Robust Geometric Deformation Extraction from Low-SNR Relaxometry Data via Constrained Manifold Projection

The Geometric Deformation metric (E) provides a robust, physics-based measure of distributional heterogeneity in biexponential decay systems such as brain white matter. E encodes tissue complexity through the curvature of the log-derivative—information that is orthogonal to signal amplitude and thus resistant to partial volume effects. However, direct computation of E requires high signal-to-noise ratio (SNR > 10,000) because differentiation and logarithm operations amplify noise. We present the Constrained Manifold Projection method, which enables robust E extraction at clinical SNR (≈70) from as few as 10 pixels. The key insight is that valid biological signals must lie on the biexponential manifold; noise does not. By projecting noisy data onto the nearest physically valid curve before measuring E, we filter noise while preserving the geometric properties that encode tissue heterogeneity. A critical methodological innovation is the R₁ Taylor Skeleton Invariant for monoexponential/biexponential discrimination. The Turán ratio R₁ = (a₀·a₂)/a₁², computed analytically from fit parameters, equals exactly 1.0 for monoexponential decay and exceeds 1.0 for any biexponential mixture. This coefficient-space approach achieves 100% discrimination between cerebrospinal fluid (CSF) and pathological tissue—solving the partial volume problem that plagued earlier threshold-based methods. Validation demonstrates: (1) Monte Carlo simulation shows >99% bias reduction and CNR >10 for Alzheimer's disease discrimination; (2) Digital Twin phantom imaging confirms spatial E-mapping with lesion detection at clinical noise levels. The dual-feature fingerprint (E vs R₁) provides complete tissue separation. This technology can be deployed as a software update to existing MRI scanners worldwide.