A Closed-Form Solution of Rotation Invariant Spherical Harmonic Features in Diffusion MRI

  • Mauro ZucchelliEmail author
  • Samuel Deslauriers-Gauthier
  • Rachid Deriche
Conference paper
Part of the Mathematics and Visualization book series (MATHVISUAL)


Rotation invariant features are an indispensable tool for characterizing diffusion Magnetic Resonance Imaging (MRI) and in particular for brain tissue microstructure estimation. In this work, we propose a new mathematical framework for efficiently calculating a complete set of such invariants from any spherical function. Specifically, our method is based on the spherical harmonics series expansion of a given function of any order and can be applied directly to the resulting coefficients by performing a simple integral operation analytically. This enable us to derive a general closed-form equation for the invariants. We test our invariants on the diffusion MRI fiber orientation distribution function obtained from the diffusion signal both in-vivo and in synthetic data. Results show how it is possible to use these invariants for characterizing the white matter using a small but complete set of features.


Diffusion MRI Rotation invariant Gaunt coefficients Spherical harmonics 



This work has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (ERC Advanced Grant agreement No 694665 : CoBCoM—Computational Brain Connectivity Mapping). Data were provided by the Human Connectome Project, WU-Minn Consortium (Principal Investigators: David Van Essen and Kamil Ugurbil; 1U54MH091657) funded by the 16 NIH Institutes and Centers that support the NIH Blueprint for Neuroscience Research; and by the McDonnell Center for Systems Neuroscience at Washington University.


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Mauro Zucchelli
    • 1
    Email author
  • Samuel Deslauriers-Gauthier
    • 1
  • Rachid Deriche
    • 1
  1. 1.Athena Project-Team, Inria Sophia Antipolis - MéditerranéeBiotFrance

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