L1 ⚛ L1-159 ⊙ Testnet

US-MRI Fusion (biopsy guidance / cardiac)

Multimodal Fusion · Real-time US co-registered to pre-acquired MRI volume

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⬇ L1-159.md

# ⚛ L1 Principle — US-MRI Fusion (biopsy guidance / cardiac) **ID:** `L1-159` · **Status:** ⊙ Testnet (genesis catalog) > **🌐 Domain:** Multimodal Fusion — *Real-time US co-registered to pre-acquired MRI volume* > **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** 2D plus 3D fused real time > **📡 Carrier:** acoustic · **🌫 Noise:** gaussian > **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41554242 --- ## 🧠 1. Introduction **US-MRI Fusion (biopsy guidance / cardiac)** is a **nonlinear inverse problem** whose unknown lives in **2D plus 3D fused real time** space, within the **Real-time US co-registered to pre-acquired MRI volume** sub-domain of **Multimodal Fusion**. Measurements consist of acoustic pressure waves recorded by transducers via a **us mri fusion** sensing mechanism. The forward operator applies, in order: L · mri volume operator; L · live ultrasound operator; L · real time registration operator; detector accumulates flux over the exposure window. Observations are corrupted by additive Gaussian noise. Existence of the recovered 2D plus 3D fused real time 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 ~= 18); real_time_registration_error dominates the stability cliff; non_rigid_tissue 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 · mri volume → L · live ultrasound → L · real time registration → Temporal integration → **y** (detector). ``` y = ∫_t dt `L.real_time_registration` `L.live_ultrasound` `L.mri_volume` x + n, n ~ 𝒩(0, σ²) ``` **Measurement DAG:** | Primitive | What it does | |---|---| | `L.mri_volume` | L · mri volume operator | | `L.live_ultrasound` | L · live ultrasound operator | | `L.real_time_registration` | L · real time registration operator | | `int.temporal` | Detector accumulates flux over the exposure window | ## 🔬 3. Physics Fingerprint | Property | Value | |---|---| | Domain | Multimodal Fusion | | Sub domain | Real-time US co-registered to pre-acquired MRI volume | | Carrier | acoustic | | Problem class | nonlinear_inverse | | Solution space | 2D_plus_3D_fused_real_time | | Noise model | gaussian | | Integration axis | temporal | | Difficulty delta | 5 | | L dag | 4 | ## 📡 4. Measurement Model Existence of the recovered 2D plus 3D fused real time 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 ~= 18); real_time_registration_error dominates the stability cliff; non_rigid_tissue 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:** `us_mri_fusion` · **Forward operator:** `us_mri_fusion_forward` **Center point:** | Parameter | Unit | Value | |---|---|---| | Fps | — | 10 | | H us | µs | 512 | | W us | µs | 512 | | H mri | — | 256 | | W mri | — | 256 | | Z mri | — | 128 | | Snr db | dB | 22 | | Patient motion | — | 0 | | Non rigid tissue | — | 0 | | Acoustic shadowing | — | 0 | | Real time registration error | — | 0 | **Allowed bounds:** | Parameter | Unit | Range | |---|---|---| | Fps | — | 1 – 60 | | H us | µs | 128 – 2048 | | W us | µs | 128 – 2048 | | Z mri | — | 32 – 512 | | Snr db | dB | 0.0 – 35.0 | | Patient motion | — | 0.0 – 0.5 | | Non rigid tissue | — | 0.0 – 0.5 | | Acoustic shadowing | — | 0.0 – 0.5 | | Real time registration error | — | 0.0 – 5.0 | ## 🎯 6. Tolerance (ε) **Center tolerance:** 20.0 | Metric | Range | |---|---| | Psnr db | 5.0 – 40.0 | ## ⚖ 7. Hardness Function Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `360` and δ = `5`. ## 💾 8. Reference Dataset - **primary** · weight 1.0 · IPFS _(not pinned yet)_ ## 9. On-chain Registration - **Chain hash:** `0x2e58505b64f99b4de4b1f714ccbefce8543fc9795cc4a6fa62122b31a3d98f7f` - **Chain tx hash:** `0x128cfe3e541d1d27fe7e311ff7b8cb5992536765c6706a20d166d2bdc30b1743` - **Chain block:** `41554242` --- ## File Mapping This bundle consists of: `L1-159.md`, `L1-159.json`. | File | Role | How to regenerate | |------|------|-------------------| | `L1-159.md` | Source of truth — edit this | Human or LLM | | `L1-159.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 `L1-159`._ _Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._

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Read the attached Markdown (L1-159.md).
Regenerate L1-159.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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L2-159-001 US-MRI Fusion (biopsy guidance / cardiac) Mismatch-Only Spec

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📋 JSON metadata

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  "chain_block": 41554242,
  "chain_hash": "0x2e58505b64f99b4de4b1f714ccbefce8543fc9795cc4a6fa62122b31a3d98f7f",
  "chain_tx_hash": "0x128cfe3e541d1d27fe7e311ff7b8cb5992536765c6706a20d166d2bdc30b1743",
  "domain": "Multimodal Fusion",
  "hardness_fn": {
    "delta": 5,
    "kappa": 360,
    "metric": "PSNR_dB",
    "type": "epsilon_fn"
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      "license_hash": null,
      "name": "primary",
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    "regime": "Existence of the recovered 2D plus 3D fused real time 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 ~= 18); real_time_registration_error dominates the stability cliff; non_rigid_tissue 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": {
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    "carrier": "acoustic",
    "difficulty_delta": 5,
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    "integration_axis": "temporal",
    "noise_model": "gaussian",
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      "L.live_ultrasound",
      "L.real_time_registration",
      "int.temporal"
    ],
    "problem_class": "nonlinear_inverse",
    "sensing_mechanism": "us_mri_fusion",
    "solution_space": "2D_plus_3D_fused_real_time",
    "sub_domain": "Real-time US co-registered to pre-acquired MRI volume",
    "title": "US-MRI Fusion (biopsy guidance / cardiac)"
  },
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      "W_us",
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      "W_mri",
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      "non_rigid_tissue",
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      "acoustic_shadowing",
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      "non_rigid_tissue",
      "patient_motion",
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    ],
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    ],
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      "input_format": "measurement_only",
      "omega": {
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        "H_us": 512,
        "SNR_dB": 22,
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        "W_us": 512,
        "Z_mri": 128,
        "acoustic_shadowing": 0.0,
        "fps": 10,
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        "patient_motion": 0.0,
        "real_time_registration_error": 0.0
      },
      "problem_class": "us_mri_fusion"
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    "epsilon_bounds": {
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      ]
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        2048
      ],
      "SNR_dB": [
        0.0,
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      ],
      "W_us": [
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        2048
      ],
      "Z_mri": [
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        512
      ],
      "acoustic_shadowing": [
        0.0,
        0.5
      ],
      "fps": [
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      ],
      "non_rigid_tissue": [
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        0.5
      ],
      "patient_motion": [
        0.0,
        0.5
      ],
      "real_time_registration_error": [
        0.0,
        5.0
      ]
    }
  },
  "staked_pwm": 0.0,
  "status": "testnet",
  "sub_domain": "Real-time US co-registered to pre-acquired MRI volume",
  "title": "US-MRI Fusion (biopsy guidance / cardiac)"
}

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