US-MRI Fusion (biopsy guidance / cardiac) L1-159
Unclaimed Principle — open for contribution
This Principle is declared in the catalog but has no reference solver, no pinned dataset, and is not registered on-chain. There is no reward pool. Submitting a cert against this Principle today will record the cert for reproducibility but pay zero PWM.
To claim it as a Bounty #7 contribution: open a PR adding (1) a reference solver, (2) ≥1 dataset pinned to IPFS, (3) updates to the L3 manifest with dataset CIDs. After verifier-agent triple-review, the founders' 3-of-5 multisig signs PWMRegistry.register() and the Principle becomes mineable.
Forward model E
US-MRI Fusion (biopsy guidance / cardiac): us mri fusion produces the measurement through a 4-node primitive DAG L.mri_volume -> L.live_ultrasound -> L.real_time_registration -> int.temporal, with time-integrated exposure and additive Gaussian thermal/electronic noise. Recovery is posed as a non-convex inverse problem that inverts the forward operator to estimate the scene-side 2D plus 3D fused real time. Difficulty tier delta=5 with effective condition number kappa_eff~18; calibration-level mismatch (real_time_registration_error, non_rigid_tissue, patient_motion) sets the accuracy floor at the Omega boundary. See the forward_model field for the closed-form imaging equation.
L-DAG
Well-posedness W
- Existence:
- true
- Uniqueness:
- true
- Stability:
- conditional
- κ:
- 360
Existence of the recovered 2D plus 3D fused real time is guaranteed within the declared Omega bounds. Uniqueness is local rather than global (non-convex landscape); convergence depends on initialisation and priors. Stability is moderately conditioned (kappa_eff ~= 18); real_time_registration_error dominates the stability cliff; non_rigid_tissue and the remaining mismatch parameters contribute higher-order bias terms. Additive gaussian thermal/electronic noise sets the irreducible data-fidelity floor, while TV / wavelet-sparsity / deep priors stabilise recovery at the ill-conditioned end of Omega.
Solvability C
- Solver class:
- linear-operator + convex optimisation [Mutual-Info-Reg, Deformable-Reg] | linear-operator + deep neural prior [USMRI-Net]
- Convergence rate q:
- 2
- Complexity:
- O(H * W * log(...)) per iteration; learned variants: O(H W Z * F_theta_cost) per forward pass