Generalized Invariants of a 4th order tensor: Building blocks for new biomarkers in dMRI

Aurobrata Ghosh (1) , Théodore Papadopoulo (1) , Rachid Deriche (1)

1 ATHENA - Computational Imaging of the Central Nervous System

This paper presents a general and complete (up to degree 4) set of invariants of 3D 4th order tensors with respect to SO3. The invariants to SO3 for the 2nd order diffusion tensor are well known and play a crucial role in deriving important biomarkers for DTI, e.g. MD, FA, RA, etc. But DTI is limited in regions with fiber heterogeneity and DTI biomarkers severely lack specificity. 4th order tensors are both a natural extension to DTI and also form an alternate basis to spherical harmonics for spherical functions. This paper presents a systematic method for computing the SO3 invariants of 3D 4th order tensors, derives relationships between the new (generalized) invariants and existing invariants and shows results on synthetic and real data. It also present, hitherto unknown, new invariants for 4th order tensors. Analogously to DTI, these new invariants can perhaps form building blocks for new biomarkers.

Mots clés: Invariants; 4th order tensors; homogeneous forms/polynomials; SO3; basic invariants; principal invariants; biomarkers

Domaines: Imagerie médicale; Neurosciences

Aurobrata Ghosh, Théodore Papadopoulo, Rachid Deriche. Generalized Invariants of a 4th order tensor: Building blocks for new biomarkers in dMRI. Computational Diffusion MRI Workshop (CDMRI), MICCAI, 2012, Nice, France. pp.165―173. ⟨hal-00789763⟩