# ⚛  L1 Principle — Image Inpainting (masked region recovery)

**ID:** `L1-384` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Signal Processing — *Spatial hole-filling / occlusion removal*
> **🎯 Problem class:** linear inverse masked · **🧮 Solution space:** 2D image
> **📡 Carrier:** photon · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41555197

---

## 🧠 1. Introduction

**Image Inpainting (masked region recovery)** is a **linear inverse masked** whose unknown lives in **2D image** space, within the **Spatial hole-filling / occlusion removal** sub-domain of **Signal Processing**.

Measurements consist of photons collected by an optical detector via a **binary masking** sensing mechanism.

The forward operator applies, in order: D · mask · spatial operator; pixel-level spatial averaging on the detector.

Observations are corrupted by additive Gaussian noise. Strictly underdetermined inside mask; uniqueness is prior-induced (TV, non-local similarity, learned). Stability decreases super-linearly with mask_ratio and hole diameter.

## ⚙ 2. Forward Model

Physical chain: **x** → D · mask · spatial → Spatial integration → **y** (detector).

```
y = ∫_A dA `D.mask.spatial` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `D.mask.spatial` | D · mask · spatial operator |
| `int.spatial` | Pixel-level spatial averaging on the detector |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Signal Processing |
| Sub domain | Spatial hole-filling / occlusion removal |
| Carrier | photon |
| Problem class | linear_inverse_masked |
| Solution space | 2D_image |
| Noise model | gaussian |
| Integration axis | spatial |
| Difficulty delta | 3 |
| L dag | 2 |

## 📡 4. Measurement Model

Strictly underdetermined inside mask; uniqueness is prior-induced (TV, non-local similarity, learned). Stability decreases super-linearly with mask_ratio and hole diameter.

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

## 📏 5. Operating Range (Ω)

**Center problem class:** `freeform_inpainting` · **Forward operator:** `binary_mask`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| H | px | 512 |
| W | px | 512 |
| Mask ratio | — | 0.25 |
| Mask shape | — | freeform |
| Noise level | — | 0 |
| Boundary dilation | — | 0 |
| Hole diameter max | — | 64 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| H | px | 128 – 2048 |
| W | px | 128 – 2048 |
| Mask ratio | — | 0.05 – 0.75 |
| Noise level | — | 0.0 – 0.05 |
| Boundary dilation | — | 0 – 8 |
| Hole diameter max | — | 8 – 512 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 27.0 dB PSNR

| Metric | Range |
|---|---|
| Psnr db | 15.0 – 40.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `10000` and δ = `3`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x551cdda785b03feb489e1a6bac709df7aa9b1bf03d78f5a6b00ef07de2bdb994`
- **Chain tx hash:** `0x3ea98f5fe15d7f904ee4043488a94956404a5991648292fe1a3a64a99d2606f5`
- **Chain block:** `41555197`

---

## File Mapping

This bundle consists of: `L1-384.md`, `L1-384.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-384.md` | Source of truth — edit this | Human or LLM |
| `L1-384.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 `L1-384`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._