L1 ⚛ L1-054 ⊙ Testnet

Photon-Counting CT (PCCT) — spectral detector CT

Medical Imaging · Energy-resolved direct-conversion CT

⬇ Download bundle 🚀 Submit to mainnet — earn 100 PWM + 5% royalty

Markdown — human-readable source of truth

⚙ auto-generated

⬇ L1-054.md

# ⚛ L1 Principle — Photon-Counting CT (PCCT) — spectral detector CT **ID:** `L1-054` · **Status:** ⊙ Testnet (genesis catalog) > **🌐 Domain:** Medical Imaging — *Energy-resolved direct-conversion CT* > **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** 3D material decomposition > **📡 Carrier:** x_ray · **🌫 Noise:** shot poisson > **⚖ Difficulty (δ):** 10 · **⛓ Block:** 41553359 --- ## 🧠 1. Introduction **Photon-Counting CT (PCCT) — spectral detector CT** is a **nonlinear inverse problem** whose unknown lives in **3D material decomposition** space, within the **Energy-resolved direct-conversion CT** sub-domain of **Medical Imaging**. Measurements consist of X-ray photons transmitted through (or scattered by) the sample via a **photon counting ct** sensing mechanism. The forward operator applies, in order: polyenergetic X-ray emission spectrum; rotates source / detector to acquire different projections; D · pcd detector operator; L · spectral decompose 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 material decomposition 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 ~= 25); charge_sharing dominates the stability cliff; pulse_pileup 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** → X-ray source → Angular scan → D · pcd detector → L · spectral decompose → Angular integration → **y** (detector). ``` y = ∫dΩ `L.spectral_decompose` `D.pcd_detector` R(θ) I₀(E) x, measurements ~ Poisson(αy) ``` **Measurement DAG:** | Primitive | What it does | |---|---| | `L.xray_source` | Polyenergetic x-ray emission spectrum | | `S.scan.angular` | Rotates source / detector to acquire different projections | | `D.pcd_detector` | D · pcd detector operator | | `L.spectral_decompose` | L · spectral decompose 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 | Energy-resolved direct-conversion CT | | Carrier | x_ray | | Problem class | nonlinear_inverse | | Solution space | 3D_material_decomposition | | Noise model | shot_poisson | | Integration axis | angular | | Difficulty delta | 10 | | L dag | 4.5 | ## 📡 4. Measurement Model Existence of the recovered 3D material decomposition 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 ~= 25); charge_sharing dominates the stability cliff; pulse_pileup 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:** `pcct` · **Forward operator:** `pcct_forward` **Center point:** | Parameter | Unit | Value | |---|---|---| | H | px | 512 | | W | px | 512 | | Z | — | 256 | | Mas | — | 100 | | Pixel mm | mm | 0.25 | | Pulse pileup | — | 0 | | N energy bins | — | 5 | | N projections | — | 720 | | Beam hardening | — | 0 | | Charge sharing | — | 0.05 | | K edge calibration | — | 0 | **Allowed bounds:** | Parameter | Unit | Range | |---|---|---| | H | px | 256 | | W | px | 256 | | Z | — | 64 – 2048 | | Mas | — | 1 – 500 | | Pixel mm | mm | 0.05 – 1.0 | | Pulse pileup | — | 0.0 – 0.5 | | N energy bins | — | 2 – 16 | | N projections | — | 180 – 2880 | | Beam hardening | — | 0.0 – 0.3 | | Charge sharing | — | 0.0 – 0.3 | | K edge calibration | — | 0.0 – 0.1 | ## 🎯 6. Tolerance (ε) **Center tolerance:** 28.0 | Metric | Range | |---|---| | Psnr db | 5.0 – 45.0 | ## ⚖ 7. Hardness Function Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `500` and δ = `10`. ## 💾 8. Reference Dataset - **primary** · weight 1.0 · IPFS _(not pinned yet)_ ## 9. On-chain Registration - **Chain hash:** `0xf3a17a4cef566a2d434a04aa0a839ae9bcb8d1f11af8cd9ef236ef70d4e7d136` - **Chain tx hash:** `0x2af601a4544eeb51a482ce435bb1dca49cdf0af5b543684bd5a39809d06724ec` - **Chain block:** `41553359` --- ## File Mapping This bundle consists of: `L1-054.md`, `L1-054.json`. | File | Role | How to regenerate | |------|------|-------------------| | `L1-054.md` | Source of truth — edit this | Human or LLM | | `L1-054.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 `L1-054`._ _Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._

📝 Polish this artifact in your LLM

Download the files on the right.
Paste the Markdown into ChatGPT / Claude / Gemini and edit freely.
Send your edited Markdown back with the prompt below.
Save the regenerated JSON, then submit via /submit.

Prompt to give your LLM

Read the attached Markdown (L1-054.md).
Regenerate L1-054.json so every field matches the Markdown.
Preserve the schema documented in the File Mapping footer at the bottom of the MD.
Output each file in its own fenced code block tagged with the filename.

⌥ Hierarchy

Upstream lineage and downstream children

↓ L2 Spec 1 child ▾

L2-054-001 Photon-Counting CT (PCCT) — spectral detector CT Mismatch-Only Spec

⬇ Download bundle

2 files · click to download

📋 JSON metadata

{
  "artifact_id": "L1-054",
  "chain_block": 41553359,
  "chain_hash": "0xf3a17a4cef566a2d434a04aa0a839ae9bcb8d1f11af8cd9ef236ef70d4e7d136",
  "chain_tx_hash": "0x2af601a4544eeb51a482ce435bb1dca49cdf0af5b543684bd5a39809d06724ec",
  "domain": "Medical Imaging",
  "hardness_fn": {
    "delta": 10,
    "kappa": 500,
    "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 3D material decomposition 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 ~= 25); charge_sharing dominates the stability cliff; pulse_pileup 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.",
    "secondary": "SSIM"
  },
  "physics_fingerprint": {
    "L_DAG": 4.5,
    "carrier": "x_ray",
    "difficulty_delta": 10,
    "domain": "Medical Imaging",
    "integration_axis": "angular",
    "noise_model": "shot_poisson",
    "primitives": [
      "L.xray_source",
      "S.scan.angular",
      "D.pcd_detector",
      "L.spectral_decompose",
      "L.backproject",
      "int.angular"
    ],
    "problem_class": "nonlinear_inverse",
    "sensing_mechanism": "photon_counting_ct",
    "solution_space": "3D_material_decomposition",
    "sub_domain": "Energy-resolved direct-conversion CT",
    "title": "Photon-Counting CT (PCCT) \u2014 spectral detector CT"
  },
  "size_tiers": {
    "allowed_forward_operators": [
      "pcct_forward"
    ],
    "allowed_omega_dimensions": [
      "H",
      "W",
      "Z",
      "N_projections",
      "N_energy_bins",
      "mAs",
      "pixel_mm",
      "charge_sharing",
      "pulse_pileup",
      "beam_hardening",
      "k_edge_calibration",
      "charge_sharing",
      "pulse_pileup",
      "beam_hardening",
      "k_edge_calibration"
    ],
    "allowed_problem_classes": [
      "pcct"
    ],
    "center_spec": {
      "epsilon_fn_center": "28.0",
      "forward_operator": "pcct_forward",
      "input_format": "measurement_only",
      "omega": {
        "H": 512,
        "N_energy_bins": 5,
        "N_projections": 720,
        "W": 512,
        "Z": 256,
        "beam_hardening": 0.0,
        "charge_sharing": 0.05,
        "k_edge_calibration": 0.0,
        "mAs": 100,
        "pixel_mm": 0.25,
        "pulse_pileup": 0.0
      },
      "problem_class": "pcct"
    },
    "epsilon_bounds": {
      "psnr_db": [
        5.0,
        45.0
      ]
    },
    "omega_bounds": {
      "H": 256,
      "N_energy_bins": [
        2,
        16
      ],
      "N_projections": [
        180,
        2880
      ],
      "W": 256,
      "Z": [
        64,
        2048
      ],
      "beam_hardening": [
        0.0,
        0.3
      ],
      "charge_sharing": [
        0.0,
        0.3
      ],
      "k_edge_calibration": [
        0.0,
        0.1
      ],
      "mAs": [
        1,
        500
      ],
      "pixel_mm": [
        0.05,
        1.0
      ],
      "pulse_pileup": [
        0.0,
        0.5
      ]
    }
  },
  "staked_pwm": 0.0,
  "status": "testnet",
  "sub_domain": "Energy-resolved direct-conversion CT",
  "title": "Photon-Counting CT (PCCT) \u2014 spectral detector CT"
}

🎁

Earn PWM by promoting this to mainnet

• 100 PWM one-time grant on mainnet promotion
• 5% upstream royalty on every downstream L4 mining event, forever
• Your wallet is enrolled the moment the founder accepts

🚀 Submit to mainnet →

How: Download the bundle above → polish the Markdown in ChatGPT / Claude / Gemini → submit your version → founder reviews → mainnet.