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Bi-invariant Two-Sample Tests in Lie Groups for Shape Analysis

Data from the Alzheimer’s Disease Neuroimaging Initiative

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Shape in Medical Imaging (ShapeMI 2020)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 12474))

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Abstract

We propose generalizations of the Hotelling’s \(T^2\) statistic and the Bhattacharayya distance for data taking values in Lie groups. A key feature of the derived measures is that they are compatible with the group structure even for manifolds that do not admit any bi-invariant metric. This property, e.g. assures analysis that does not depend on the reference shape, thus, preventing bias due to arbitrary choices thereof. Furthermore, the generalizations agree with the common definitions for the special case of flat vector spaces guaranteeing consistency. Employing a permutation test setup, we further obtain nonparametric, two-sample testing procedures that themselves are bi-invariant and consistent. We validate our method in group tests revealing significant differences in hippocampal shape between individuals with mild cognitive impairment and normal controls.

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Notes

  1. 1.

    adni.loni.usc.edu.

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Acknowledgments

M. Hanik is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy - The Berlin Mathematics Research Center MATH+ (EXC-2046/1, project ID: 390685689). We are grateful for the open-access dataset of the Alzheimer’s Disease Neuroimaging Initiative (ADNI) (Data collection and sharing for this project was funded by the ADNI (National Institutes of Health Grant U01 AG024904) and DOD ADNI (Department of Defense award number W81XWH-12-2-0012). ADNI is funded by the National Institute on Aging, the National Institute of Biomedical Imaging and Bioengineering, and through generous contributions from the following: AbbVie, Alzheimer’s Association; Alzheimer’s Drug Discovery Foundation; Araclon Biotech; BioClinica, Inc.; Biogen; Bristol-Myers Squibb Company; CereSpir, Inc.; Cogstate; Eisai Inc.; Elan Pharmaceuticals, Inc.; Eli Lilly and Company; EuroImmun; F. Hoffmann-La Roche Ltd and its affiliated company Genentech, Inc.; Fujirebio; GE Healthcare; IXICO Ltd.; Janssen Alzheimer Immunotherapy Research & Development, LLC.; Johnson & Johnson Pharmaceutical Research & Development LLC.; Lumosity; Lundbeck; Merck & Co., Inc.; Meso Scale Diagnostics, LLC.; NeuroRx Research; Neurotrack Technologies; Novartis Pharmaceuticals Corporation; Pfizer Inc.; Piramal Imaging; Servier; Takeda Pharmaceutical Company; and Transition Therapeutics. The Canadian Institutes of Health Research is providing funds to support ADNI clinical sites in Canada. Private sector contributions are facilitated by the Foundation for the National Institutes of Health (www.fnih.org). The grantee organization is the Northern California Institute for Research and Education, and the study is coordinated by the Alzheimer’s Therapeutic Research Institute at the University of Southern California. ADNI data are disseminated by the Laboratory for Neuro Imaging at the University of Southern California.) as well as for F. Ambellan’s help in establishing dense correspondences of the hippocampal surface meshes.

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Authors and Affiliations

  1. Zuse Institute Berlin, Berlin, Germany

    Martin Hanik, Hans-Christian Hege & Christoph von Tycowicz

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  1. Martin Hanik
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  2. Hans-Christian Hege
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  3. Christoph von Tycowicz
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Correspondence to Martin Hanik .

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Editors and Affiliations

  1. Harvard Medical School, Boston, USA

    Martin Reuter

  2. Klinikum der Universität München, LMU München, München, Germany

    Christian Wachinger

  3. Inria Sophia-Antipolis, Valbonne, France

    Hervé Lombaert

  4. University of North Carolina at Chapel Hill, Kitware Inc., Carrboro, NC, USA

    Beatriz Paniagua

  5. ETH Zurich, Zürich, Switzerland

    Orcun Goksel

  6. Istanbul Technical University, Istanbul, Turkey

    Islem Rekik

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Hanik, M., Hege, HC., Tycowicz, C.v. (2020). Bi-invariant Two-Sample Tests in Lie Groups for Shape Analysis. In: Reuter, M., Wachinger, C., Lombaert, H., Paniagua, B., Goksel, O., Rekik, I. (eds) Shape in Medical Imaging. ShapeMI 2020. Lecture Notes in Computer Science(), vol 12474. Springer, Cham. https://doi.org/10.1007/978-3-030-61056-2_4

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  • DOI: https://doi.org/10.1007/978-3-030-61056-2_4

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