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 $D^{+}(3)$) parts. The metric is defined as

$$g_S(\delta S, \delta S) = k(\Lambda)\,g_{SO(3)}(\Omega, \Omega) + g_{D^{+}(3)}(d\Lambda, d\Lambda)$$

where:

• $\Omega = U^{\top} \delta U \in \mathfrak{so}(3)$ is the infinitesimal generator of rotation,
• $$d\Lambda = \operatorname{diag}(U^{\top}\delta S U) \in T_\Lambda D^{+}(3)$$,
• $$g_{SO(3)}(\Omega, \Omega) = \frac{1}{2} \operatorname{tr} (\Omega^\top \Omega)$$,
• $$g_{D^{+}(3)}(d\Lambda, d\Lambda) = \sum_{i=1}^3 (d\lambda_i/\lambda_i)^2$$,
• $k(\Lambda) \in [0,1]$ is a smooth increasing function of tensor anisotropy.

Anisotropy is measured by the Hilbert index:

$HA(S) := \log(\lambda_1/\lambda_3)$

For two tensors $S_1$, $S_2$, the weight is

$$k(HA_1, HA_2) = \frac{1}{2}\left[1 + \tanh(3\,HA_1\,HA_2 - 7)\right]$$

The induced Riemannian distance is given by

$$d^2(S_1, S_2) = k(\Lambda_1, \Lambda_2) d^2_{SO(3)}(U_1, U_2) + d^2_{D^{+}(3)}(\Lambda_1, \Lambda_2)$$

with

• $d_{SO(3)}(U_1, U_2) = \|\log (U_1^\top U_2)\|_F$,
• $$d_{D^{+}(3)}(\Lambda_1, \Lambda_2) = \sqrt{\sum_{i} \log^2(\lambda_{1,i}/\lambda_{2,i})}$$, where the logarithm for diagonal $\Lambda$ is entry-wise.

2. Geodesic and Interpolation Procedures

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

• Spectrum: For $0 \leq t \leq 1$,

$$\lambda_i(t) = \exp\big[(1-t)\log \lambda_{1,i} + t\log \lambda_{2,i}\big],\quad \Lambda(t) = \operatorname{diag}(\lambda_1(t),\lambda_2(t),\lambda_3(t))$$

• Orientation: $U_1$, $U_2$ are converted to quaternions $q_1$, $q_2$; $q_2^\star$ is selected from the 8 covers of $SO(3)/G$ to maximize $q_1 \cdot q_2$; interpolation is

$$q_m = (1-t)q_1 + t q_2^\star,\qquad q(t) = \frac{q_m}{\|q_m\|}$$

The SPD tensor at $t$ is synthesized as

$S(t) = R(q(t)) \Lambda(t) R(q(t))^\top$

where $R(q)$ is the $SO(3)$ matrix corresponding to quaternion $q$.

3. Anisotropy Preservation Property

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

$$HA(S_\mu) = \log\left(\frac{\lambda_{\mu,1}}{\lambda_{\mu,3}}\right) = w_1\,HA_1 + w_2\,HA_2$$

where $w_1, w_2$ 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 $O(1)$ quaternion dot-products and normalizations. No iterative procedures are required for means or geodesics. Comparative timing for $1000$ 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 ($SO(3)$ log) | 0.65 | Matrix-log distance on $SO(3)$ | | Spectral-quaternions | 0.11 | Quaternion chordal, fastest |

The spectral-quaternions approach is approximately $1.5\times$ faster than Log-Euclidean and $5\times$ faster than affine-invariant in this benchmark.

5. Algorithmic Workflow

A high-level computational workflow is as follows:

1. Compute eigendecompositions: $(\Lambda_1, U_1)$ from $S_1$, $(\Lambda_2, U_2)$ from $S_2$.
2. Calculate spectrum distance: $$d_{\mathrm{spec}}^2 = \sum_i [\log(\lambda_{1,i}/\lambda_{2,i})]^2$$.
3. Convert $U_1, U_2$ to quaternions $q_1, \{q_2\}$ (8 covers).
4. Align quaternions: $$q_2^\star = \operatorname{argmax}_{q \in \mathcal{Q}_2} (q_1 \cdot q)$$.
5. Quaternion distance: $d_{\mathrm{rot}}^2 = \| q_1 - q_2^\star \|^2$.
6. Compute anisotropy indices and $k$ as above.
7. Combine for $$d_{SD} = \sqrt{ k d_{\mathrm{rot}}^2 + d_{\mathrm{spec}}^2 }$$.

Interpolation for $t \in [0, 1]$ proceeds by interpolating spectra and quaternions as above, and reconstructing $S(t)$.

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 $t$ moves from $0$ to $1$; 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 $$g^{AIM}_S(\xi, \eta)=\operatorname{tr}(S^{-1} \xi\,S^{-1} \eta)$$ and geodesic $$S(t)=S_1^{1/2}(S_1^{-1/2} S_2 S_1^{-1/2})^t S_1^{1/2}$$, 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).