# ⚛  L1 Principle — Image Scanning Microscopy (ISM) — pixel reassignment super-resolution

**ID:** `L1-016` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Microscopy — *Confocal with small-pinhole detector array*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** 2D intensity
> **📡 Carrier:** photon · **🌫 Noise:** poisson gaussian
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41554156

---

## 🧠 1. Introduction

**Image Scanning Microscopy (ISM) — pixel reassignment super-resolution** is a **linear inverse problem** whose unknown lives in **2D intensity** space, within the **Confocal with small-pinhole detector array** sub-domain of **Microscopy**.

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

The forward operator applies, in order: K · psf · confocal operator; ordered pixel-by-pixel sampling; L · pixel reassign operator; detector accumulates flux over the exposure window.

Observations are corrupted by Poisson shot noise plus Gaussian read-out noise. Existence of the recovered 2D 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 well-conditioned (kappa_eff ~= 9); detector_registration dominates the stability cliff; pinhole_size_AU and the remaining mismatch parameters contribute higher-order bias terms. Poisson signal noise + gaussian read noise sets the irreducible data-fidelity floor, while mild Tikhonov or analytic inversion is sufficient at the nominal Omega point.

## ⚙ 2. Forward Model

Physical chain: **x** → K · psf · confocal → Raster scan → L · pixel reassign → Temporal integration → **y** (detector).

```
y = ∫_t dt `L.pixel_reassign` S_raster `K.psf.confocal` x + n,    Poisson + 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `K.psf.confocal` | K · psf · confocal operator |
| `S.scan.raster` | Ordered pixel-by-pixel sampling |
| `L.pixel_reassign` | L · pixel reassign operator |
| `int.temporal` | Detector accumulates flux over the exposure window |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Microscopy |
| Sub domain | Confocal with small-pinhole detector array |
| Carrier | photon |
| Problem class | linear_inverse |
| Solution space | 2D_intensity |
| Noise model | poisson_gaussian |
| Integration axis | temporal |
| Difficulty delta | 3 |
| L dag | 3.3 |

## 📡 4. Measurement Model

Existence of the recovered 2D 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 well-conditioned (kappa_eff ~= 9); detector_registration dominates the stability cliff; pinhole_size_AU and the remaining mismatch parameters contribute higher-order bias terms. Poisson signal noise + gaussian read noise sets the irreducible data-fidelity floor, while mild Tikhonov or analytic inversion is sufficient at the nominal Omega point.

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

## 📏 5. Operating Range (Ω)

**Center problem class:** `ism` · **Forward operator:** `ism_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| H | px | 1024 |
| W | px | 1024 |
| Na | — | 1.4 |
| N det | — | 32 |
| Pixel nm | nm | 40 |
| Pinhole au | — | 0.2 |
| Peak photons | photons | 300 |
| Detector registration | — | 0 |
| Reassignment kernel drift | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| H | px | 256 – 4096 |
| W | px | 256 – 4096 |
| Na | — | 1.0 – 1.49 |
| N det | — | 4 – 100 |
| Pixel nm | nm | 20 – 100 |
| Pinhole au | — | 0.1 – 1.0 |
| Peak photons | photons | 30 – 3000 |
| Detector registration | — | 0.0 – 0.1 |
| Reassignment kernel drift | — | 0.0 – 0.2 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 30.0

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

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `180` and δ = `3`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x7733790aa6235469f6240481753e2e0d6acb727353fe8f75919cad7c305955c6`
- **Chain tx hash:** `0x0d3681ee5f259ff05977e7df1e15429adbc5fdb9646cb74f7f6869f9766ab629`
- **Chain block:** `41554156`

---

## File Mapping

This bundle consists of: `L1-016.md`, `L1-016.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-016.md` | Source of truth — edit this | Human or LLM |
| `L1-016.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 `L1-016`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._