# ⚛  L1 Principle — Dual-Energy X-ray Absorptiometry (DEXA)

**ID:** `L1-037` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Medical Imaging — *Bone mineral density via two-energy decomposition*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** 2D bone mineral density
> **📡 Carrier:** x_ray · **🌫 Noise:** shot poisson
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41552301

---

## 🧠 1. Introduction

**Dual-Energy X-ray Absorptiometry (DEXA)** is a **linear inverse problem** whose unknown lives in **2D bone mineral density** space, within the **Bone mineral density via two-energy decomposition** sub-domain of **Medical Imaging**.

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

The forward operator applies, in order: L · xray source dual operator; exponential attenuation along the propagation path; L · two material decompose operator; pixel-level spatial averaging on the detector.

Observations are corrupted by Poisson shot noise from quantum-limited detection. Existence of the recovered 2D bone mineral density 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); beam_hardening dominates the stability cliff; patient_alignment 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** → L · xray source dual → Beer-Lambert attenuation → L · two material decompose → Spatial integration → **y** (detector).

```
y = ∫_A dA `L.two_material_decompose` exp(-∫µ dl) `L.xray_source_dual` x,    measurements ~ Poisson(αy)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.xray_source_dual` | L · xray source dual operator |
| `L.beer_lambert` | Exponential attenuation along the propagation path |
| `L.two_material_decompose` | L · two material decompose operator |
| `int.spatial` | Pixel-level spatial averaging on the detector |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Medical Imaging |
| Sub domain | Bone mineral density via two-energy decomposition |
| Carrier | x_ray |
| Problem class | linear_inverse |
| Solution space | 2D_bone_mineral_density |
| Noise model | shot_poisson |
| Integration axis | spatial |
| Difficulty delta | 3 |
| L dag | 3.2 |

## 📡 4. Measurement Model

Existence of the recovered 2D bone mineral density 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); beam_hardening dominates the stability cliff; patient_alignment 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:** `dexa` · **Forward operator:** `dexa_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| H | px | 1024 |
| W | px | 512 |
| Mas | — | 10 |
| Kvp hi | — | 140 |
| Kvp lo | — | 70 |
| Pixel mm | mm | 0.5 |
| Beam hardening | — | 0.05 |
| Patient alignment | — | 0 |
| Fat fraction variation | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| H | px | 512 |
| W | px | 256 |
| Mas | — | 1 – 100 |
| Kvp hi | — | 100 – 150 |
| Kvp lo | — | 60 – 80 |
| Pixel mm | mm | 0.1 – 2.0 |
| Beam hardening | — | 0.0 – 0.2 |
| Calibration drift | — | 0.0 – 0.05 |
| Patient alignment | — | 0.0 – 2.0 |
| Fat fraction variation | — | 0.0 – 0.3 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 30.0

| 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:** `0x1b180345715c22e98dce5616f472c3135d9d037edf910135ff5401bfb049afa0`
- **Chain tx hash:** `0xa2c4275ad933053a705d24233547692e6480513eda72372e3e28e98e515152cb`
- **Chain block:** `41552301`

---

## File Mapping

This bundle consists of: `L1-037.md`, `L1-037.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-037.md` | Source of truth — edit this | Human or LLM |
| `L1-037.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 `L1-037`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._