# ⚛  L1 Principle — Computed Tomography (CT) — fan/cone-beam X-ray reconstruction

**ID:** `L1-029` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Medical Imaging — *Multi-angle X-ray tomographic imaging*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** 3D attenuation
> **📡 Carrier:** x_ray · **🌫 Noise:** shot poisson
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41552300

---

## 🧠 1. Introduction

**Computed Tomography (CT) — fan/cone-beam X-ray reconstruction** is a **linear inverse problem** whose unknown lives in **3D attenuation** space, within the **Multi-angle X-ray tomographic imaging** 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; rotates source / detector to acquire different projections; exponential attenuation along the propagation path; 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 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 ~= 18); beam_hardening 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 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 → Beer-Lambert attenuation → Angular integration → **y** (detector).

```
y = ∫dΩ exp(-∫µ dl) 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 |
| `L.beer_lambert` | Exponential attenuation along the propagation path |
| `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 | Multi-angle X-ray tomographic imaging |
| Carrier | x_ray |
| Problem class | linear_inverse |
| Solution space | 3D_attenuation |
| Noise model | shot_poisson |
| Integration axis | angular |
| Difficulty delta | 5 |
| L dag | 3.8 |

## 📡 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 ~= 18); beam_hardening 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 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:** `ct` · **Forward operator:** `ct_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| H | px | 512 |
| W | px | 512 |
| Z | — | 256 |
| Kvp | — | 120 |
| Mas | — | 200 |
| Scatter | — | 0 |
| Pixel mm | mm | 0.5 |
| N projections | — | 720 |
| Beam hardening | — | 0 |
| Patient motion | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| H | px | 256 – 2048 |
| W | px | 256 – 2048 |
| Z | — | 32 – 4096 |
| Kvp | — | 60 – 140 |
| Mas | — | 1 – 1000 |
| Scatter | — | 0.0 – 0.3 |
| Pixel mm | mm | 0.1 – 5.0 |
| Truncation | — | 0.0 – 0.2 |
| N projections | — | 60 – 2880 |
| Beam hardening | — | 0.0 – 0.3 |
| Metal artifact | — | 0.0 – 0.5 |
| Patient motion | — | 0.0 – 0.3 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 32.0

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

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `360` and δ = `5`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x2b6738af2abc49d87290e62682eb7c8eedddfd08d121b696601a4af8c13c5e0c`
- **Chain tx hash:** `0x71f1b68c3ddd1a8995ffd8d557eb9f991cedefc97cade68b8d2261945b189473`
- **Chain block:** `41552300`

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## File Mapping

This bundle consists of: `L1-029.md`, `L1-029.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-029.md` | Source of truth — edit this | Human or LLM |
| `L1-029.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 `L1-029`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._