# ⚛  L1 Principle — Single-Photon Emission Computed Tomography (SPECT)

**ID:** `L1-034` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Medical Imaging — *Gamma-camera rotating tomography*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** 3D activity concentration
> **📡 Carrier:** photon · **🌫 Noise:** shot poisson
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41553358

---

## 🧠 1. Introduction

**Single-Photon Emission Computed Tomography (SPECT)** is a **linear inverse problem** whose unknown lives in **3D activity concentration** space, within the **Gamma-camera rotating tomography** sub-domain of **Medical Imaging**.

Measurements consist of photons collected by an optical detector via a **spect gamma camera** sensing mechanism.

The forward operator applies, in order: L · gamma emission operator; L · collimator 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 activity concentration 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 ~= 16); collimator_blur dominates the stability cliff; attenuation_correction_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 · gamma emission → L · collimator → Angular scan → Angular integration → **y** (detector).

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

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.gamma_emission` | L · gamma emission operator |
| `L.collimator` | L · collimator 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 | Medical Imaging |
| Sub domain | Gamma-camera rotating tomography |
| Carrier | photon |
| Problem class | linear_inverse |
| Solution space | 3D_activity_concentration |
| Noise model | shot_poisson |
| Integration axis | angular |
| Difficulty delta | 5 |
| L dag | 3.5 |

## 📡 4. Measurement Model

Existence of the recovered 3D activity concentration 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 ~= 16); collimator_blur dominates the stability cliff; attenuation_correction_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:** `spect` · **Forward operator:** `spect_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| H | px | 128 |
| W | px | 128 |
| Z | — | 64 |
| Snr db | dB | 18 |
| Scatter | — | 0.2 |
| N angles | — | 64 |
| Patient motion | — | 0 |
| Collimator blur | — | 0 |
| Counts per pixel | — | 100 |
| Attenuation correction error | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| H | px | 64 |
| W | px | 64 |
| Z | — | 16 – 512 |
| Snr db | dB | 0.0 – 35.0 |
| Scatter | — | 0.0 – 0.5 |
| N angles | — | 8 – 256 |
| Patient motion | — | 0.0 – 0.3 |
| Collimator blur | — | 0.0 – 0.3 |
| Counts per pixel | — | 1 – 10000 |
| Attenuation correction error | — | 0.0 – 0.3 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 22.0

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

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `320` and δ = `5`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0xd8fe453230341be5dad2cc48a43e99b867d718d9af5d1016413fcff7adaf0c08`
- **Chain tx hash:** `0xabe2942ee94669f848d13adce3f4cb4f36d37d691190e4c1e0bef96bf1648212`
- **Chain block:** `41553358`

---

## File Mapping

This bundle consists of: `L1-034.md`, `L1-034.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-034.md` | Source of truth — edit this | Human or LLM |
| `L1-034.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 `L1-034`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._