# ⚛  L1 Principle — Amplitude-Modulated Continuous-Wave Time-of-Flight (AMCW-ToF) Depth

**ID:** `L1-095` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Depth Imaging — *Indirect ToF via phase*
> **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** 2D depth map
> **📡 Carrier:** photon · **🌫 Noise:** shot poisson
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41547811

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## 🧠 1. Introduction

**Amplitude-Modulated Continuous-Wave Time-of-Flight (AMCW-ToF) Depth** is a **nonlinear inverse problem** whose unknown lives in **2D depth map** space, within the **Indirect ToF via phase** sub-domain of **Depth Imaging**.

Measurements consist of photons collected by an optical detector via a **amcw modulated illumination** sensing mechanism.

The forward operator applies, in order: L · modulate · amcw operator; L · reflect scene operator; L · demodulate operator; detector accumulates flux over the exposure window.

Observations are corrupted by Poisson shot noise from quantum-limited detection. Well-posed with multi-frequency unwrap (3 f_mod sufficient for Chinese Remainder Theorem unwrap). Precision sigma_d ~ c / (4*pi*f_mod * sqrt(SNR)). Multipath interference (MPI) at concave corners and translucent surfaces biases depth; motion during 4-bucket integration causes phase artifacts.

## ⚙ 2. Forward Model

Physical chain: **x** → L · modulate · amcw → L · reflect scene → L · demodulate → Temporal integration → **y** (detector).

```
y = ∫_t dt `L.demodulate` `L.reflect_scene` `L.modulate.amcw` x,    measurements ~ Poisson(αy)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.modulate.amcw` | L · modulate · amcw operator |
| `L.reflect_scene` | L · reflect scene operator |
| `L.demodulate` | L · demodulate operator |
| `int.temporal` | Detector accumulates flux over the exposure window |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Depth Imaging |
| Sub domain | Indirect ToF via phase |
| Carrier | photon |
| Problem class | nonlinear_inverse |
| Solution space | 2D_depth_map |
| Noise model | shot_poisson |
| Integration axis | temporal |
| Difficulty delta | 3 |
| L dag | 3.2 |

## 📡 4. Measurement Model

Well-posed with multi-frequency unwrap (3 f_mod sufficient for Chinese Remainder Theorem unwrap). Precision sigma_d ~ c / (4*pi*f_mod * sqrt(SNR)). Multipath interference (MPI) at concave corners and translucent surfaces biases depth; motion during 4-bucket integration causes phase artifacts.

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

## 📏 5. Operating Range (Ω)

**Center problem class:** `amcw_tof_depth` · **Forward operator:** `four_bucket_tof_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| H | px | 424 |
| W | px | 512 |
| N freq | — | 3 |
| T int ms | ms | 10 |
| K buckets | — | 4 |
| F mod mhz | MHz | 80 |
| Photon count | — | 500 |
| Ambient light | — | 0 |
| Motion jitter | — | 0 |
| Multipath interference | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| H | px | 240 – 1080 |
| W | px | 320 – 1920 |
| N freq | — | 1 – 5 |
| T int ms | ms | 1 – 100 |
| K buckets | — | 3 – 9 |
| F mod mhz | MHz | 10 – 200 |
| Photon count | — | 20 – 5000 |
| Ambient light | — | 0.0 – 50000 |
| Motion jitter | — | 0.0 – 0.05 |
| Temperature drift | — | 0.0 – 0.05 |
| Mixed pixels at edges | — | 0.0 – 0.1 |
| Multipath interference | — | 0.0 – 0.3 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 8.0 mm RMSE

| Metric | Range |
|---|---|
| Depth rmse mm | 2.0 – 100.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **depth_RMSE_mm**, with κ = `700` and δ = `3`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x69287c38a8582f25577dd3476514ee4e1c4751cf7707a96c9107a986c6779138`
- **Chain tx hash:** `0xb790c39792c2ac236327e21a9896c3e70db6c3e5137b11b4861a93756237fd5d`
- **Chain block:** `41547811`

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## File Mapping

This bundle consists of: `L1-095.md`, `L1-095.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-095.md` | Source of truth — edit this | Human or LLM |
| `L1-095.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 `L1-095`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._