L1 ⚛ L1-069 ⊙ Testnet

Magnetic Particle Imaging (MPI)

Medical Imaging · Superparamagnetic iron-oxide tracer imaging

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⬇ L1-069.md

# ⚛ L1 Principle — Magnetic Particle Imaging (MPI) **ID:** `L1-069` · **Status:** ⊙ Testnet (genesis catalog) > **🌐 Domain:** Medical Imaging — *Superparamagnetic iron-oxide tracer imaging* > **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** 3D tracer concentration > **📡 Carrier:** radio_wave · **🌫 Noise:** gaussian > **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41552304 --- ## 🧠 1. Introduction **Magnetic Particle Imaging (MPI)** is a **nonlinear inverse problem** whose unknown lives in **3D tracer concentration** space, within the **Superparamagnetic iron-oxide tracer imaging** sub-domain of **Medical Imaging**. Measurements consist of radio-frequency electromagnetic waves via a **mpi ffp scan** sensing mechanism. The forward operator applies, in order: L · drive field operator; L · selection field ffp operator; D · receive coil operator; detector accumulates flux over the exposure window. Observations are corrupted by additive Gaussian noise. Existence of the recovered 3D tracer concentration 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 ~= 20); spatial_relaxation dominates the stability cliff; tracer_distribution 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 · drive field → L · selection field ffp → D · receive coil → Temporal integration → **y** (detector). ``` y = ∫_t dt `D.receive_coil` `L.selection_field_ffp` `L.drive_field` x + n, n ~ 𝒩(0, σ²) ``` **Measurement DAG:** | Primitive | What it does | |---|---| | `L.drive_field` | L · drive field operator | | `L.selection_field_ffp` | L · selection field ffp operator | | `D.receive_coil` | D · receive coil operator | | `int.temporal` | Detector accumulates flux over the exposure window | ## 🔬 3. Physics Fingerprint | Property | Value | |---|---| | Domain | Medical Imaging | | Sub domain | Superparamagnetic iron-oxide tracer imaging | | Carrier | radio_wave | | Problem class | nonlinear_inverse | | Solution space | 3D_tracer_concentration | | Noise model | gaussian | | Integration axis | temporal | | Difficulty delta | 5 | | L dag | 4 | ## 📡 4. Measurement Model Existence of the recovered 3D tracer concentration 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 ~= 20); spatial_relaxation dominates the stability cliff; tracer_distribution 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:** `mpi` · **Forward operator:** `mpi_forward` **Center point:** | Parameter | Unit | Value | |---|---|---| | H | px | 128 | | W | px | 128 | | Z | — | 64 | | Snr db | dB | 20 | | N harmonics | — | 20 | | F drive khz | kHz | 25 | | Spatial relaxation | — | 0 | | Tracer distribution | — | 0 | | Drive field harmonics | — | 0 | | Coil sensitivity error | — | 0 | **Allowed bounds:** | Parameter | Unit | Range | |---|---|---| | H | px | 32 | | W | px | 32 | | Z | — | 8 – 256 | | Snr db | dB | 0.0 – 35.0 | | N harmonics | — | 5 – 100 | | F drive khz | kHz | 5 – 50 | | Spatial relaxation | — | 0.0 – 0.3 | | Tracer distribution | — | 0.0 – 0.3 | | Drive field harmonics | — | 0.0 – 0.3 | | Coil sensitivity error | — | 0.0 – 0.3 | ## 🎯 6. Tolerance (ε) **Center tolerance:** 20.0 | Metric | Range | |---|---| | Psnr db | 5.0 – 45.0 | ## ⚖ 7. Hardness Function Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `400` and δ = `5`. ## 💾 8. Reference Dataset - **primary** · weight 1.0 · IPFS _(not pinned yet)_ ## 9. On-chain Registration - **Chain hash:** `0xd667e923b42a7e570729f61b099f1850c7402ccdf2a5ef4672bf583fe102c1d6` - **Chain tx hash:** `0x35fda28a37fe8bdbe112439e5d1a5b3c668b1ede4349679e82f908eea582971c` - **Chain block:** `41552304` --- ## File Mapping This bundle consists of: `L1-069.md`, `L1-069.json`. | File | Role | How to regenerate | |------|------|-------------------| | `L1-069.md` | Source of truth — edit this | Human or LLM | | `L1-069.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 `L1-069`._ _Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._

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Read the attached Markdown (L1-069.md).
Regenerate L1-069.json so every field matches the Markdown.
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Output each file in its own fenced code block tagged with the filename.

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L2-069-001 Magnetic Particle Imaging (MPI) Mismatch-Only Spec

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📋 JSON metadata

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  "chain_block": 41552304,
  "chain_hash": "0xd667e923b42a7e570729f61b099f1850c7402ccdf2a5ef4672bf583fe102c1d6",
  "chain_tx_hash": "0x35fda28a37fe8bdbe112439e5d1a5b3c668b1ede4349679e82f908eea582971c",
  "domain": "Medical Imaging",
  "hardness_fn": {
    "delta": 5,
    "kappa": 400,
    "metric": "PSNR_dB",
    "type": "epsilon_fn"
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      "license_hash": null,
      "name": "primary",
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  "observable_profile": {
    "metric": "PSNR_dB",
    "regime": "Existence of the recovered 3D tracer concentration 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 ~= 20); spatial_relaxation dominates the stability cliff; tracer_distribution 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.drive_field",
      "L.selection_field_ffp",
      "D.receive_coil",
      "int.temporal"
    ],
    "problem_class": "nonlinear_inverse",
    "sensing_mechanism": "mpi_ffp_scan",
    "solution_space": "3D_tracer_concentration",
    "sub_domain": "Superparamagnetic iron-oxide tracer imaging",
    "title": "Magnetic Particle Imaging (MPI)"
  },
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      "N_harmonics",
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      "SNR_dB",
      "spatial_relaxation",
      "tracer_distribution",
      "drive_field_harmonics",
      "coil_sensitivity_error",
      "spatial_relaxation",
      "tracer_distribution",
      "drive_field_harmonics",
      "coil_sensitivity_error"
    ],
    "allowed_problem_classes": [
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    "center_spec": {
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      "forward_operator": "mpi_forward",
      "input_format": "measurement_only",
      "omega": {
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        "N_harmonics": 20,
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        "Z": 64,
        "coil_sensitivity_error": 0.0,
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        "f_drive_kHz": 25,
        "spatial_relaxation": 0.0,
        "tracer_distribution": 0.0
      },
      "problem_class": "mpi"
    },
    "epsilon_bounds": {
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    "omega_bounds": {
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      "N_harmonics": [
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      "SNR_dB": [
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      ],
      "W": 32,
      "Z": [
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      ],
      "coil_sensitivity_error": [
        0.0,
        0.3
      ],
      "drive_field_harmonics": [
        0.0,
        0.3
      ],
      "f_drive_kHz": [
        5,
        50
      ],
      "spatial_relaxation": [
        0.0,
        0.3
      ],
      "tracer_distribution": [
        0.0,
        0.3
      ]
    }
  },
  "staked_pwm": 0.0,
  "status": "testnet",
  "sub_domain": "Superparamagnetic iron-oxide tracer imaging",
  "title": "Magnetic Particle Imaging (MPI)"
}

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