# ⚛  L1 Principle — Diffuse Optical Tomography (DOT)

**ID:** `L1-040` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Medical Imaging — *Near-IR scattering-based functional imaging*
> **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** 3D optical absorption
> **📡 Carrier:** photon · **🌫 Noise:** shot poisson
> **⚖ Difficulty (δ):** 10 · **⛓ Block:** 41553385

---

## 🧠 1. Introduction

**Diffuse Optical Tomography (DOT)** is a **nonlinear inverse problem** whose unknown lives in **3D optical absorption** space, within the **Near-IR scattering-based functional imaging** sub-domain of **Medical Imaging**.

Measurements consist of photons collected by an optical detector via a **near infrared diffusion** sensing mechanism.

The forward operator applies, in order: L · nir source operator; L · diffusion propagation operator; S · scan · source detector operator; pixel-level spatial averaging on the detector.

Observations are corrupted by Poisson shot noise from quantum-limited detection. Existence of the recovered 3D optical absorption is guaranteed within the declared Omega bounds. Uniqueness is local rather than global (non-convex landscape); convergence depends on initialisation and priors. Stability is moderately conditioned (kappa_eff ~= 30); tissue_scattering dominates the stability cliff; anatomical_prior_error 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 · nir source → L · diffusion propagation → S · scan · source detector → Spatial integration → **y** (detector).

```
y = ∫_A dA `S.scan.source_detector` `L.diffusion_propagation` `L.nir_source` x,    measurements ~ Poisson(αy)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.nir_source` | L · nir source operator |
| `L.diffusion_propagation` | L · diffusion propagation operator |
| `S.scan.source_detector` | S · scan · source detector operator |
| `int.spatial` | Pixel-level spatial averaging on the detector |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Medical Imaging |
| Sub domain | Near-IR scattering-based functional imaging |
| Carrier | photon |
| Problem class | nonlinear_inverse |
| Solution space | 3D_optical_absorption |
| Noise model | shot_poisson |
| Integration axis | spatial |
| Difficulty delta | 10 |
| L dag | 4 |

## 📡 4. Measurement Model

Existence of the recovered 3D optical absorption is guaranteed within the declared Omega bounds. Uniqueness is local rather than global (non-convex landscape); convergence depends on initialisation and priors. Stability is moderately conditioned (kappa_eff ~= 30); tissue_scattering dominates the stability cliff; anatomical_prior_error 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:** `dot` · **Forward operator:** `dot_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| Snr db | dB | 25 |
| Motion | — | 0 |
| N sources | — | 32 |
| Lambda nm | nm | 760 |
| N detectors | — | 32 |
| Partial volume | — | 0.1 |
| Tissue scattering | — | 10 |
| Source distance cm | — | 3.5 |
| Anatomical prior error | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| Snr db | dB | 0.0 – 35.0 |
| Motion | — | 0.0 – 0.3 |
| N sources | — | 8 – 256 |
| Lambda nm | nm | 650 – 900 |
| N detectors | — | 8 – 256 |
| Partial volume | — | 0.0 – 0.5 |
| Tissue scattering | — | 5.0 – 20.0 |
| Source distance cm | — | 1.0 – 10.0 |
| Anatomical prior error | — | 0.0 – 0.5 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 18.0

| Metric | Range |
|---|---|
| Psnr db | 5.0 – 45.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `600` and δ = `10`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0xb4872c1d2d2dcb7a5ef7e22a621bb2298ea0ab7a90ad8acec35cf81509135a4b`
- **Chain tx hash:** `0x45e960be5d35ae5d2adaeb980d395acdb6a2a60427236e96d9b4892a99ccf402`
- **Chain block:** `41553385`

---

## File Mapping

This bundle consists of: `L1-040.md`, `L1-040.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-040.md` | Source of truth — edit this | Human or LLM |
| `L1-040.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 `L1-040`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._