L1 ⚛ L1-157 ⊙ Testnet

PET-MR Fusion (molecular + multi-contrast soft tissue)

Multimodal Fusion · Simultaneous or sequential PET + multi-contrast MR imaging

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⬇ L1-157.md

# ⚛ L1 Principle — PET-MR Fusion (molecular + multi-contrast soft tissue) **ID:** `L1-157` · **Status:** ⊙ Testnet (genesis catalog) > **🌐 Domain:** Multimodal Fusion — *Simultaneous or sequential PET + multi-contrast MR imaging* > **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** 4D pet mr fused > **📡 Carrier:** photon · **🌫 Noise:** gaussian > **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41554241 --- ## 🧠 1. Introduction **PET-MR Fusion (molecular + multi-contrast soft tissue)** is a **nonlinear inverse problem** whose unknown lives in **4D pet mr fused** space, within the **Simultaneous or sequential PET + multi-contrast MR imaging** sub-domain of **Multimodal Fusion**. Measurements consist of photons collected by an optical detector via a **pet mr fusion** sensing mechanism. The forward operator applies, in order: L · mr multi contrast operator; L · pet activity map operator; L · registration operator; L · attenuation from mr operator; pixel-level spatial averaging on the detector. Observations are corrupted by additive Gaussian noise. Existence of the recovered 4D pet mr fused 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 ~= 16); mr_based_attenuation_error dominates the stability cliff; motion_between_scans 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. ## ⚙ 2. Forward Model Physical chain: **x** → L · mr multi contrast → L · pet activity map → L · registration → L · attenuation from mr → Spatial integration → **y** (detector). ``` y = ∫_A dA `L.attenuation_from_mr` `L.registration` `L.pet_activity_map` `L.mr_multi_contrast` x + n, n ~ 𝒩(0, σ²) ``` **Measurement DAG:** | Primitive | What it does | |---|---| | `L.mr_multi_contrast` | L · mr multi contrast operator | | `L.pet_activity_map` | L · pet activity map operator | | `L.registration` | L · registration operator | | `L.attenuation_from_mr` | L · attenuation from mr operator | | `int.spatial` | Pixel-level spatial averaging on the detector | ## 🔬 3. Physics Fingerprint | Property | Value | |---|---| | Domain | Multimodal Fusion | | Sub domain | Simultaneous or sequential PET + multi-contrast MR imaging | | Carrier | photon | | Problem class | nonlinear_inverse | | Solution space | 4D_pet_mr_fused | | Noise model | gaussian | | Integration axis | spatial | | Difficulty delta | 5 | | L dag | 3.8 | ## 📡 4. Measurement Model Existence of the recovered 4D pet mr fused 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 ~= 16); mr_based_attenuation_error dominates the stability cliff; motion_between_scans 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. | Metric | Value | |---|---| | Metric | PSNR_dB | | Secondary | SSIM | ## 📏 5. Operating Range (Ω) **Center problem class:** `pet_mr_fusion` · **Forward operator:** `pet_mr_fusion_forward` **Center point:** | Parameter | Unit | Value | |---|---|---| | H | px | 256 | | W | px | 256 | | Z | — | 128 | | Snr db | dB | 22 | | Truncation | — | 0 | | Motion between scans | — | 0 | | Susceptibility distortion | — | 0 | | Mr based attenuation error | — | 0 | **Allowed bounds:** | Parameter | Unit | Range | |---|---|---| | H | px | 128 | | W | px | 128 | | Z | — | 32 – 512 | | Snr db | dB | 0.0 – 35.0 | | Truncation | — | 0.0 – 0.3 | | Motion between scans | — | 0.0 – 10.0 | | Susceptibility distortion | — | 0.0 – 0.3 | | Mr based attenuation error | — | 0.0 – 0.3 | ## 🎯 6. Tolerance (ε) **Center tolerance:** 24.0 | Metric | Range | |---|---| | Psnr db | 5.0 – 40.0 | ## ⚖ 7. Hardness Function Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `320` and δ = `5`. ## 💾 8. Reference Dataset - **primary** · weight 1.0 · IPFS _(not pinned yet)_ ## 9. On-chain Registration - **Chain hash:** `0xadbe9829004e25702f4a55073114dea1c8b4534c9fe88e7db27e79a6757dfc74` - **Chain tx hash:** `0xf1449e0212c8dcee45ae95d39c2d02393a7c66c210006488134e9abaf0f4bffd` - **Chain block:** `41554241` --- ## File Mapping This bundle consists of: `L1-157.md`, `L1-157.json`. | File | Role | How to regenerate | |------|------|-------------------| | `L1-157.md` | Source of truth — edit this | Human or LLM | | `L1-157.json` | Structured metadata for the registry | LLM regenerates from the sections above | **Prompt for your LLM after editing this Markdown:** > Read the attached Markdown. Regenerate the sibling `.json` so every field matches. > Preserve the schema documented in the rows above. > Output each file in its own fenced code block tagged with the filename. > Output only the JSON object. _This Markdown was auto-synthesized from the catalog row for `L1-157`._ _Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._

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Read the attached Markdown (L1-157.md).
Regenerate L1-157.json so every field matches the Markdown.
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L2-157-001 PET-MR Fusion (molecular + multi-contrast soft tissue) Mismatch-Only Spec

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📋 JSON metadata

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  "domain": "Multimodal Fusion",
  "hardness_fn": {
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    "solution_space": "4D_pet_mr_fused",
    "sub_domain": "Simultaneous or sequential PET + multi-contrast MR imaging",
    "title": "PET-MR Fusion (molecular + multi-contrast soft tissue)"
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  "staked_pwm": 0.0,
  "status": "testnet",
  "sub_domain": "Simultaneous or sequential PET + multi-contrast MR imaging",
  "title": "PET-MR Fusion (molecular + multi-contrast soft tissue)"
}

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