L1 ⚛ L1-033 ⊙ Testnet

Positron Emission Tomography (PET)

Medical Imaging · Coincidence-based functional PET

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⬇ L1-033.md

# ⚛ L1 Principle — Positron Emission Tomography (PET) **ID:** `L1-033` · **Status:** ⊙ Testnet (genesis catalog) > **🌐 Domain:** Medical Imaging — *Coincidence-based functional PET* > **🎯 Problem class:** linear inverse · **🧮 Solution space:** 3D activity concentration > **📡 Carrier:** photon · **🌫 Noise:** shot poisson > **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41552300 --- ## 🧠 1. Introduction **Positron Emission Tomography (PET)** is a **linear inverse problem** whose unknown lives in **3D activity concentration** space, within the **Coincidence-based functional PET** sub-domain of **Medical Imaging**. Measurements consist of photons collected by an optical detector via a **pet coincidence** sensing mechanism. The forward operator applies, in order: L · annihilation 511keV operator; D · coincidence detector operator; spreads measurements back along source rays (adjoint operator); integration over the solid angle of incidence/emission. Observations are corrupted by Poisson shot noise from quantum-limited detection. Existence of the recovered 3D activity concentration 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 ~= 14); random_coincidences dominates the stability cliff; scatter 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** → L · annihilation 511keV → D · coincidence detector → Angular integration → **y** (detector). ``` y = ∫dΩ `D.coincidence_detector` `L.annihilation_511keV` x, measurements ~ Poisson(αy) ``` **Measurement DAG:** | Primitive | What it does | |---|---| | `L.annihilation_511keV` | L · annihilation 511kev operator | | `D.coincidence_detector` | D · coincidence detector operator | | `int.angular` | Integration over the solid angle of incidence/emission | **🛠 Solver components** _(used inside the solver, not in the forward equation)_: | Primitive | What it does | |---|---| | `L.backproject` | Spreads measurements back along source rays (adjoint operator) | ## 🔬 3. Physics Fingerprint | Property | Value | |---|---| | Domain | Medical Imaging | | Sub domain | Coincidence-based functional PET | | Carrier | photon | | Problem class | linear_inverse | | Solution space | 3D_activity_concentration | | Noise model | shot_poisson | | Integration axis | angular | | Difficulty delta | 5 | | L dag | 3.6 | ## 📡 4. Measurement Model Existence of the recovered 3D activity concentration 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 ~= 14); random_coincidences dominates the stability cliff; scatter 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:** `pet` · **Forward operator:** `pet_forward` **Center point:** | Parameter | Unit | Value | |---|---|---| | H | px | 256 | | W | px | 256 | | Z | — | 64 | | N lors | — | 100000000 | | Snr db | dB | 20 | | Scatter | — | 0.3 | | Counts per lor | — | 10 | | Patient motion | — | 0 | | Random coincidences | — | 0.1 | | Attenuation correction error | — | 0 | **Allowed bounds:** | Parameter | Unit | Range | |---|---|---| | H | px | 128 | | W | px | 128 | | Z | — | 16 – 512 | | N lors | — | 1000000 – 10000000000 | | Snr db | dB | 0.0 – 35.0 | | Scatter | — | 0.0 – 0.5 | | Counts per lor | — | 0.1 – 1000 | | Patient motion | — | 0.0 – 0.3 | | Tof timing jitter | — | 0.0 – 0.5 | | Random coincidences | — | 0.0 – 0.5 | | Attenuation correction error | — | 0.0 – 0.3 | ## 🎯 6. Tolerance (ε) **Center tolerance:** 25.0 | Metric | Range | |---|---| | Psnr db | 5.0 – 45.0 | ## ⚖ 7. Hardness Function Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `280` and δ = `5`. ## 💾 8. Reference Dataset - **primary** · weight 1.0 · IPFS _(not pinned yet)_ ## 9. On-chain Registration - **Chain hash:** `0xb0fee7279f4fb8f72dc0613a217ce040eac84b6bae89248905ccad2b5e60d5bb` - **Chain tx hash:** `0x7440ebfb62f7f3d37f47cf9adb30aa59e8710656d8666a930787a10907ff576c` - **Chain block:** `41552300` --- ## File Mapping This bundle consists of: `L1-033.md`, `L1-033.json`. | File | Role | How to regenerate | |------|------|-------------------| | `L1-033.md` | Source of truth — edit this | Human or LLM | | `L1-033.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 `L1-033`._ _Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._

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Read the attached Markdown (L1-033.md).
Regenerate L1-033.json so every field matches the Markdown.
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L2-033-001 Positron Emission Tomography (PET) Mismatch-Only Spec

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📋 JSON metadata

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  "chain_tx_hash": "0x7440ebfb62f7f3d37f47cf9adb30aa59e8710656d8666a930787a10907ff576c",
  "domain": "Medical Imaging",
  "hardness_fn": {
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    "kappa": 280,
    "metric": "PSNR_dB",
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    "regime": "Existence of the recovered 3D activity concentration 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 ~= 14); random_coincidences dominates the stability cliff; scatter 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"
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  "physics_fingerprint": {
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    "carrier": "photon",
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    "integration_axis": "angular",
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    "sensing_mechanism": "pet_coincidence",
    "solution_space": "3D_activity_concentration",
    "sub_domain": "Coincidence-based functional PET",
    "title": "Positron Emission Tomography (PET)"
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        "scatter": 0.3
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      "TOF_timing_jitter": [
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      "W": 128,
      "Z": [
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      ],
      "attenuation_correction_error": [
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      "counts_per_LOR": [
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      "patient_motion": [
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      ],
      "random_coincidences": [
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      ],
      "scatter": [
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      ]
    }
  },
  "staked_pwm": 0.0,
  "status": "testnet",
  "sub_domain": "Coincidence-based functional PET",
  "title": "Positron Emission Tomography (PET)"
}

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