# ⚛  L1 Principle — Microwave Imaging — permittivity imaging for biomed/NDE

**ID:** `L1-255` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Electromagnetics — *Microwave tomography (biomedical/NDT)*
> **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** 3D permittivity conductivity
> **📡 Carrier:** em · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41554001

---

## 🧠 1. Introduction

**Microwave Imaging — permittivity imaging for biomed/NDE** is a **nonlinear inverse problem** whose unknown lives in **3D permittivity conductivity** space, within the **Microwave tomography (biomedical/NDT)** sub-domain of **Electromagnetics**.

Measurements consist of electromagnetic field measurements via a **antenna array** sensing mechanism.

The forward operator applies, in order: L · incident field operator; L · scatter equation operator; L · born contrast operator; integration over a band of frequencies.

Observations are corrupted by additive Gaussian noise. Ill-posed, non-linear; limited data coverage typical.

## ⚙ 2. Forward Model

Physical chain: **x** → L · incident field → L · scatter equation → L · born contrast → Frequency integration → **y** (detector).

```
y = ∫dω `L.born_contrast` `L.scatter_equation` `L.incident_field` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.incident_field` | L · incident field operator |
| `L.scatter_equation` | L · scatter equation operator |
| `L.born_contrast` | L · born contrast operator |
| `int.frequency` | Integration over a band of frequencies |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Electromagnetics |
| Sub domain | Microwave tomography (biomedical/NDT) |
| Carrier | em |
| Problem class | nonlinear_inverse |
| Solution space | 3D_permittivity_conductivity |
| Noise model | gaussian |
| Integration axis | frequency |
| Difficulty delta | 5 |
| L dag | 3.5 |

## 📡 4. Measurement Model

Ill-posed, non-linear; limited data coverage typical.

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

## 📏 5. Operating Range (Ω)

**Center problem class:** `microwave_tomo` · **Forward operator:** `microwave_tomo_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| H | px | 128 |
| W | px | 128 |
| Z | — | 32 |
| Snr db | dB | 25 |
| N freqs | — | 16 |
| N antennas | — | 16 |
| Antenna pos error | — | 0 |
| Skin effect error | — | 0 |
| Coupling medium error | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| H | px | 32 – 1024 |
| W | px | 32 – 1024 |
| Z | — | 8 – 512 |
| Snr db | dB | -5 – 40 |
| N freqs | — | 1 – 256 |
| N antennas | — | 4 – 256 |
| Antenna pos error | — | 0.0 – 0.05 |
| Skin effect error | — | 0.0 – 0.3 |
| Coupling medium error | — | 0.0 – 0.2 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 20.0

| Metric | Range |
|---|---|
| Psnr db | 10.0 – 45.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `800` and δ = `5`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x0d32f5f98bad358432597ff247ae7dcdf16e8bc1def752c1ab7d0bf1be4e4b8a`
- **Chain tx hash:** `0x20f7ac378c74b2f0823163e759a924415974a215b15006a9d92d934c0f2c47c6`
- **Chain block:** `41554001`

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## File Mapping

This bundle consists of: `L1-255.md`, `L1-255.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-255.md` | Source of truth — edit this | Human or LLM |
| `L1-255.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 `L1-255`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._