# ⚛  L1 Principle — Optical Diffraction Tomography (ODT) — 3D refractive-index imaging

**ID:** `L1-073` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Coherent Imaging — *Multi-angle coherent tomography*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** 3D refractive index
> **📡 Carrier:** photon · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41554170

---

## 🧠 1. Introduction

**Optical Diffraction Tomography (ODT) — 3D refractive-index imaging** is a **linear inverse problem** whose unknown lives in **3D refractive index** space, within the **Multi-angle coherent tomography** sub-domain of **Coherent Imaging**.

Measurements consist of photons collected by an optical detector via a **multi angle coherent holography** sensing mechanism.

The forward operator applies, in order: rotates source / detector to acquire different projections; F · propagation · helmholtz operator; L · projection · ewald cap operator; integration over the solid angle of incidence/emission.

Observations are corrupted by additive Gaussian noise. Ill-posed along the missing-cone axis (cone of k-space unreachable by finite-NA illumination). Regularized inversion (TV, deep priors) fills the cone with smoothness assumptions. Non-linear Beyond-Born solvers (LS-LT, SEAGLE) handle multiple scattering but become strongly non-convex.

## ⚙ 2. Forward Model

Physical chain: **x** → Angular scan → F · propagation · helmholtz → L · projection · ewald cap → Angular integration → **y** (detector).

```
y = ∫dΩ `L.projection.ewald_cap` `F.propagation.helmholtz` R(θ) x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `S.scan.angular` | Rotates source / detector to acquire different projections |
| `F.propagation.helmholtz` | F · propagation · helmholtz operator |
| `L.projection.ewald_cap` | L · projection · ewald cap operator |
| `int.angular` | Integration over the solid angle of incidence/emission |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Coherent Imaging |
| Sub domain | Multi-angle coherent tomography |
| Carrier | photon |
| Problem class | linear_inverse |
| Solution space | 3D_refractive_index |
| Noise model | gaussian |
| Integration axis | angular |
| Difficulty delta | 5 |
| L dag | 3.8 |

## 📡 4. Measurement Model

Ill-posed along the missing-cone axis (cone of k-space unreachable by finite-NA illumination). Regularized inversion (TV, deep priors) fills the cone with smoothness assumptions. Non-linear Beyond-Born solvers (LS-LT, SEAGLE) handle multiple scattering but become strongly non-convex.

| Metric | Value |
|---|---|
| Metric | 3D_PSNR_dB |
| Secondary | refractive_index_L2 |

## 📏 5. Operating Range (Ω)

**Center problem class:** `3d_refractive_index_recovery` · **Forward operator:** `rytov_ewald_cap`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| D | — | 128 |
| H | px | 512 |
| W | px | 512 |
| N 0 | — | 1.33 |
| K angles | — | 90 |
| Na illum | — | 1 |
| Delta n max | — | 0.02 |
| Wavelength nm | nm | 532 |
| Illumination angle error | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| D | — | 32 – 512 |
| H | px | 128 – 2048 |
| W | px | 128 – 2048 |
| N 0 | — | 1.0 – 1.5 |
| K angles | — | 30 – 360 |
| Na illum | — | 0.4 – 1.4 |
| Delta n max | — | 0.001 – 0.1 |
| Wavelength nm | nm | 400 – 800 |
| Aberration zernike | — | 0.0 – 0.5 |
| Limited angle range | — | 90 – 360 |
| Illumination angle error | — | 0.0 – 2.0 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 27.0 dB 3D PSNR

| Metric | Range |
|---|---|
| Psnr db | 15.0 – 38.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **3D_PSNR_dB**, with κ = `6000` and δ = `5`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0xeeaf2db215d84a2c5cda06002d3890c1437abd73213fae9bb2ff76b241ff699d`
- **Chain tx hash:** `0x1a84c7012fb789e01e13ef7d96b01b540866eff455a10fd61c1837ba13f316f5`
- **Chain block:** `41554170`

---

## File Mapping

This bundle consists of: `L1-073.md`, `L1-073.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-073.md` | Source of truth — edit this | Human or LLM |
| `L1-073.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 `L1-073`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._