# ⚛  L1 Principle — Digital Breast Tomosynthesis (DBT)

**ID:** `L1-063` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Medical Imaging — *Limited-angle tomography of the breast*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** 3D attenuation limited angle
> **📡 Carrier:** x_ray · **🌫 Noise:** shot poisson
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41553372

---

## 🧠 1. Introduction

**Digital Breast Tomosynthesis (DBT)** is a **linear inverse problem** whose unknown lives in **3D attenuation limited angle** space, within the **Limited-angle tomography of the breast** 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; S · scan · limited angle operator; L · sart reconstruct 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 limited angle 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); limited_angle_artifacts 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 → S · scan · limited angle → L · sart reconstruct → Angular integration → **y** (detector).

```
y = ∫dΩ `L.sart_reconstruct` `S.scan.limited_angle` I₀(E) x,    measurements ~ Poisson(αy)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.xray_source` | Polyenergetic x-ray emission spectrum |
| `S.scan.limited_angle` | S · scan · limited angle operator |
| `L.sart_reconstruct` | L · sart reconstruct operator |
| `int.angular` | Integration over the solid angle of incidence/emission |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Medical Imaging |
| Sub domain | Limited-angle tomography of the breast |
| Carrier | x_ray |
| Problem class | linear_inverse |
| Solution space | 3D_attenuation_limited_angle |
| Noise model | shot_poisson |
| Integration axis | angular |
| Difficulty delta | 5 |
| L dag | 3.8 |

## 📡 4. Measurement Model

Existence of the recovered 3D attenuation limited angle 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); limited_angle_artifacts 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:** `dbt` · **Forward operator:** `dbt_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| H | px | 3328 |
| W | px | 2560 |
| Z | — | 60 |
| Kvp | — | 28 |
| Mas | — | 60 |
| Scatter | — | 0.15 |
| Pixel um | µm | 85 |
| N projections | — | 15 |
| Patient motion | — | 0 |
| Tilt range deg | deg | 15 |
| Compression variation | — | 0 |
| Limited angle artifacts | — | 0.3 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| H | px | 1664 |
| W | px | 1280 |
| Z | — | 16 – 200 |
| Kvp | — | 22 – 34 |
| Mas | — | 10 – 400 |
| Scatter | — | 0.0 – 0.3 |
| Pixel um | µm | 50 – 200 |
| N projections | — | 9 – 25 |
| Patient motion | — | 0.0 – 0.3 |
| Tilt range deg | deg | 7.5 – 30 |
| Compression variation | — | 0.0 – 0.2 |
| Limited angle artifacts | — | 0.1 – 0.7 |

## 🎯 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 = `360` and δ = `5`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x474b716b33bc75c4888fb038fc166b96487641a2f5f5f5ac9c4e33187eb94927`
- **Chain tx hash:** `0xa0078995185ac02bd42d72717bde04a4112362140797523c8aa33acca2287903`
- **Chain block:** `41553372`

---

## File Mapping

This bundle consists of: `L1-063.md`, `L1-063.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-063.md` | Source of truth — edit this | Human or LLM |
| `L1-063.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 `L1-063`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._