# ⚛  L1 Principle — Cone-Beam CT (CBCT) — dental / interventional

**ID:** `L1-038` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Medical Imaging — *Flat-panel cone-beam CT reconstruction*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** 3D attenuation
> **📡 Carrier:** x_ray · **🌫 Noise:** shot poisson
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41553358

---

## 🧠 1. Introduction

**Cone-Beam CT (CBCT) — dental / interventional** is a **linear inverse problem** whose unknown lives in **3D attenuation** space, within the **Flat-panel cone-beam CT reconstruction** sub-domain of **Medical Imaging**.

Measurements consist of X-ray photons transmitted through (or scattered by) the sample via a **xray ct** sensing mechanism.

The forward operator applies, in order: polyenergetic X-ray emission spectrum; L · cone beam geometry operator; rotates source / detector to acquire different projections; L · FDK backproject 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 attenuation 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 ~= 20); scatter dominates the stability cliff; truncation 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 → L · cone beam geometry → Angular scan → L · FDK backproject → Angular integration → **y** (detector).

```
y = ∫dΩ `L.FDK_backproject` R(θ) `L.cone_beam_geometry` I₀(E) x,    measurements ~ Poisson(αy)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.xray_source` | Polyenergetic x-ray emission spectrum |
| `L.cone_beam_geometry` | L · cone beam geometry operator |
| `S.scan.angular` | Rotates source / detector to acquire different projections |
| `L.FDK_backproject` | L · fdk backproject operator |
| `int.angular` | Integration over the solid angle of incidence/emission |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Medical Imaging |
| Sub domain | Flat-panel cone-beam CT reconstruction |
| Carrier | x_ray |
| Problem class | linear_inverse |
| Solution space | 3D_attenuation |
| Noise model | shot_poisson |
| Integration axis | angular |
| Difficulty delta | 5 |
| L dag | 4 |

## 📡 4. Measurement Model

Existence of the recovered 3D attenuation 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 ~= 20); scatter dominates the stability cliff; truncation 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:** `cbct` · **Forward operator:** `cbct_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| H | px | 512 |
| W | px | 512 |
| Z | — | 512 |
| Kvp | — | 90 |
| Mas | — | 20 |
| Scatter | — | 0.2 |
| Pixel mm | mm | 0.3 |
| Truncation | — | 0 |
| N projections | — | 400 |
| Patient motion | — | 0 |
| Geometric misalignment | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| H | px | 256 |
| W | px | 256 |
| Z | — | 64 – 2048 |
| Kvp | — | 60 – 120 |
| Mas | — | 1 – 200 |
| Scatter | — | 0.0 – 0.5 |
| Pixel mm | mm | 0.1 – 2.0 |
| Truncation | — | 0.0 – 0.3 |
| N projections | — | 100 – 2000 |
| Beam hardening | — | 0.0 – 0.3 |
| Patient motion | — | 0.0 – 0.3 |
| Geometric misalignment | — | 0.0 – 2.0 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 27.0

| Metric | Range |
|---|---|
| Psnr db | 5.0 – 45.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `400` and δ = `5`.

## 💾 8. Reference Dataset

- **primary** · weight 1.0 · IPFS _(not pinned yet)_

## 9. On-chain Registration

- **Chain hash:** `0x0167c8e97c6664b8519368f278d9d1fff3ace452d1d642090aec60c2f7e14b27`
- **Chain tx hash:** `0x6026b2517c3101582d089bad659fd833ef65a08756ca8a85cf59e7e145352218`
- **Chain block:** `41553358`

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## File Mapping

This bundle consists of: `L1-038.md`, `L1-038.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-038.md` | Source of truth — edit this | Human or LLM |
| `L1-038.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 `L1-038`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._