# ⚛  L1 Principle — Talbot-Lau Grating Interferometry (X-ray phase-contrast imaging)

**ID:** `L1-074` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Coherent Imaging — *Grating-based phase and dark-field imaging*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** triplet 2D fields
> **📡 Carrier:** x_ray · **🌫 Noise:** shot poisson
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41554170

---

## 🧠 1. Introduction

**Talbot-Lau Grating Interferometry (X-ray phase-contrast imaging)** is a **linear inverse problem** whose unknown lives in **triplet 2D fields** space, within the **Grating-based phase and dark-field imaging** sub-domain of **Coherent Imaging**.

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

The forward operator applies, in order: K · transmit · grating operator; F · propagation · fresnel operator; L · modulation · analyzer operator; pixel-level spatial averaging on the detector.

Observations are corrupted by Poisson shot noise from quantum-limited detection. Well-conditioned linear Fourier extraction of three contrast channels from K phase-stepped frames (K >= 3 required; K = 8 typical for noise averaging). Stability dominated by grating alignment and vibration; dark-field channel degrades when visibility V < 0.15.

## ⚙ 2. Forward Model

Physical chain: **x** → K · transmit · grating → F · propagation · fresnel → L · modulation · analyzer → Spatial integration → **y** (detector).

```
y = ∫_A dA `L.modulation.analyzer` `F.propagation.fresnel` `K.transmit.grating` x,    measurements ~ Poisson(αy)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `K.transmit.grating` | K · transmit · grating operator |
| `F.propagation.fresnel` | F · propagation · fresnel operator |
| `L.modulation.analyzer` | L · modulation · analyzer operator |
| `int.spatial` | Pixel-level spatial averaging on the detector |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Coherent Imaging |
| Sub domain | Grating-based phase and dark-field imaging |
| Carrier | x_ray |
| Problem class | linear_inverse |
| Solution space | triplet_2D_fields |
| Noise model | shot_poisson |
| Integration axis | temporal |
| Difficulty delta | 3 |
| L dag | 3.4 |

## 📡 4. Measurement Model

Well-conditioned linear Fourier extraction of three contrast channels from K phase-stepped frames (K >= 3 required; K = 8 typical for noise averaging). Stability dominated by grating alignment and vibration; dark-field channel degrades when visibility V < 0.15.

| Metric | Value |
|---|---|
| Metric | phase_PSNR_dB |
| Secondary | darkfield_SNR |

## 📏 5. Operating Range (Ω)

**Center problem class:** `grating_phase_contrast_imaging` · **Forward operator:** `talbot_lau_three_grating`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| H | px | 1024 |
| W | px | 1024 |
| K steps | — | 8 |
| V center | — | 0.3 |
| E mean kev | — | 40 |
| Alignment tilt | — | 0 |
| Grating period um | µm | 4.8 |
| Grating period drift | — | 0 |
| Photon count per pixel | — | 10000 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| H | px | 256 – 4096 |
| W | px | 256 – 4096 |
| K steps | — | 3 – 16 |
| V center | — | 0.05 – 0.6 |
| E mean kev | — | 10.0 – 120.0 |
| Alignment tilt | — | 0.0 – 0.02 |
| Grating period um | µm | 1.0 – 10.0 |
| Phase step jitter | — | 0.0 – 0.1 |
| Grating period drift | — | 0.0 – 0.05 |
| Source coherence loss | — | 0.0 – 0.5 |
| Photon count per pixel | — | 100 – 1000000 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 30.0 dB phase PSNR

| Metric | Range |
|---|---|
| Psnr db | 18.0 – 42.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **phase_PSNR_dB**, with κ = `400` and δ = `3`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x0ba11dfeb8a6b1207f64364b2c77952da6b41f189d4405907b04d3c6973de6f1`
- **Chain tx hash:** `0xacbb7730764cd90afc9010534172c04ecfbee959079c125f5f74cd721093ab9a`
- **Chain block:** `41554170`

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## File Mapping

This bundle consists of: `L1-074.md`, `L1-074.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-074.md` | Source of truth — edit this | Human or LLM |
| `L1-074.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 `L1-074`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._