CHAIRI Metric: Riemannian Framework in DTI

• CHAIRI metric is a Riemannian framework for DTI that preserves tensor anisotropy through spectral decomposition and quaternion-based interpolation.
• It decouples eigenvalue and orientation interpolation, ensuring computational efficiency and numerical stability in handling SPD tensors.
• Compared to affine-invariant and Log-Euclidean metrics, it exactly maintains anisotropy, offering improved accuracy in diffusion tensor imaging studies.

The CHAIRI metric—Editor's term for "Computationally tractable, H­Hilbert‐Anisotropy‐preserving, RiemannIan metric"—is a Riemannian framework introduced for statistical analysis and processing of Diffusion Tensor Imaging (DTI) data. The construction, also known as the spectral-quaternion or $d_{SD}$ metric, explicitly preserves anisotropy during interpolation and averaging of symmetric positive-definite (SPD) tensors, a feature not shared by classical affine-invariant or Log-Euclidean metrics. Developed by Collard et al., the metric is grounded in a geometric decomposition of SPD tensors into spectral (eigenvalues) and orientational (eigenvectors) components, with core interpolation steps executed via closed-form expressions and numeric operations that confer favorable computational complexity and stability (Collard et al., 2012).

1. Mathematical Foundations

Let $S^{+}(3)$ denote the manifold of $3 \times 3$ SPD matrices, with focus restricted to the open subset $$S^{+}_{\rangle}(3) = \{ S \in S^{+}(3) \mid 0 < \lambda_3 < \lambda_2 < \lambda_1 \}$$, where each $S$ admits a unique spectral decomposition $S = U \Lambda U^{\top}$ with $U \in SO(3)/G$ encoding orientation and $$\Lambda = \mathrm{diag}(\lambda_1, \lambda_2, \lambda_3)$$ the ordered spectrum.

At each $S=U\Lambda U^{\top}$, the tangent space splits into orientation (tangent to $SO(3)/G$) and scale (tangent to $S^{+}(3)$0) parts. The metric is defined as

$S^{+}(3)$1

where:

• $S^{+}(3)$2 is the infinitesimal generator of rotation,
• $S^{+}(3)$3,
• $S^{+}(3)$4,
• $S^{+}(3)$5,
• $S^{+}(3)$6 is a smooth increasing function of tensor anisotropy.

Anisotropy is measured by the Hilbert index:

$S^{+}(3)$7

For two tensors $S^{+}(3)$8, $S^{+}(3)$9, the weight is

$3 \times 3$0

The induced Riemannian distance is given by

$3 \times 3$1

with

• $3 \times 3$2,
• $3 \times 3$3, where the logarithm for diagonal $3 \times 3$4 is entry-wise.

2. Geodesic and Interpolation Procedures

Closed-form geodesic approximation is achieved by decoupled interpolation of spectrum and orientation:

• Spectrum: For $3 \times 3$5,

$3 \times 3$6

• Orientation: $3 \times 3$7, $3 \times 3$8 are converted to quaternions $3 \times 3$9, $$S^{+}_{\rangle}(3) = \{ S \in S^{+}(3) \mid 0 < \lambda_3 < \lambda_2 < \lambda_1 \}$$0; $$S^{+}_{\rangle}(3) = \{ S \in S^{+}(3) \mid 0 < \lambda_3 < \lambda_2 < \lambda_1 \}$$1 is selected from the 8 covers of $$S^{+}_{\rangle}(3) = \{ S \in S^{+}(3) \mid 0 < \lambda_3 < \lambda_2 < \lambda_1 \}$$2 to maximize $$S^{+}_{\rangle}(3) = \{ S \in S^{+}(3) \mid 0 < \lambda_3 < \lambda_2 < \lambda_1 \}$$3; interpolation is

$$S^{+}_{\rangle}(3) = \{ S \in S^{+}(3) \mid 0 < \lambda_3 < \lambda_2 < \lambda_1 \}$$4

The SPD tensor at $$S^{+}_{\rangle}(3) = \{ S \in S^{+}(3) \mid 0 < \lambda_3 < \lambda_2 < \lambda_1 \}$$5 is synthesized as

$$S^{+}_{\rangle}(3) = \{ S \in S^{+}(3) \mid 0 < \lambda_3 < \lambda_2 < \lambda_1 \}$$6

where $$S^{+}_{\rangle}(3) = \{ S \in S^{+}(3) \mid 0 < \lambda_3 < \lambda_2 < \lambda_1 \}$$7 is the $$S^{+}_{\rangle}(3) = \{ S \in S^{+}(3) \mid 0 < \lambda_3 < \lambda_2 < \lambda_1 \}$$8 matrix corresponding to quaternion $$S^{+}_{\rangle}(3) = \{ S \in S^{+}(3) \mid 0 < \lambda_3 < \lambda_2 < \lambda_1 \}$$9.

3. Anisotropy Preservation Property

A defining feature is that the Hilbert anisotropy index commutes with spectrum interpolation:

$S$0

where $S$1 are the interpolation weights. Thus, the anisotropy of the mean is exactly the weighted arithmetic mean of the input anisotropies. The affine-invariant and Log-Euclidean means do not satisfy this property, as averaging under these metrics reduces anisotropy (“washes out” the anisotropy signal). Empirically, classical fractional anisotropy (FA) is better preserved under spectral-quaternion interpolation than under alternatives (Collard et al., 2012).

4. Computational Complexity and Numerical Aspects

The metric achieves tractability through closed-form formulas and efficient quaternion operations. The principal steps involve one diagonal logarithm per tensor, one quaternion conversion (in closed form), and $S$2 quaternion dot-products and normalizations. No iterative procedures are required for means or geodesics. Comparative timing for $S$3 random distances: | Method | Time (s) | Notes | |------------------------|------------|-------------------------------------------| | Affine-invariant | 0.47 | Repeated 3×3 eigendecomp., log/exp, iter. | | Log-Euclidean | 0.17 | One log/exp per tensor, still 3×3 | | Spectral ($S$4 log) | 0.65 | Matrix-log distance on $S$5 | | Spectral-quaternions | 0.11 | Quaternion chordal, fastest |

The spectral-quaternions approach is approximately $S$6 faster than Log-Euclidean and $S$7 faster than affine-invariant in this benchmark.

5. Algorithmic Workflow

A high-level computational workflow is as follows:

1. Compute eigendecompositions: $S$8 from $S$9, $S = U \Lambda U^{\top}$0 from $S = U \Lambda U^{\top}$1.
2. Calculate spectrum distance: $S = U \Lambda U^{\top}$2.
3. Convert $S = U \Lambda U^{\top}$3 to quaternions $S = U \Lambda U^{\top}$4 (8 covers).
4. Align quaternions: $S = U \Lambda U^{\top}$5.
5. Quaternion distance: $S = U \Lambda U^{\top}$6.
6. Compute anisotropy indices and $S = U \Lambda U^{\top}$7 as above.
7. Combine for $S = U \Lambda U^{\top}$8.

Interpolation for $S = U \Lambda U^{\top}$9 proceeds by interpolating spectra and quaternions as above, and reconstructing $U \in SO(3)/G$0.

6. Empirical and Comparative Evaluation

Empirical results indicate several key behaviors:

• Distance Consistency: Log-Euclidean and spectral-quaternion behave identically when eigenvalues vary; with varying orientations, spectral-quaternion produces a smooth, nearly linear curve, while Log-Euclidean is jagged and overly sensitive.
• Anisotropy Under Averaging: Log-Euclidean systematically reduces Hilbert and FA as interpolation parameter $U \in SO(3)/G$1 moves from $U \in SO(3)/G$2 to $U \in SO(3)/G$3; spectral-quaternion preserves Hilbert anisotropy exactly, and substantially better preserves FA.
• Global Rotational Invariance: Both approaches are invariant under simultaneous global rotation of endpoints, but only spectral-quaternions retain shape and anisotropy.
• Four-corner Interpolation: The spectral method maintains equal anisotropy at interior points, whereas Log-Euclidean produces “swelling” and substantial anisotropy loss.

7. Comparison with Other Riemannian Metrics

The affine-invariant metric (AIM) uses $U \in SO(3)/G$4 and geodesic $U \in SO(3)/G$5, inherently coupling orientation and spectrum. The spectral-quaternion metric explicitly splits these components, facilitating explicit anisotropy weighting and independent control. While AIM and Log-Euclidean metrics decrease anisotropy under averaging, the spectral-quaternion metric ensures exact “anisotropy commutes with averaging.” Additionally, spectral-quaternions provide closed-form expressions for means and geodesics with minimal computational overhead, contrasting with the higher complexity of legacy Riemannian metrics (Collard et al., 2012).