# ⚛  L1 Principle — Lattice Light Sheet Microscopy (LLSM) — Bessel-lattice 3D imaging

**ID:** `L1-020` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Microscopy — *High-resolution 3D live imaging via optical lattice*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** 3D intensity
> **📡 Carrier:** photon · **🌫 Noise:** poisson gaussian
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41554168

---

## 🧠 1. Introduction

**Lattice Light Sheet Microscopy (LLSM) — Bessel-lattice 3D imaging** is a **linear inverse problem** whose unknown lives in **3D intensity** space, within the **High-resolution 3D live imaging via optical lattice** sub-domain of **Microscopy**.

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

The forward operator applies, in order: L · illumination · lattice operator; K · psf · detection operator; S · scan · axial operator; detector accumulates flux over the exposure window.

Observations are corrupted by Poisson shot noise plus Gaussian read-out noise. Existence of the recovered 3D intensity is guaranteed within the declared Omega bounds. Uniqueness holds on the measurement-supported subspace; out-of-support modes are controlled by the declared priors. Stability is moderately conditioned (kappa_eff ~= 18); lattice_registration dominates the stability cliff; dithering_phase_error and the remaining mismatch parameters contribute higher-order bias terms. Poisson signal noise + gaussian read noise 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** → L · illumination · lattice → K · psf · detection → S · scan · axial → Temporal integration → **y** (detector).

```
y = ∫_t dt `S.scan.axial` `K.psf.detection` `L.illumination.lattice` x + n,    Poisson + 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.illumination.lattice` | L · illumination · lattice operator |
| `K.psf.detection` | K · psf · detection operator |
| `S.scan.axial` | S · scan · axial operator |
| `int.temporal` | Detector accumulates flux over the exposure window |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Microscopy |
| Sub domain | High-resolution 3D live imaging via optical lattice |
| Carrier | photon |
| Problem class | linear_inverse |
| Solution space | 3D_intensity |
| Noise model | poisson_gaussian |
| Integration axis | axial |
| Difficulty delta | 5 |
| L dag | 3.8 |

## 📡 4. Measurement Model

Existence of the recovered 3D intensity is guaranteed within the declared Omega bounds. Uniqueness holds on the measurement-supported subspace; out-of-support modes are controlled by the declared priors. Stability is moderately conditioned (kappa_eff ~= 18); lattice_registration dominates the stability cliff; dithering_phase_error and the remaining mismatch parameters contribute higher-order bias terms. Poisson signal noise + gaussian read noise 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:** `llsm` · **Forward operator:** `llsm_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| H | px | 2048 |
| W | px | 512 |
| Z | — | 100 |
| Na det | — | 1.1 |
| Na illum | — | 0.6 |
| Peak photons | photons | 200 |
| Lattice pitch nm | nm | 1500 |
| Lattice registration | — | 0 |
| Dithering phase error | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| H | px | 512 – 4096 |
| W | px | 256 – 2048 |
| Z | — | 32 – 512 |
| Na det | — | 0.8 – 1.2 |
| Na illum | — | 0.3 – 0.8 |
| Peak photons | photons | 30 – 2000 |
| Lattice pitch nm | nm | 600 – 3000 |
| Lattice registration | — | 0.0 – 0.1 |
| Dithering phase error | — | 0.0 – 0.2 |
| Sample induced aberration | — | 0.0 – 0.3 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 28.5

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

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `360` and δ = `5`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x5e1bbdd7f64417cf30eb12955afe6c6bc58dfcdbc5e8a77537b40926f17389b6`
- **Chain tx hash:** `0x7d009b5ece6f1a1570af66a425cbd9edae78ba5990d88498df65b33dad323f8a`
- **Chain block:** `41554168`

---

## File Mapping

This bundle consists of: `L1-020.md`, `L1-020.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-020.md` | Source of truth — edit this | Human or LLM |
| `L1-020.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 `L1-020`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._