# ⚛  L1 Principle — Structured-Light 3D Imaging (projector-camera triangulation)

**ID:** `L1-097` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Depth Imaging — *Active triangulation with coded patterns*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** 2D depth map
> **📡 Carrier:** photon · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41547811

---

## 🧠 1. Introduction

**Structured-Light 3D Imaging (projector-camera triangulation)** is a **linear inverse problem** whose unknown lives in **2D depth map** space, within the **Active triangulation with coded patterns** sub-domain of **Depth Imaging**.

Measurements consist of photons collected by an optical detector via a **projector camera triangulation** sensing mechanism.

The forward operator applies, in order: S · project · pattern operator; L · reflect lambertian operator; L · triangulate operator; pixel-level spatial averaging on the detector.

Observations are corrupted by additive Gaussian noise. Well-posed when K projector-pattern bits exceed log2(W_p) + phase-unwrap margin and surface is Lambertian. Degrades on specular, translucent, dark, and high-frequency surfaces (e.g. hair), in strong ambient light, and at occlusions (camera sees but projector cannot).

## ⚙ 2. Forward Model

Physical chain: **x** → S · project · pattern → L · reflect lambertian → L · triangulate → Spatial integration → **y** (detector).

```
y = ∫_A dA `L.triangulate` `L.reflect_lambertian` `S.project.pattern` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `S.project.pattern` | S · project · pattern operator |
| `L.reflect_lambertian` | L · reflect lambertian operator |
| `L.triangulate` | L · triangulate operator |
| `int.spatial` | Pixel-level spatial averaging on the detector |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Depth Imaging |
| Sub domain | Active triangulation with coded patterns |
| Carrier | photon |
| Problem class | linear_inverse |
| Solution space | 2D_depth_map |
| Noise model | gaussian |
| Integration axis | temporal |
| Difficulty delta | 3 |
| L dag | 3.2 |

## 📡 4. Measurement Model

Well-posed when K projector-pattern bits exceed log2(W_p) + phase-unwrap margin and surface is Lambertian. Degrades on specular, translucent, dark, and high-frequency surfaces (e.g. hair), in strong ambient light, and at occlusions (camera sees but projector cannot).

| Metric | Value |
|---|---|
| Metric | depth_RMSE_mm |
| Secondary | depth_MAE_mm |

## 📏 5. Operating Range (Ω)

**Center problem class:** `structured_light_depth` · **Forward operator:** `phase_shift_gray_code`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| H | px | 1080 |
| W | px | 1920 |
| W p | — | 1024 |
| K patterns | — | 24 |
| Baseline mm | mm | 120 |
| Photon count | — | 500 |
| Ambient light | — | 0 |
| Specular fraction | — | 0 |
| Projector camera pose drift | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| H | px | 480 – 4000 |
| W | px | 640 – 4000 |
| W p | — | 256 – 4096 |
| K patterns | — | 4 – 64 |
| Baseline mm | mm | 30 – 500 |
| Photon count | — | 50 – 10000 |
| Ambient light | — | 0.0 – 100000 |
| Specular fraction | — | 0.0 – 0.5 |
| Occlusion fraction | — | 0.0 – 0.3 |
| Subsurface scattering | — | 0.0 – 0.3 |
| Projector camera pose drift | — | 0.0 – 0.002 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 0.5 mm depth RMSE

| Metric | Range |
|---|---|
| Depth rmse mm | 0.1 – 20.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **depth_RMSE_mm**, with κ = `600` and δ = `3`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0xb0f648b5592f9e7acd3f02ed5540ca01b3a470930c9815fe5bcad170aebe27ad`
- **Chain tx hash:** `0xaf6d7c9dd89cedbd85561c6fe7bca07d2b5f8137e6714bbc680ff5f91f89f5f7`
- **Chain block:** `41547811`

---

## File Mapping

This bundle consists of: `L1-097.md`, `L1-097.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-097.md` | Source of truth — edit this | Human or LLM |
| `L1-097.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 `L1-097`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._