Complete Set of Invariants of a 4th Order Tensor: The 12 Tasks of HARDI from Ternary Quartics

  • Théo Papadopoulo
  • Aurobrata Ghosh
  • Rachid Deriche
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8675)


Invariants play a crucial role in Diffusion MRI. In DTI (2 nd order tensors), invariant scalars (FA, MD) have been successfully used in clinical applications. But DTI has limitations and HARDI models (e.g. 4 th order tensors) have been proposed instead. These, however, lack invariant features and computing them systematically is challenging.

We present a simple and systematic method to compute a functionally complete set of invariants of a non-negative 3D 4 th order tensor with respect to SO 3. Intuitively, this transforms the tensor’s non-unique ternary quartic (TQ) decomposition (from Hilbert’s theorem) to a unique canonical representation independent of orientation – the invariants.

The method consists of two steps. In the first, we reduce the 18 degrees-of-freedom (DOF) of a TQ representation by 3-DOFs via an orthogonal transformation. This transformation is designed to enhance a rotation-invariant property of choice of the 3D 4 th order tensor. In the second, we further reduce 3-DOFs via a 3D rotation transformation of coordinates to arrive at a canonical set of invariants to SO 3 of the tensor.

The resulting invariants are, by construction, (i) functionally complete, (ii) functionally irreducible (if desired), (iii) computationally efficient and (iv) reversible (mappable to the TQ coefficients or shape); which is the novelty of our contribution in comparison to prior work.

Results from synthetic and real data experiments validate the method and indicate its importance.


Invariants SO3 4th order tensors ternary quartics orthogonal & rotation transforms canonical representation 


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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Théo Papadopoulo
    • 1
  • Aurobrata Ghosh
    • 1
  • Rachid Deriche
    • 1
  1. 1.Athena Project TeamInria Sophia Antipolis - MéditerranéeFrance

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