# ⚛  L1 Principle — Fourier Ptychographic Microscopy (FPM)

**ID:** `L1-008` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Microscopy — *Synthetic-aperture low-NA wide-FOV microscopy*
> **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** complex 2D
> **📡 Carrier:** photon · **🌫 Noise:** shot poisson
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41554154

---

## 🧠 1. Introduction

**Fourier Ptychographic Microscopy (FPM)** is a **nonlinear inverse problem** whose unknown lives in **complex 2D** space, within the **Synthetic-aperture low-NA wide-FOV microscopy** sub-domain of **Microscopy**.

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

The forward operator applies, in order: S · scan · led angle operator; L · fourier aperture operator; phase is lost; only intensity is measured; integration over the solid angle of incidence/emission.

Observations are corrupted by Poisson shot noise from quantum-limited detection. Existence of the recovered complex 2D is guaranteed within the declared Omega bounds. Uniqueness is local rather than global (non-convex landscape); convergence depends on initialisation and priors. Stability is ill-conditioned (kappa_eff ~= 80); LED_position_error dominates the stability cliff; LED_intensity_variation and the remaining mismatch parameters contribute higher-order bias terms. Photon-shot-noise-limited (poisson counting) sets the irreducible data-fidelity floor, while TV / wavelet-sparsity / deep priors stabilise recovery at the ill-conditioned end of Omega.

## ⚙ 2. Forward Model

Physical chain: **x** → S · scan · led angle → L · fourier aperture → Magnitude-squared → Angular integration → **y** (detector).

```
y = ∫dΩ |·|² `L.fourier_aperture` `S.scan.led_angle` x,    measurements ~ Poisson(αy)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `S.scan.led_angle` | S · scan · led angle operator |
| `L.fourier_aperture` | L · fourier aperture operator |
| `L.modulus_squared` | Phase is lost; only intensity is measured |
| `int.angular` | Integration over the solid angle of incidence/emission |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Microscopy |
| Sub domain | Synthetic-aperture low-NA wide-FOV microscopy |
| Carrier | photon |
| Problem class | nonlinear_inverse |
| Solution space | complex_2D |
| Noise model | shot_poisson |
| Integration axis | angular |
| Difficulty delta | 5 |
| L dag | 4 |

## 📡 4. Measurement Model

Existence of the recovered complex 2D is guaranteed within the declared Omega bounds. Uniqueness is local rather than global (non-convex landscape); convergence depends on initialisation and priors. Stability is ill-conditioned (kappa_eff ~= 80); LED_position_error dominates the stability cliff; LED_intensity_variation and the remaining mismatch parameters contribute higher-order bias terms. Photon-shot-noise-limited (poisson counting) sets the irreducible data-fidelity floor, while TV / wavelet-sparsity / deep priors stabilise recovery at the ill-conditioned end of Omega.

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

## 📏 5. Operating Range (Ω)

**Center problem class:** `fpm` · **Forward operator:** `fpm_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| H | px | 512 |
| W | px | 512 |
| N led | — | 169 |
| Na obj | — | 0.1 |
| Overlap | — | 0.65 |
| Peak photons | photons | 1000 |
| Partial coherence | — | 1 |
| Led position error | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| H | px | 256 – 4096 |
| W | px | 256 – 4096 |
| N led | — | 49 – 625 |
| Na obj | — | 0.05 – 0.3 |
| Overlap | — | 0.35 – 0.85 |
| Peak photons | photons | 50 – 10000 |
| Partial coherence | — | 0.7 – 1.0 |
| Led position error | — | 0.0 – 2.0 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 26.0

| Metric | Range |
|---|---|
| Psnr db | 12.0 – 42.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `1600` and δ = `5`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x9beaa2d1b5a6dee00726849542f6ce3cfde09620cc9e6c20a0344ac5dc09c199`
- **Chain tx hash:** `0x0e226b8e0395a1ee955c02b8fd4f7c033631ca2c65f9b86acf2193e789e729f0`
- **Chain block:** `41554154`

---

## File Mapping

This bundle consists of: `L1-008.md`, `L1-008.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-008.md` | Source of truth — edit this | Human or LLM |
| `L1-008.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 `L1-008`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._