L1 ⚛ L1-055 ⊙ Testnet

Magnetic Resonance Elastography (MRE)

Medical Imaging · Tissue stiffness via synchronized MR motion encoding

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⬇ L1-055.md

# ⚛ L1 Principle — Magnetic Resonance Elastography (MRE) **ID:** `L1-055` · **Status:** ⊙ Testnet (genesis catalog) > **🌐 Domain:** Medical Imaging — *Tissue stiffness via synchronized MR motion encoding* > **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** 2D tissue stiffness > **📡 Carrier:** radio_wave · **🌫 Noise:** gaussian > **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41553372 --- ## 🧠 1. Introduction **Magnetic Resonance Elastography (MRE)** is a **nonlinear inverse problem** whose unknown lives in **2D tissue stiffness** space, within the **Tissue stiffness via synchronized MR motion encoding** sub-domain of **Medical Imaging**. Measurements consist of radio-frequency electromagnetic waves via a **mri motion encode** sensing mechanism. The forward operator applies, in order: Bloch-equation tip of the magnetisation vector; L · motion encoding gradient operator; L · inverse elasticity operator; detector accumulates flux over the exposure window. Observations are corrupted by additive Gaussian noise. Existence of the recovered 2D tissue stiffness 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); wave_reflections dominates the stability cliff; tissue_anisotropy 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** → RF excitation pulse → L · motion encoding gradient → L · inverse elasticity → Temporal integration → **y** (detector). ``` y = ∫_t dt `L.inverse_elasticity` `L.motion_encoding_gradient` B₁(t) x + n, n ~ 𝒩(0, σ²) ``` **Measurement DAG:** | Primitive | What it does | |---|---| | `L.rf_excitation` | Bloch-equation tip of the magnetisation vector | | `L.motion_encoding_gradient` | L · motion encoding gradient operator | | `L.inverse_elasticity` | L · inverse elasticity operator | | `int.temporal` | Detector accumulates flux over the exposure window | ## 🔬 3. Physics Fingerprint | Property | Value | |---|---| | Domain | Medical Imaging | | Sub domain | Tissue stiffness via synchronized MR motion encoding | | Carrier | radio_wave | | Problem class | nonlinear_inverse | | Solution space | 2D_tissue_stiffness | | Noise model | gaussian | | Integration axis | temporal | | Difficulty delta | 5 | | L dag | 4 | ## 📡 4. Measurement Model Existence of the recovered 2D tissue stiffness 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); wave_reflections dominates the stability cliff; tissue_anisotropy 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:** `mre` · **Forward operator:** `mre_forward` **Center point:** | Parameter | Unit | Value | |---|---|---| | H | px | 256 | | W | px | 256 | | Z | — | 32 | | Snr db | dB | 20 | | F mech hz | Hz | 60 | | Motion artifact | — | 0 | | Wave reflections | — | 0 | | Tissue anisotropy | — | 0 | | Inversion reg error | — | 0 | **Allowed bounds:** | Parameter | Unit | Range | |---|---|---| | H | px | 64 | | W | px | 64 | | Z | — | 8 – 128 | | Snr db | dB | 0.0 – 35.0 | | F mech hz | Hz | 20 – 200 | | Motion artifact | — | 0.0 – 0.3 | | Wave reflections | — | 0.0 – 0.5 | | Tissue anisotropy | — | 0.0 – 0.3 | | Inversion reg error | — | 0.0 – 0.3 | ## 🎯 6. Tolerance (ε) **Center tolerance:** 19.0 | Metric | Range | |---|---| | Psnr db | 5.0 – 45.0 | ## ⚖ 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:** `0x448629572152127fa5dac2a6d337f6a554490bba400bcab8943f3e720ee5811f` - **Chain tx hash:** `0x5138183ae33ae2e600caf07ff4e4ab2887bce639e099d8434f136f6dd670cf19` - **Chain block:** `41553372` --- ## File Mapping This bundle consists of: `L1-055.md`, `L1-055.json`. | File | Role | How to regenerate | |------|------|-------------------| | `L1-055.md` | Source of truth — edit this | Human or LLM | | `L1-055.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 `L1-055`._ _Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._

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Regenerate L1-055.json so every field matches the Markdown.
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L2-055-001 Magnetic Resonance Elastography (MRE) Mismatch-Only Spec

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📋 JSON metadata

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  "chain_tx_hash": "0x5138183ae33ae2e600caf07ff4e4ab2887bce639e099d8434f136f6dd670cf19",
  "domain": "Medical Imaging",
  "hardness_fn": {
    "delta": 5,
    "kappa": 320,
    "metric": "PSNR_dB",
    "type": "epsilon_fn"
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    "regime": "Existence of the recovered 2D tissue stiffness 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); wave_reflections dominates the stability cliff; tissue_anisotropy 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.",
    "secondary": "SSIM"
  },
  "physics_fingerprint": {
    "L_DAG": 4.0,
    "carrier": "radio_wave",
    "difficulty_delta": 5,
    "domain": "Medical Imaging",
    "integration_axis": "temporal",
    "noise_model": "gaussian",
    "primitives": [
      "L.rf_excitation",
      "L.motion_encoding_gradient",
      "L.inverse_elasticity",
      "int.temporal"
    ],
    "problem_class": "nonlinear_inverse",
    "sensing_mechanism": "mri_motion_encode",
    "solution_space": "2D_tissue_stiffness",
    "sub_domain": "Tissue stiffness via synchronized MR motion encoding",
    "title": "Magnetic Resonance Elastography (MRE)"
  },
  "size_tiers": {
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    "allowed_omega_dimensions": [
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      "W",
      "Z",
      "f_mech_Hz",
      "SNR_dB",
      "wave_reflections",
      "tissue_anisotropy",
      "inversion_reg_error",
      "motion_artifact",
      "wave_reflections",
      "tissue_anisotropy",
      "inversion_reg_error",
      "motion_artifact"
    ],
    "allowed_problem_classes": [
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    "center_spec": {
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      "forward_operator": "mre_forward",
      "input_format": "measurement_only",
      "omega": {
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        "SNR_dB": 20,
        "W": 256,
        "Z": 32,
        "f_mech_Hz": 60,
        "inversion_reg_error": 0.0,
        "motion_artifact": 0.0,
        "tissue_anisotropy": 0.0,
        "wave_reflections": 0.0
      },
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    },
    "epsilon_bounds": {
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        45.0
      ]
    },
    "omega_bounds": {
      "H": 64,
      "SNR_dB": [
        0.0,
        35.0
      ],
      "W": 64,
      "Z": [
        8,
        128
      ],
      "f_mech_Hz": [
        20,
        200
      ],
      "inversion_reg_error": [
        0.0,
        0.3
      ],
      "motion_artifact": [
        0.0,
        0.3
      ],
      "tissue_anisotropy": [
        0.0,
        0.3
      ],
      "wave_reflections": [
        0.0,
        0.5
      ]
    }
  },
  "staked_pwm": 0.0,
  "status": "testnet",
  "sub_domain": "Tissue stiffness via synchronized MR motion encoding",
  "title": "Magnetic Resonance Elastography (MRE)"
}

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