L1 ⚛ L1-059 ⊙ Testnet

Susceptibility-Weighted Imaging (SWI)

Medical Imaging · T2*/phase venous and microbleed imaging

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⬇ L1-059.md

# ⚛ L1 Principle — Susceptibility-Weighted Imaging (SWI) **ID:** `L1-059` · **Status:** ⊙ Testnet (genesis catalog) > **🌐 Domain:** Medical Imaging — *T2*/phase venous and microbleed imaging* > **🎯 Problem class:** linear inverse · **🧮 Solution space:** 3D susceptibility > **📡 Carrier:** radio_wave · **🌫 Noise:** gaussian > **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41553372 --- ## 🧠 1. Introduction **Susceptibility-Weighted Imaging (SWI)** is a **linear inverse problem** whose unknown lives in **3D susceptibility** space, within the **T2*/phase venous and microbleed imaging** sub-domain of **Medical Imaging**. Measurements consist of radio-frequency electromagnetic waves via a **mri fourier encoding** sensing mechanism. The forward operator applies, in order: Bloch-equation tip of the magnetisation vector; L · gradient echo operator; L · phase mask operator; pixel-level spatial averaging on the detector. Observations are corrupted by additive Gaussian noise. Existence of the recovered 3D susceptibility is guaranteed within the declared Omega bounds. Uniqueness holds on the measurement-supported subspace; out-of-support modes are controlled by the declared priors. Stability is moderately conditioned (kappa_eff ~= 12); B0_inhomogeneity dominates the stability cliff; air_tissue_boundaries and the remaining mismatch parameters contribute higher-order bias terms. Additive gaussian thermal/electronic noise sets the irreducible data-fidelity floor, while mild Tikhonov or analytic inversion is sufficient at the nominal Omega point. ## ⚙ 2. Forward Model Physical chain: **x** → RF excitation pulse → L · gradient echo → L · phase mask → Spatial integration → **y** (detector). ``` y = ∫_A dA `L.phase_mask` `L.gradient_echo` 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.gradient_echo` | L · gradient echo operator | | `L.phase_mask` | L · phase mask operator | | `int.spatial` | Pixel-level spatial averaging on the detector | ## 🔬 3. Physics Fingerprint | Property | Value | |---|---| | Domain | Medical Imaging | | Sub domain | T2*/phase venous and microbleed imaging | | Carrier | radio_wave | | Problem class | linear_inverse | | Solution space | 3D_susceptibility | | Noise model | gaussian | | Integration axis | spatial | | Difficulty delta | 3 | | L dag | 3.3 | ## 📡 4. Measurement Model Existence of the recovered 3D susceptibility is guaranteed within the declared Omega bounds. Uniqueness holds on the measurement-supported subspace; out-of-support modes are controlled by the declared priors. Stability is moderately conditioned (kappa_eff ~= 12); B0_inhomogeneity dominates the stability cliff; air_tissue_boundaries and the remaining mismatch parameters contribute higher-order bias terms. Additive gaussian thermal/electronic noise sets the irreducible data-fidelity floor, while mild Tikhonov or analytic inversion is sufficient at the nominal Omega point. | Metric | Value | |---|---| | Metric | PSNR_dB | | Secondary | SSIM | ## 📏 5. Operating Range (Ω) **Center problem class:** `swi` · **Forward operator:** `swi_forward` **Center point:** | Parameter | Unit | Value | |---|---|---| | H | px | 256 | | W | px | 256 | | Z | — | 64 | | Te ms | ms | 25 | | Snr db | dB | 22 | | Phase wrap | — | 0 | | B0 inhomogeneity | — | 0 | | Coil phase error | — | 0 | | Air tissue boundaries | — | 0 | **Allowed bounds:** | Parameter | Unit | Range | |---|---|---| | H | px | 128 | | W | px | 128 | | Z | — | 16 – 256 | | Te ms | ms | 10 – 80 | | Snr db | dB | 0.0 – 35.0 | | Phase wrap | — | 0.0 – 0.3 | | B0 inhomogeneity | — | 0.0 – 0.3 | | Coil phase error | — | 0.0 – 0.3 | | Air tissue boundaries | — | 0.0 – 0.5 | ## 🎯 6. Tolerance (ε) **Center tolerance:** 24.0 | Metric | Range | |---|---| | Psnr db | 5.0 – 45.0 | ## ⚖ 7. Hardness Function Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `240` and δ = `3`. ## 💾 8. Reference Dataset - **primary** · weight 1.0 · IPFS _(not pinned yet)_ ## 9. On-chain Registration - **Chain hash:** `0x7c31accd30f1408fb0e3db6db812a0527d69b7e44555a513bf46669fca678c29` - **Chain tx hash:** `0x5cf36f36e62eb7f46881513bd99a522650d79ec98123d10b9b4e49699b92a342` - **Chain block:** `41553372` --- ## File Mapping This bundle consists of: `L1-059.md`, `L1-059.json`. | File | Role | How to regenerate | |------|------|-------------------| | `L1-059.md` | Source of truth — edit this | Human or LLM | | `L1-059.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 `L1-059`._ _Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._

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Read the attached Markdown (L1-059.md).
Regenerate L1-059.json so every field matches the Markdown.
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L2-059-001 Susceptibility-Weighted Imaging (SWI) Mismatch-Only Spec

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📋 JSON metadata

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  "chain_tx_hash": "0x5cf36f36e62eb7f46881513bd99a522650d79ec98123d10b9b4e49699b92a342",
  "domain": "Medical Imaging",
  "hardness_fn": {
    "delta": 3,
    "kappa": 240,
    "metric": "PSNR_dB",
    "type": "epsilon_fn"
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      "name": "primary",
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    "regime": "Existence of the recovered 3D susceptibility is guaranteed within the declared Omega bounds. Uniqueness holds on the measurement-supported subspace; out-of-support modes are controlled by the declared priors. Stability is moderately conditioned (kappa_eff ~= 12); B0_inhomogeneity dominates the stability cliff; air_tissue_boundaries and the remaining mismatch parameters contribute higher-order bias terms. Additive gaussian thermal/electronic noise sets the irreducible data-fidelity floor, while mild Tikhonov or analytic inversion is sufficient at the nominal Omega point.",
    "secondary": "SSIM"
  },
  "physics_fingerprint": {
    "L_DAG": 3.3,
    "carrier": "radio_wave",
    "difficulty_delta": 3,
    "domain": "Medical Imaging",
    "integration_axis": "spatial",
    "noise_model": "gaussian",
    "primitives": [
      "L.rf_excitation",
      "L.gradient_echo",
      "L.phase_mask",
      "int.spatial"
    ],
    "problem_class": "linear_inverse",
    "sensing_mechanism": "mri_fourier_encoding",
    "solution_space": "3D_susceptibility",
    "sub_domain": "T2*/phase venous and microbleed imaging",
    "title": "Susceptibility-Weighted Imaging (SWI)"
  },
  "size_tiers": {
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      "W",
      "Z",
      "TE_ms",
      "SNR_dB",
      "B0_inhomogeneity",
      "air_tissue_boundaries",
      "phase_wrap",
      "coil_phase_error",
      "B0_inhomogeneity",
      "air_tissue_boundaries",
      "phase_wrap",
      "coil_phase_error"
    ],
    "allowed_problem_classes": [
      "swi"
    ],
    "center_spec": {
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      "forward_operator": "swi_forward",
      "input_format": "measurement_only",
      "omega": {
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        "H": 256,
        "SNR_dB": 22,
        "TE_ms": 25,
        "W": 256,
        "Z": 64,
        "air_tissue_boundaries": 0.0,
        "coil_phase_error": 0.0,
        "phase_wrap": 0.0
      },
      "problem_class": "swi"
    },
    "epsilon_bounds": {
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        45.0
      ]
    },
    "omega_bounds": {
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        0.0,
        0.3
      ],
      "H": 128,
      "SNR_dB": [
        0.0,
        35.0
      ],
      "TE_ms": [
        10,
        80
      ],
      "W": 128,
      "Z": [
        16,
        256
      ],
      "air_tissue_boundaries": [
        0.0,
        0.5
      ],
      "coil_phase_error": [
        0.0,
        0.3
      ],
      "phase_wrap": [
        0.0,
        0.3
      ]
    }
  },
  "staked_pwm": 0.0,
  "status": "testnet",
  "sub_domain": "T2*/phase venous and microbleed imaging",
  "title": "Susceptibility-Weighted Imaging (SWI)"
}

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