# ⚛  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`

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## 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._