Quantitative Susceptibility Mapping (QSM) L1-503

Medical ImagingQuantitative MR susceptibility recovery (multi-physics joint inverse)δ=5 · advancedL_DAG = 7.4📋 Stub — not mineable

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Unclaimed Principle — open for contribution

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Forward model E

p hi (r) = g amma \cdot T E \cdot B_{z} (r); B_{z} (r) - B_{0} = D_{k} er n e l (\cdot) c hi (r); j o in tl y in v er tp hi (r) \to c hi (r) v ia d i p o l e - co n v o l u t i o n o p er a t or (k - s p a ce w a v e - co n er e g u l a r i z a t i o n or m o d e l - ba se d M E D I / T G V - QS M)

Quantitative Susceptibility Mapping (QSM): joint multi-physics forward couples gradient-echo MR phase encoding (Larmor precession under static + perturbed field) with Maxwell magnetostatic dipole-field perturbation arising from spatially varying tissue magnetic susceptibility chi(r). The forward DAG has 7 primitives with two coupling constraints (n_c = 2): (i) phase-to-field linear coupling phi = gamma * TE * (B_z - B_0); (ii) field-to-susceptibility dipole convolution (B_z - B_0) = D_kernel(*) chi where D_kernel(k) = (1/3) - k_z^2/|k|^2 is the Lorentzian dipole response in k-space. Recovery is posed as the joint linear inverse problem that recovers chi(r) from measured complex GRE signal across multiple TE. Difficulty tier delta = 5 with raw condition number kappa ~ 240 (zero-cone singularity at the magic angle ~ 54.7 degrees) and effective kappa_eff ~ 30 after wave-cone regularization. Mismatch parameters: B0_inhomogeneity, brain_segmentation_error, phase_unwrap_error, dipole_kernel_truncation_error, background_field_removal_error. Additive Gaussian thermal/electronic noise sets the data-fidelity floor. See forward_model field for the closed-form joint imaging equation.

L-DAG

L.rf_excitation -> L.gradient_echo -> L.larmor_precession -> L.maxwell_magnetostatic -> L.dipole_convolution -> L.phase_extract -> int.spatial

L.rf_excitationL.gradient_echoL.larmor_precessionL.maxwell_magnetostaticL.dipole_convolutionL.phase_extractint.spatial

Well-posedness W

Existence:: true
Uniqueness:: conditional
Stability:: conditional
κ:: 240

Existence of recovered 3D quantitative susceptibility chi(r) is guaranteed within the declared Omega bounds. Uniqueness holds on the wave-cone-supported subspace; modes inside the magic-angle zero cone are recovered only through priors (MEDI L1-edge, TGV second-order, deep-prior). Stability is moderately conditioned (kappa_eff ~ 30 after wave-cone regularization) — B0_inhomogeneity and background_field_removal_error dominate the stability cliff; phase_unwrap_error and dipole_kernel_truncation contribute higher-order bias terms. Joint Hadamard well-posedness for the coupled phase-field-susceptibility forward is established by Liu, de Rochefort, Wharton, Bilgic literature 2009-2018 (see references).

Solvability C

Solver class:: linear-operator + convex optimisation [MEDI, TKD, iLSQR, TGV-QSM] | linear-operator + deep neural prior [QSMnet, xQSM]
Convergence rate q:: 2
Complexity:: O(H * W * Z * log(H * W * Z)) per iteration via FFT-based dipole-convolution forward / adjoint; learned variants O(H W Z * F_theta_cost) per forward pass

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