with images and shapes in medicine. This paper describes a method for estimating a Gaussian model for collections of images or shapes that exhibit underlying correlations, e.g., in the form of smoothness. The proposed method incorporates a Gaussian-process noise model within a generative formulation. For optimization, we derive a novel expectation maximization (EM) algorithm. We demonstrate the efficacy of the method on synthetic examples and on anatomical shape data.

Supported by the National Institutes of Health under R21-EB026061, NIBIB-U24EB029011, NIAMS-R01AR076120, NHLBI-R01HL135568, NIBIB-R01EB016701, and NIGMS-P41GM103545. Authors thank Erin Anstadt, MD and Jesse A Goldstein, MD for providing cranial shapes. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health.

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  • 21 September 2021

    A correction has been published.

References

  1. Atkins, P.R., et al.: Quantitative comparison of cortical bone thickness using correspondence-based shape modeling in patients with cam femoroacetabular impingement. J. Orthopaedic Res. 35(8), 1743–1753 (2017)

    Article  MathSciNet  Google Scholar 

  2. Bhalodia, R., Dvoracek, L.A., Ayyash, A.M., Kavan, L., Whitaker, R., Goldstein, J.A.: Quantifying the severity of metopic craniosynostosis: a pilot study application of machine learning in craniofacial surgery. J. Craniofac. Surg. 31(3), 697–701 (2020)

    Article  Google Scholar 

  3. Cates, J., Fletcher, P.T., Styner, M., Shenton, M., Whitaker, R.: Shape modeling and analysis with entropy-based particle systems. In: Karssemeijer, N., Lelieveldt, B. (eds.) IPMI 2007. LNCS, vol. 4584, pp. 333–345. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-73273-0_28

    Chapter  Google Scholar 

  4. Chen, Y., Wiesel, A., Eldar, Y.C., Hero, A.O.: Shrinkage algorithms for MMSE covariance estimation. IEEE Trans. Signal Process. 58(10), 5016–5029 (2010)

    Article  MathSciNet  Google Scholar 

  5. Doan, N.T., van Lew, B., Lelieveldt, B., van Buchem, M.A., Reiber, J.H.C., Milles, J.: Deformation texture-based features for classification in Alzheimer’s disease. In: Medical Imaging 2013: Image Processing, vol. 8669, pp. 601–607. International Society for Optics and Photonics, SPIE (2013)

    Google Scholar 

  6. Fletcher, P.T., Lu, C., Pizer, S.M., Joshi, S.: Principal geodesic analysis for the study of nonlinear statistics of shape. IEEE Trans. Med. Imaging 23(8), 995–1005 (2004)

    Article  Google Scholar 

  7. Gerig, T., Shahim, K., Reyes, M., Vetter, T., Lüthi, M.: Spatially varying registration using Gaussian processes. In: Golland, P., Hata, N., Barillot, C., Hornegger, J., Howe, R. (eds.) MICCAI 2014. LNCS, vol. 8674, pp. 413–420. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-10470-6_52

    Chapter  Google Scholar 

  8. Gu, M., Shen, W.: Generalized probabilistic principal component analysis of correlated data. J. Mach. Learn. Res. 21(13), 1–41 (2020)

    MathSciNet  MATH  Google Scholar 

  9. Joshi, S.C., Miller, M.I., Grenander, U.: On the geometry and shape of brain sub-manifolds. Int. J. Pattern Recogn. Artif. Intell. 11(08), 1317–1343 (1997)

    Article  Google Scholar 

  10. Kellogg, R., Allori, A.C., Rogers, G.F., Marcus, J.R.: Interfrontal angle for characterization of trigonocephaly: part 1: development and validation of a tool for diagnosis of metopic synostosis. J. Craniofac. Surg. 23(3), 799–804 (2012)

    Article  Google Scholar 

  11. Kingma, D.P., Ba, J.: Adam: a method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014)

  12. Lawrence, N., Hyvärinen, A.: Probabilistic non-linear principal component analysis with Gaussian process latent variable models. J. Mach. Learn. Res. 6(11), 1783–1816 (2005)

    MathSciNet  MATH  Google Scholar 

  13. Lê, M., Unkelbach, J., Ayache, N., Delingette, H.: GPSSI: Gaussian process for sampling segmentations of images. In: Navab, N., Hornegger, J., Wells, W.M., Frangi, A.F. (eds.) MICCAI 2015. LNCS, vol. 9351, pp. 38–46. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-24574-4_5

    Chapter  Google Scholar 

  14. Ledoit, O., Wolf, M.: A well-conditioned estimator for large-dimensional covariance matrices. J. Multivariate Anal. 88(2), 365–411 (2004)

    Article  MathSciNet  Google Scholar 

  15. Leys, C., Klein, O., Dominicy, Y., Ley, C.: Detecting multivariate outliers: use a robust variant of the Mahalanobis distance. J. Exp. Soc. Psychol. 74, 150–156 (2018)

    Article  Google Scholar 

  16. Liu, P.J.: Using Gaussian process regression to denoise images and remove artefacts from microarray data. University of Toronto (2007)

    Google Scholar 

  17. Lüthi, M., Gerig, T., Jud, C., Vetter, T.: Gaussian process morphable models. IEEE Trans. Pattern Anal. Mach. Intell. 40(8), 1860–1873 (2017)

    Article  Google Scholar 

  18. Oktay, O., et al.: Anatomically constrained neural networks (ACNNs): application to cardiac image enhancement and segmentation. IEEE Trans. Med. Imaging 37(2), 384–395 (2018). https://doi.org/10.1109/TMI.2017.2743464

  19. Purisha, Z., Jidling, C., Wahlström, N., Schön, T.B., Särkkä, S.: Probabilistic approach to limited-data computed tomography reconstruction. Inverse Probl. 35(10), 105004 (2019)

    Article  MathSciNet  Google Scholar 

  20. Roweis, S.: EM algorithms for PCA and SPCA. In: Proceedings of the 10th International Conference on Neural Information Processing Systems, pp. 626–632, NIPS 1997. MIT Press, Cambridge (1997)

    Google Scholar 

  21. Rueckert, D., Frangi, A.F., Schnabel, J.A.: Automatic construction of 3-D statistical deformation models of the brain using nonrigid registration. IEEE Trans. Med. Imaging 22(8), 1014–1025 (2003)

    Article  Google Scholar 

  22. Styner, M., et al.: Framework for the statistical shape analysis of brain structures using SPHARM-PDM. Insight J. 1071, 242–250 (2006)

    Google Scholar 

  23. Styner, M., Gerig, G.: Medial models incorporating object variability for 3D shape analysis. In: Insana, M.F., Leahy, R.M. (eds.) IPMI 2001. LNCS, vol. 2082, pp. 502–516. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-45729-1_53

    Chapter  Google Scholar 

  24. Beiging, E.T., Morris, A., Wilson, B.D., McGann, C.J., Marrouche, N.F., Cates, J.: Left atrial shape predicts recurrence after atrial fibrillation catheter ablation. J. Cardiovasc. Electrophysiol. 29(7), 966–972 (2018). https://doi.org/10.1111/jce.13641

    Article  Google Scholar 

  25. Tipping, M.E., Bishop, C.M.: Probabilistic principal component analysis. J. R. Stat. Soc. Ser. B 61(3), 611–622 (1999)

    Article  MathSciNet  Google Scholar 

  26. Uebersax, J.S., Grove, W.M.: A latent trait finite mixture model for the analysis of rating agreement. Biometrics 49(3), 823–835 (1993)

    Article  MathSciNet  Google Scholar 

  27. Wood, B.C., et al.: What’s in a name? Accurately diagnosing metopic craniosynostosis using a computational approach. Plastic Reconstr. Surg. 137(1), 205–213 (2016)

    Article  Google Scholar 

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Tao, W., Bhalodia, R., Whitaker, R. (2021). A Gaussian Process Model for Unsupervised Analysis of High Dimensional Shape Data. In: Lian, C., Cao, X., Rekik, I., Xu, X., Yan, P. (eds) Machine Learning in Medical Imaging. MLMI 2021. Lecture Notes in Computer Science(), vol 12966. Springer, Cham. https://doi.org/10.1007/978-3-030-87589-3_37

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  • DOI: https://doi.org/10.1007/978-3-030-87589-3_37

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