L1 ⚛ L1-036 ⊙ Testnet

Digital Mammography

Medical Imaging · Low-energy X-ray breast imaging

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⬇ L1-036.md

# ⚛ L1 Principle — Digital Mammography **ID:** `L1-036` · **Status:** ⊙ Testnet (genesis catalog) > **🌐 Domain:** Medical Imaging — *Low-energy X-ray breast imaging* > **🎯 Problem class:** linear inverse · **🧮 Solution space:** 2D breast projection > **📡 Carrier:** x_ray · **🌫 Noise:** shot poisson > **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41553385 --- ## 🧠 1. Introduction **Digital Mammography** is a **linear inverse problem** whose unknown lives in **2D breast projection** space, within the **Low-energy X-ray breast imaging** sub-domain of **Medical Imaging**. Measurements consist of X-ray photons transmitted through (or scattered by) the sample via a **mammography xray** sensing mechanism. The forward operator applies, in order: polyenergetic X-ray emission spectrum; L · compression operator; exponential attenuation along the propagation path; pixel-level spatial averaging on the detector. Observations are corrupted by Poisson shot noise from quantum-limited detection. Existence of the recovered 2D breast projection 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 well-conditioned (kappa_eff ~= 9); scatter dominates the stability cliff; compression_variation and the remaining mismatch parameters contribute higher-order bias terms. Photon-shot-noise-limited (poisson counting) 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** → X-ray source → L · compression → Beer-Lambert attenuation → Spatial integration → **y** (detector). ``` y = ∫_A dA exp(-∫µ dl) `L.compression` I₀(E) x, measurements ~ Poisson(αy) ``` **Measurement DAG:** | Primitive | What it does | |---|---| | `L.xray_source` | Polyenergetic x-ray emission spectrum | | `L.compression` | L · compression operator | | `L.beer_lambert` | Exponential attenuation along the propagation path | | `int.spatial` | Pixel-level spatial averaging on the detector | ## 🔬 3. Physics Fingerprint | Property | Value | |---|---| | Domain | Medical Imaging | | Sub domain | Low-energy X-ray breast imaging | | Carrier | x_ray | | Problem class | linear_inverse | | Solution space | 2D_breast_projection | | Noise model | shot_poisson | | Integration axis | spatial | | Difficulty delta | 3 | | L dag | 2.8 | ## 📡 4. Measurement Model Existence of the recovered 2D breast projection 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 well-conditioned (kappa_eff ~= 9); scatter dominates the stability cliff; compression_variation and the remaining mismatch parameters contribute higher-order bias terms. Photon-shot-noise-limited (poisson counting) 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:** `mammography` · **Forward operator:** `mammography_forward` **Center point:** | Parameter | Unit | Value | |---|---|---| | H | px | 4096 | | W | px | 3328 | | Kvp | — | 28 | | Mas | — | 75 | | Scatter | — | 0.1 | | Pixel um | µm | 50 | | Dense tissue overlap | — | 0 | | Compression variation | — | 0 | **Allowed bounds:** | Parameter | Unit | Range | |---|---|---| | H | px | 2048 | | W | px | 1664 | | Kvp | — | 22 – 34 | | Mas | — | 10 – 400 | | Scatter | — | 0.0 – 0.3 | | Pixel um | µm | 25 – 100 | | Dense tissue overlap | — | 0.0 – 0.5 | | Compression variation | — | 0.0 – 0.2 | | Geometric magnification | — | 1.0 – 2.0 | ## 🎯 6. Tolerance (ε) **Center tolerance:** 35.0 | Metric | Range | |---|---| | Psnr db | 5.0 – 45.0 | ## ⚖ 7. Hardness Function Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `180` and δ = `3`. ## 💾 8. Reference Dataset - **primary** · weight 1.0 · IPFS _(not pinned yet)_ ## 9. On-chain Registration - **Chain hash:** `0xeff332ec150d64d8ce07d27d7f3d4370132bc437f35c6726f04a0a9fb59d816d` - **Chain tx hash:** `0x1cb31a095e9536b02a750de30141eb5f77f2e96740c572822bb3960a1c2d5bba` - **Chain block:** `41553385` --- ## File Mapping This bundle consists of: `L1-036.md`, `L1-036.json`. | File | Role | How to regenerate | |------|------|-------------------| | `L1-036.md` | Source of truth — edit this | Human or LLM | | `L1-036.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 `L1-036`._ _Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._

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Read the attached Markdown (L1-036.md).
Regenerate L1-036.json so every field matches the Markdown.
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L2-036-001 Digital Mammography Mismatch-Only Spec

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📋 JSON metadata

{
  "artifact_id": "L1-036",
  "chain_block": 41553385,
  "chain_hash": "0xeff332ec150d64d8ce07d27d7f3d4370132bc437f35c6726f04a0a9fb59d816d",
  "chain_tx_hash": "0x1cb31a095e9536b02a750de30141eb5f77f2e96740c572822bb3960a1c2d5bba",
  "domain": "Medical Imaging",
  "hardness_fn": {
    "delta": 3,
    "kappa": 180,
    "metric": "PSNR_dB",
    "type": "epsilon_fn"
  },
  "initiator_dataset": [
    {
      "ipfs_cid": null,
      "license_hash": null,
      "name": "primary",
      "weight": 1.0
    }
  ],
  "layer": "L1",
  "observable_profile": {
    "metric": "PSNR_dB",
    "regime": "Existence of the recovered 2D breast projection 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 well-conditioned (kappa_eff ~= 9); scatter dominates the stability cliff; compression_variation and the remaining mismatch parameters contribute higher-order bias terms. Photon-shot-noise-limited (poisson counting) 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": 2.8,
    "carrier": "x_ray",
    "difficulty_delta": 3,
    "domain": "Medical Imaging",
    "integration_axis": "spatial",
    "noise_model": "shot_poisson",
    "primitives": [
      "L.xray_source",
      "L.compression",
      "L.beer_lambert",
      "int.spatial"
    ],
    "problem_class": "linear_inverse",
    "sensing_mechanism": "mammography_xray",
    "solution_space": "2D_breast_projection",
    "sub_domain": "Low-energy X-ray breast imaging",
    "title": "Digital Mammography"
  },
  "size_tiers": {
    "allowed_forward_operators": [
      "mammography_forward"
    ],
    "allowed_omega_dimensions": [
      "H",
      "W",
      "kVp",
      "mAs",
      "pixel_um",
      "scatter",
      "compression_variation",
      "dense_tissue_overlap",
      "scatter",
      "compression_variation",
      "dense_tissue_overlap",
      "geometric_magnification"
    ],
    "allowed_problem_classes": [
      "mammography"
    ],
    "center_spec": {
      "epsilon_fn_center": "35.0",
      "forward_operator": "mammography_forward",
      "input_format": "measurement_only",
      "omega": {
        "H": 4096,
        "W": 3328,
        "compression_variation": 0.0,
        "dense_tissue_overlap": 0.0,
        "kVp": 28,
        "mAs": 75,
        "pixel_um": 50,
        "scatter": 0.1
      },
      "problem_class": "mammography"
    },
    "epsilon_bounds": {
      "psnr_db": [
        5.0,
        45.0
      ]
    },
    "omega_bounds": {
      "H": 2048,
      "W": 1664,
      "compression_variation": [
        0.0,
        0.2
      ],
      "dense_tissue_overlap": [
        0.0,
        0.5
      ],
      "geometric_magnification": [
        1.0,
        2.0
      ],
      "kVp": [
        22,
        34
      ],
      "mAs": [
        10,
        400
      ],
      "pixel_um": [
        25,
        100
      ],
      "scatter": [
        0.0,
        0.3
      ]
    }
  },
  "staked_pwm": 0.0,
  "status": "testnet",
  "sub_domain": "Low-energy X-ray breast imaging",
  "title": "Digital Mammography"
}

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