# ⚛  L1 Principle — Single-Pixel Imaging (random basis compressive sensing)

**ID:** `L1-026` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Compressive Imaging — *Single-pixel / single-detector imaging*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** 2D spatial
> **📡 Carrier:** photon · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41547811

---

## 🧠 1. Introduction

**Single-Pixel Imaging (random basis compressive sensing)** is a **linear inverse problem** whose unknown lives in **2D spatial** space, within the **Single-pixel / single-detector imaging** sub-domain of **Compressive Imaging**.

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

The forward operator applies, in order: S · pattern · random operator; L · inner product operator; pixel-level spatial averaging on the detector; D · scalar operator.

Observations are corrupted by additive Gaussian noise. Underdetermined (m << n) compressive recovery; random sensing matrices with i.i.d. Bernoulli/Gaussian entries satisfy RIP w.h.p. when m >= C * s * log(n/s) for s-sparse signals in a known basis.

## ⚙ 2. Forward Model

Physical chain: **x** → S · pattern · random → L · inner product → Spatial integration → D · scalar → **y** (detector).

```
y = `D.scalar` ∫_A dA `L.inner_product` `S.pattern.random` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `S.pattern.random` | S · pattern · random operator |
| `L.inner_product` | L · inner product operator |
| `int.spatial` | Pixel-level spatial averaging on the detector |
| `D.scalar` | D · scalar operator |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Compressive Imaging |
| Sub domain | Single-pixel / single-detector imaging |
| Carrier | photon |
| Problem class | linear_inverse |
| Solution space | 2D_spatial |
| Noise model | gaussian |
| Integration axis | spatial |
| Difficulty delta | 3 |
| L dag | 3 |

## 📡 4. Measurement Model

Underdetermined (m << n) compressive recovery; random sensing matrices with i.i.d. Bernoulli/Gaussian entries satisfy RIP w.h.p. when m >= C * s * log(n/s) for s-sparse signals in a known basis.

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

## 📏 5. Operating Range (Ω)

**Center problem class:** `compressed_sensing_recovery` · **Forward operator:** `random_projection`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| N pixels | — | 4096 |
| Gain alpha | — | 0 |
| Illum sigma | — | 0 |
| Noise level | — | 0.01 |
| Sampling ratio | — | 0.25 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| N pixels | — | 1024 – 65536 |
| Gain alpha | — | 0.0 – 0.003 |
| Dark offset | — | 0.0 – 0.02 |
| Illum sigma | — | 0.0 – 15.0 |
| Noise level | — | 0.001 – 0.05 |
| Sampling ratio | — | 0.05 – 0.5 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 27.0 dB PSNR

| Metric | Range |
|---|---|
| Psnr db | 18.0 – 40.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `4000` and δ = `3`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x9530ddad080c15de3e2194d2487f390339dd06a6f57560f5c6196e85c3c655cb`
- **Chain tx hash:** `0xdb29b0671dfe88bafb7b42624f90e28dd67bf064cd72ced85046d7feea520814`
- **Chain block:** `41547811`

---

## File Mapping

This bundle consists of: `L1-026.md`, `L1-026.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-026.md` | Source of truth — edit this | Human or LLM |
| `L1-026.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 `L1-026`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._