# ⚛  L1 Principle — X-ray Angiography (DSA)

**ID:** `L1-039` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Medical Imaging — *Subtraction-based vascular imaging*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** 2D vessel map
> **📡 Carrier:** x_ray · **🌫 Noise:** shot poisson
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41553371

---

## 🧠 1. Introduction

**X-ray Angiography (DSA)** is a **linear inverse problem** whose unknown lives in **2D vessel map** space, within the **Subtraction-based vascular imaging** sub-domain of **Medical Imaging**.

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

The forward operator applies, in order: polyenergetic X-ray emission spectrum; L · contrast bolus operator; L · digital subtraction operator; detector accumulates flux over the exposure window.

Observations are corrupted by Poisson shot noise from quantum-limited detection. Existence of the recovered 2D vessel map 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 ~= 10); contrast_timing dominates the stability cliff; patient_motion and the remaining mismatch parameters contribute higher-order bias terms. Photon-shot-noise-limited (poisson counting) sets the irreducible data-fidelity floor, while mild Tikhonov or analytic inversion is sufficient at the nominal Omega point.

## ⚙ 2. Forward Model

Physical chain: **x** → X-ray source → L · contrast bolus → L · digital subtraction → Temporal integration → **y** (detector).

```
y = ∫_t dt `L.digital_subtraction` `L.contrast_bolus` I₀(E) x,    measurements ~ Poisson(αy)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.xray_source` | Polyenergetic x-ray emission spectrum |
| `L.contrast_bolus` | L · contrast bolus operator |
| `L.digital_subtraction` | L · digital subtraction operator |
| `int.temporal` | Detector accumulates flux over the exposure window |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Medical Imaging |
| Sub domain | Subtraction-based vascular imaging |
| Carrier | x_ray |
| Problem class | linear_inverse |
| Solution space | 2D_vessel_map |
| Noise model | shot_poisson |
| Integration axis | temporal |
| Difficulty delta | 3 |
| L dag | 3 |

## 📡 4. Measurement Model

Existence of the recovered 2D vessel map 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 ~= 10); contrast_timing dominates the stability cliff; patient_motion and the remaining mismatch parameters contribute higher-order bias terms. Photon-shot-noise-limited (poisson counting) sets the irreducible data-fidelity floor, while mild Tikhonov or analytic inversion is sufficient at the nominal Omega point.

| Metric | Value |
|---|---|
| Metric | PSNR_dB |
| Secondary | SSIM |

## 📏 5. Operating Range (Ω)

**Center problem class:** `xray_angio` · **Forward operator:** `xray_angio_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| H | px | 1024 |
| W | px | 1024 |
| Fps | — | 15 |
| Kvp | — | 80 |
| Pixel um | µm | 150 |
| Patient motion | — | 0 |
| Contrast timing | — | 0 |
| Misregistration | — | 0 |
| Dose per frame ugy | — | 5 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| H | px | 512 |
| W | px | 512 |
| Fps | — | 1 – 60 |
| Kvp | — | 60 – 120 |
| Scatter | — | 0.0 – 0.3 |
| Pixel um | µm | 50 – 300 |
| Patient motion | — | 0.0 – 0.3 |
| Contrast timing | — | 0.0 – 0.5 |
| Misregistration | — | 0.0 – 5.0 |
| Dose per frame ugy | — | 0.5 – 50 |

## 🎯 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 = `200` and δ = `3`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x08c99ed291c11a03c8c90d01c7e235e6b6fb39c334818028fa527bfef37a1efa`
- **Chain tx hash:** `0xf091da089367fb0c8633564faf2d28ff0f11f09ffa956bba6e011430b671f1d4`
- **Chain block:** `41553371`

---

## File Mapping

This bundle consists of: `L1-039.md`, `L1-039.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-039.md` | Source of truth — edit this | Human or LLM |
| `L1-039.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 `L1-039`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._