L1 ⚛ L1-046 ⊙ Testnet

Doppler Ultrasound (color / spectral)

Medical Imaging · Blood-flow velocity via Doppler shift

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⬇ L1-046.md

# ⚛ L1 Principle — Doppler Ultrasound (color / spectral) **ID:** `L1-046` · **Status:** ⊙ Testnet (genesis catalog) > **🌐 Domain:** Medical Imaging — *Blood-flow velocity via Doppler shift* > **🎯 Problem class:** linear inverse · **🧮 Solution space:** 2D velocity map > **📡 Carrier:** acoustic · **🌫 Noise:** gaussian > **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41553358 --- ## 🧠 1. Introduction **Doppler Ultrasound (color / spectral)** is a **linear inverse problem** whose unknown lives in **2D velocity map** space, within the **Blood-flow velocity via Doppler shift** sub-domain of **Medical Imaging**. Measurements consist of acoustic pressure waves recorded by transducers via a **doppler ultrasound** sensing mechanism. The forward operator applies, in order: L · emit · acoustic pulse operator; L · doppler autocorrelation operator; detector accumulates flux over the exposure window. Observations are corrupted by additive Gaussian noise. Existence of the recovered 2D velocity map 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 ~= 10); angle_correction dominates the stability cliff; wall_filter_bleedthrough 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** → L · emit · acoustic pulse → L · doppler autocorrelation → Temporal integration → **y** (detector). ``` y = ∫_t dt `L.doppler_autocorrelation` `L.emit.acoustic_pulse` x + n, n ~ 𝒩(0, σ²) ``` **Measurement DAG:** | Primitive | What it does | |---|---| | `L.emit.acoustic_pulse` | L · emit · acoustic pulse operator | | `L.doppler_autocorrelation` | L · doppler autocorrelation operator | | `int.temporal` | Detector accumulates flux over the exposure window | ## 🔬 3. Physics Fingerprint | Property | Value | |---|---| | Domain | Medical Imaging | | Sub domain | Blood-flow velocity via Doppler shift | | Carrier | acoustic | | Problem class | linear_inverse | | Solution space | 2D_velocity_map | | Noise model | gaussian | | Integration axis | temporal | | Difficulty delta | 3 | | L dag | 3.2 | ## 📡 4. Measurement Model Existence of the recovered 2D velocity map 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 ~= 10); angle_correction dominates the stability cliff; wall_filter_bleedthrough 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:** `doppler_us` · **Forward operator:** `doppler_us_forward` **Center point:** | Parameter | Unit | Value | |---|---|---| | F mhz | MHz | 5 | | Snr db | dB | 22 | | Prf khz | kHz | 5 | | Aliasing | — | 0 | | Depth cm | — | 10 | | N elements | — | 128 | | Angle correction | — | 0 | | Wall filter bleedthrough | — | 0 | **Allowed bounds:** | Parameter | Unit | Range | |---|---|---| | F mhz | MHz | 1 – 15 | | Snr db | dB | 0.0 – 35.0 | | Prf khz | kHz | 0.5 – 20 | | Aliasing | — | 0.0 – 0.5 | | Depth cm | — | 1 – 25 | | N elements | — | 32 – 256 | | Angle correction | — | 0.0 – 0.5 | | Low velocity sensitivity | — | 0.0 – 0.3 | | Wall filter bleedthrough | — | 0.0 – 0.3 | ## 🎯 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 = `200` and δ = `3`. ## 💾 8. Reference Dataset - **primary** · weight 1.0 · IPFS _(not pinned yet)_ ## 9. On-chain Registration - **Chain hash:** `0xd7ddf7569a81138fc26592b42a8e7866993195425b5c6627d0097f0cd9dfce57` - **Chain tx hash:** `0xa1a7462d4c71ae904d936f9b1c2e59a6dfe84bfb389b4d6dcb886eee36f5bdef` - **Chain block:** `41553358` --- ## File Mapping This bundle consists of: `L1-046.md`, `L1-046.json`. | File | Role | How to regenerate | |------|------|-------------------| | `L1-046.md` | Source of truth — edit this | Human or LLM | | `L1-046.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 `L1-046`._ _Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._

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Read the attached Markdown (L1-046.md).
Regenerate L1-046.json so every field matches the Markdown.
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L2-046-001 Doppler Ultrasound (color / spectral) Mismatch-Only Spec

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📋 JSON metadata

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  "chain_block": 41553358,
  "chain_hash": "0xd7ddf7569a81138fc26592b42a8e7866993195425b5c6627d0097f0cd9dfce57",
  "chain_tx_hash": "0xa1a7462d4c71ae904d936f9b1c2e59a6dfe84bfb389b4d6dcb886eee36f5bdef",
  "domain": "Medical Imaging",
  "hardness_fn": {
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    "kappa": 200,
    "metric": "PSNR_dB",
    "type": "epsilon_fn"
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    "metric": "PSNR_dB",
    "regime": "Existence of the recovered 2D velocity map 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 ~= 10); angle_correction dominates the stability cliff; wall_filter_bleedthrough 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": 3.2,
    "carrier": "acoustic",
    "difficulty_delta": 3,
    "domain": "Medical Imaging",
    "integration_axis": "temporal",
    "noise_model": "gaussian",
    "primitives": [
      "L.emit.acoustic_pulse",
      "L.doppler_autocorrelation",
      "int.temporal"
    ],
    "problem_class": "linear_inverse",
    "sensing_mechanism": "doppler_ultrasound",
    "solution_space": "2D_velocity_map",
    "sub_domain": "Blood-flow velocity via Doppler shift",
    "title": "Doppler Ultrasound (color / spectral)"
  },
  "size_tiers": {
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    ],
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      "PRF_kHz",
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      "SNR_dB",
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      "aliasing",
      "angle_correction",
      "wall_filter_bleedthrough",
      "aliasing",
      "low_velocity_sensitivity"
    ],
    "allowed_problem_classes": [
      "doppler_us"
    ],
    "center_spec": {
      "epsilon_fn_center": "24.0",
      "forward_operator": "doppler_us_forward",
      "input_format": "measurement_only",
      "omega": {
        "N_elements": 128,
        "PRF_kHz": 5,
        "SNR_dB": 22,
        "aliasing": 0.0,
        "angle_correction": 0.0,
        "depth_cm": 10,
        "f_MHz": 5,
        "wall_filter_bleedthrough": 0.0
      },
      "problem_class": "doppler_us"
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    "epsilon_bounds": {
      "psnr_db": [
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        45.0
      ]
    },
    "omega_bounds": {
      "N_elements": [
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        256
      ],
      "PRF_kHz": [
        0.5,
        20
      ],
      "SNR_dB": [
        0.0,
        35.0
      ],
      "aliasing": [
        0.0,
        0.5
      ],
      "angle_correction": [
        0.0,
        0.5
      ],
      "depth_cm": [
        1,
        25
      ],
      "f_MHz": [
        1,
        15
      ],
      "low_velocity_sensitivity": [
        0.0,
        0.3
      ],
      "wall_filter_bleedthrough": [
        0.0,
        0.3
      ]
    }
  },
  "staked_pwm": 0.0,
  "status": "testnet",
  "sub_domain": "Blood-flow velocity via Doppler shift",
  "title": "Doppler Ultrasound (color / spectral)"
}

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