L1 ⚛ L1-034 ⊙ Testnet

Single-Photon Emission Computed Tomography (SPECT)

Medical Imaging · Gamma-camera rotating tomography

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⬇ L1-034.md

# ⚛ L1 Principle — Single-Photon Emission Computed Tomography (SPECT) **ID:** `L1-034` · **Status:** ⊙ Testnet (genesis catalog) > **🌐 Domain:** Medical Imaging — *Gamma-camera rotating tomography* > **🎯 Problem class:** linear inverse · **🧮 Solution space:** 3D activity concentration > **📡 Carrier:** photon · **🌫 Noise:** shot poisson > **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41553358 --- ## 🧠 1. Introduction **Single-Photon Emission Computed Tomography (SPECT)** is a **linear inverse problem** whose unknown lives in **3D activity concentration** space, within the **Gamma-camera rotating tomography** sub-domain of **Medical Imaging**. Measurements consist of photons collected by an optical detector via a **spect gamma camera** sensing mechanism. The forward operator applies, in order: L · gamma emission operator; L · collimator operator; rotates source / detector to acquire different projections; 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 ~= 16); collimator_blur dominates the stability cliff; attenuation_correction_error and the remaining mismatch parameters contribute higher-order bias terms. Photon-shot-noise-limited (poisson counting) 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 · gamma emission → L · collimator → Angular scan → Angular integration → **y** (detector). ``` y = ∫dΩ R(θ) `L.collimator` `L.gamma_emission` x, measurements ~ Poisson(αy) ``` **Measurement DAG:** | Primitive | What it does | |---|---| | `L.gamma_emission` | L · gamma emission operator | | `L.collimator` | L · collimator operator | | `S.scan.angular` | Rotates source / detector to acquire different projections | | `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 | Gamma-camera rotating tomography | | Carrier | photon | | Problem class | linear_inverse | | Solution space | 3D_activity_concentration | | Noise model | shot_poisson | | Integration axis | angular | | Difficulty delta | 5 | | L dag | 3.5 | ## 📡 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 ~= 16); collimator_blur dominates the stability cliff; attenuation_correction_error and the remaining mismatch parameters contribute higher-order bias terms. Photon-shot-noise-limited (poisson counting) 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:** `spect` · **Forward operator:** `spect_forward` **Center point:** | Parameter | Unit | Value | |---|---|---| | H | px | 128 | | W | px | 128 | | Z | — | 64 | | Snr db | dB | 18 | | Scatter | — | 0.2 | | N angles | — | 64 | | Patient motion | — | 0 | | Collimator blur | — | 0 | | Counts per pixel | — | 100 | | Attenuation correction error | — | 0 | **Allowed bounds:** | Parameter | Unit | Range | |---|---|---| | H | px | 64 | | W | px | 64 | | Z | — | 16 – 512 | | Snr db | dB | 0.0 – 35.0 | | Scatter | — | 0.0 – 0.5 | | N angles | — | 8 – 256 | | Patient motion | — | 0.0 – 0.3 | | Collimator blur | — | 0.0 – 0.3 | | Counts per pixel | — | 1 – 10000 | | Attenuation correction error | — | 0.0 – 0.3 | ## 🎯 6. Tolerance (ε) **Center tolerance:** 22.0 | Metric | Range | |---|---| | Psnr db | 5.0 – 45.0 | ## ⚖ 7. Hardness Function Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `320` and δ = `5`. ## 💾 8. Reference Dataset - **primary** · weight 1.0 · IPFS _(not pinned yet)_ ## 9. On-chain Registration - **Chain hash:** `0xd8fe453230341be5dad2cc48a43e99b867d718d9af5d1016413fcff7adaf0c08` - **Chain tx hash:** `0xabe2942ee94669f848d13adce3f4cb4f36d37d691190e4c1e0bef96bf1648212` - **Chain block:** `41553358` --- ## File Mapping This bundle consists of: `L1-034.md`, `L1-034.json`. | File | Role | How to regenerate | |------|------|-------------------| | `L1-034.md` | Source of truth — edit this | Human or LLM | | `L1-034.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 `L1-034`._ _Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._

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Regenerate L1-034.json so every field matches the Markdown.
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L2-034-001 Single-Photon Emission Computed Tomography (SPECT) Mismatch-Only Spec

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📋 JSON metadata

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  "domain": "Medical Imaging",
  "hardness_fn": {
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    "kappa": 320,
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    "sub_domain": "Gamma-camera rotating tomography",
    "title": "Single-Photon Emission Computed Tomography (SPECT)"
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      "counts_per_pixel": [
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  "staked_pwm": 0.0,
  "status": "testnet",
  "sub_domain": "Gamma-camera rotating tomography",
  "title": "Single-Photon Emission Computed Tomography (SPECT)"
}

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