Diffusion Tensor Imaging with Riemannian geometry. Curvature-based anisotropy (CA) complements standard FA/MD for white matter analysis.
Geometric features: 76.4% vs 69.2% raw tensors (+7.2pp). Benchmark: DOI 10.5281/zenodo.19192330
Diffusion Tensor Imaging (DTI) maps water diffusion in brain tissue to reveal white matter structure. It's critical for diagnosing multiple sclerosis, stroke, traumatic brain injury, and neurodegenerative diseases.
Standard DTI metrics (FA, MD) treat 3×3 diffusion tensors as flat vectors. This creates artifacts: arithmetic averaging produces eigenvalue swelling, linear interpolation can violate positive-definiteness, and standard metrics miss geometric tissue signatures.
Diffusion tensors are symmetric positive-definite matrices living on GL+(3)/SO(3). The affine-invariant (geodesic) distance naturally respects this constraint. Geodesic interpolation preserves positive-definiteness at every intermediate point.
The novel curvature anisotropy (CA) metric uses Ricci curvature to weight shape sensitivity — detecting tissue changes in the deviatoric (fiber direction) sector that FA and MD cannot distinguish.
Geometric DTI is especially relevant for:
Geodesic distances on GL+(3)/SO(3) respect the positive-definite constraint. No Euclidean artifacts like swelling.
CA uses Ricci curvature to weight shape sensitivity — detecting tissue changes invisible to FA alone.
Separates deviatoric (shape, 5D) from volumetric (size, 1D) using the DeWitt metric decomposition.
Automatic classification from diffusion tensor eigenvalues.
λ = [1.7, 0.3, 0.3] × 10-3
Highly anisotropic — coherent axonal bundles
λ = [1.0, 0.7, 0.7] × 10-3
Mildly anisotropic — cell bodies
λ = [3.0, 3.0, 3.0] × 10-3
Isotropic — free water diffusion
2D cross-sections showing tissue anisotropy and the Euclidean swelling artifact.
Riemannian vs Euclidean: why geodesic interpolation eliminates the swelling artifact in DTI tractography.