# ⚛  L1 Principle — Electron Tomography — tilt-series 3D reconstruction

**ID:** `L1-086` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Electron Microscopy — *Tilt-series TEM for 3D cellular structure*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** 3D density
> **📡 Carrier:** electron · **🌫 Noise:** shot poisson
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41554184

---

## 🧠 1. Introduction

**Electron Tomography — tilt-series 3D reconstruction** is a **linear inverse problem** whose unknown lives in **3D density** space, within the **Tilt-series TEM for 3D cellular structure** sub-domain of **Electron Microscopy**.

Measurements consist of electrons collected by an electron detector via a **electron transmission** sensing mechanism.

The forward operator applies, in order: L · transmit sample operator; rotates source / detector to acquire different projections; spreads measurements back along source rays (adjoint operator); integration over the solid angle of incidence/emission.

Observations are corrupted by Poisson shot noise from quantum-limited detection. Existence of the recovered 3D density 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 ~= 20); tilt_angle_error dominates the stability cliff; missing_wedge 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** → L · transmit sample → Angular scan → Angular integration → **y** (detector).

```
y = ∫dΩ R(θ) `L.transmit_sample` x,    measurements ~ Poisson(αy)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.transmit_sample` | L · transmit sample operator |
| `S.scan.angular` | Rotates source / detector to acquire different projections |
| `int.angular` | Integration over the solid angle of incidence/emission |

**🛠 Solver components** _(used inside the solver, not in the forward equation)_:

| Primitive | What it does |
|---|---|
| `L.backproject` | Spreads measurements back along source rays (adjoint operator) |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Electron Microscopy |
| Sub domain | Tilt-series TEM for 3D cellular structure |
| Carrier | electron |
| Problem class | linear_inverse |
| Solution space | 3D_density |
| Noise model | shot_poisson |
| Integration axis | angular |
| Difficulty delta | 5 |
| L dag | 3.8 |

## 📡 4. Measurement Model

Existence of the recovered 3D density 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 ~= 20); tilt_angle_error dominates the stability cliff; missing_wedge 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:** `etomo_tilt_series` · **Forward operator:** `etomo_tilt_series_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| H | px | 2048 |
| W | px | 2048 |
| Z | — | 128 |
| Kv | — | 300 |
| N tilts | — | 61 |
| Missing wedge | — | 0.33 |
| Peak electrons | — | 30 |
| Tilt range deg | deg | 60 |
| Tilt angle error | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| H | px | 512 – 4096 |
| W | px | 512 – 4096 |
| Z | — | 32 – 512 |
| Kv | — | 200 – 300 |
| N tilts | — | 20 – 200 |
| Beam damage | — | 0.0 – 0.5 |
| Missing wedge | — | 0.1 – 0.6 |
| Peak electrons | — | 1 – 500 |
| Tilt range deg | deg | 40 – 80 |
| Alignment error | — | 0.0 – 5.0 |
| Tilt angle error | — | 0.0 – 0.5 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 24.0

| Metric | Range |
|---|---|
| Psnr db | 10.0 – 40.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `400` and δ = `5`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0xfb181034b956374646ef8727ee21d10ec1b88668f4aa5156494b5c8dcc3a17a7`
- **Chain tx hash:** `0xd4f0c4fe89f7a6aad1b0a7286d708e7f7fa7159edfcd30bec2178532f034bdcf`
- **Chain block:** `41554184`

---

## File Mapping

This bundle consists of: `L1-086.md`, `L1-086.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-086.md` | Source of truth — edit this | Human or LLM |
| `L1-086.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 `L1-086`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._