# ⚛  L1 Principle — PillCam-SPECTRA Optical-Property Recovery (analytical core)

**ID:** `L1-511` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Medical Imaging — *Capsule-endoscopic multispectral diffuse-reflectance optical-property recovery*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** 2D chromophore concentration
> **📡 Carrier:** photon · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41553360

---

## 🧠 1. Introduction

**PillCam-SPECTRA Optical-Property Recovery (analytical core)** is a **linear inverse problem** whose unknown lives in **2D chromophore concentration** space, within the **Capsule-endoscopic multispectral diffuse-reflectance optical-property recovery** sub-domain of **Medical Imaging**.

Measurements consist of photons collected by an optical detector via a **diffuse reflectance capsule** sensing mechanism.

The forward operator applies, in order: L · optical source operator; L · diffuse reflectance operator; exponential attenuation along the propagation path; L · detector response operator; pixel-level spatial averaging on the detector; detector sums all spectral bands.

Observations are corrupted by additive Gaussian noise. Existence of recovered chromophore concentrations is guaranteed within the declared Omega bounds. Uniqueness holds under multi-wavelength acquisition with N_wavelengths >= 5 (sufficient to resolve the typical chromophore basis); single-wavelength acquisition is non-unique (cannot decompose mu_a into chromophore components). Stability is well-conditioned (kappa_eff ~ 15) — capsule_to_mucosa_distance dominates absolute-concentration bias; mucosal_curvature contributes a geometric-correction factor; scattering_anisotropy_g uncertainty contributes a scaling factor of order ~10%. Joint Hadamard well-posedness for the diffuse-reflectance forward is established by Farrell-Patterson-Wilson 1992 (foundational), Wang-Jacques 1995 (MCML), Yudovsky-Pilon 2010 (multispectral inverse), Bjorgan-Milanic-Randeberg 2014 (hyperspectral classification), and Calin-Parasca-Savastru 2014 (medical hyperspectral).

## ⚙ 2. Forward Model

Physical chain: **x** → L · optical source → L · diffuse reflectance → Beer-Lambert attenuation → L · detector response → Spatial integration → Spectral integration → **y** (detector).

```
y = Σ_λ ∫_A dA `L.detector_response` exp(-∫µ dl) `L.diffuse_reflectance` `L.optical_source` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.optical_source` | L · optical source operator |
| `L.diffuse_reflectance` | L · diffuse reflectance operator |
| `L.beer_lambert` | Exponential attenuation along the propagation path |
| `L.detector_response` | L · detector response operator |
| `int.spatial` | Pixel-level spatial averaging on the detector |
| `int.spectral` | Detector sums all spectral bands |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Medical Imaging |
| Sub domain | Capsule-endoscopic multispectral diffuse-reflectance optical-property recovery |
| Carrier | photon |
| Problem class | linear_inverse |
| Solution space | 2D_chromophore_concentration |
| Noise model | gaussian |
| Integration axis | spectral_spatial |
| Difficulty delta | 3 |
| L dag | 4.9 |

## 📡 4. Measurement Model

Existence of recovered chromophore concentrations is guaranteed within the declared Omega bounds. Uniqueness holds under multi-wavelength acquisition with N_wavelengths >= 5 (sufficient to resolve the typical chromophore basis); single-wavelength acquisition is non-unique (cannot decompose mu_a into chromophore components). Stability is well-conditioned (kappa_eff ~ 15) — capsule_to_mucosa_distance dominates absolute-concentration bias; mucosal_curvature contributes a geometric-correction factor; scattering_anisotropy_g uncertainty contributes a scaling factor of order ~10%. Joint Hadamard well-posedness for the diffuse-reflectance forward is established by Farrell-Patterson-Wilson 1992 (foundational), Wang-Jacques 1995 (MCML), Yudovsky-Pilon 2010 (multispectral inverse), Bjorgan-Milanic-Randeberg 2014 (hyperspectral classification), and Calin-Parasca-Savastru 2014 (medical hyperspectral).

| Metric | Value |
|---|---|
| Metric | PSNR_dB |
| Secondary | RMSE_per_chromophore |

## 📏 5. Operating Range (Ω)

**Center problem class:** `pillcam_spectra_optical_recovery` · **Forward operator:** `diffuse_reflectance_capsule_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| H | px | 256 |
| W | px | 256 |
| Snr db | dB | 26 |
| Motion blur | — | 0 |
| N wavelengths | — | 8 |
| Lambda range nm | nm | 400 – 950 |
| Mucosal curvature | — | 0 |
| Working distance mm | mm | 2 |
| Illumination uniformity | — | 0 |
| Scattering anisotropy g | g | 0.85 |
| Capsule orientation drift | — | 0 |
| Capsule to mucosa distance | — | 0 |
| Source detector distance mm | mm | 5 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| H | px | 128 – 1024 |
| W | px | 128 – 1024 |
| Snr db | dB | 10.0 – 38.0 |
| Motion blur | — | 0.0 – 0.3 |
| N wavelengths | — | 3 – 16 |
| Mucosal curvature | — | 0.0 – 0.3 |
| Working distance mm | mm | 0.5 – 5.0 |
| Illumination uniformity | — | 0.0 – 0.4 |
| Scattering anisotropy g | g | 0.7 – 0.95 |
| Capsule orientation drift | — | 0.0 – 0.4 |
| Capsule to mucosa distance | — | 0.0 – 0.4 |
| Source detector distance mm | mm | 3.0 – 12.0 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 28.0

| Metric | Range |
|---|---|
| Psnr db | 10.0 – 45.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `80` and δ = `3`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x012312d456bc56bf1a0e52320db3a100003569313e9ae82eb93f91c0253d7b20`
- **Chain tx hash:** `0x0f9a522bef3d15516c0a667ac47b1fa9269d5ae319486d8d0b207085a8c4855b`
- **Chain block:** `41553360`

---

## File Mapping

This bundle consists of: `L1-511.md`, `L1-511.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-511.md` | Source of truth — edit this | Human or LLM |
| `L1-511.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 `L1-511`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._