# ⚛  L1 Principle — Diffusion MRI (DWI / DTI / HARDI)

**ID:** `L1-045` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Medical Imaging — *Water-diffusion-weighted MRI for microstructure/tractography*
> **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** 4D diffusion tensor or ODF
> **📡 Carrier:** radio_wave · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41552301

---

## 🧠 1. Introduction

**Diffusion MRI (DWI / DTI / HARDI)** is a **nonlinear inverse problem** whose unknown lives in **4D diffusion tensor or ODF** space, within the **Water-diffusion-weighted MRI for microstructure/tractography** sub-domain of **Medical Imaging**.

Measurements consist of radio-frequency electromagnetic waves via a **mri fourier encoding** sensing mechanism.

The forward operator applies, in order: Bloch-equation tip of the magnetisation vector; L · diffusion gradient operator; L · gradient encoding operator; integration over the solid angle of incidence/emission.

Observations are corrupted by additive Gaussian noise. Existence of the recovered 4D diffusion tensor or ODF 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 ~= 18); eddy_currents dominates the stability cliff; motion and the remaining mismatch parameters contribute higher-order bias terms. Additive gaussian thermal/electronic noise 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** → RF excitation pulse → L · diffusion gradient → L · gradient encoding → Angular integration → **y** (detector).

```
y = ∫dΩ `L.gradient_encoding` `L.diffusion_gradient` B₁(t) x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.rf_excitation` | Bloch-equation tip of the magnetisation vector |
| `L.diffusion_gradient` | L · diffusion gradient operator |
| `L.gradient_encoding` | L · gradient encoding operator |
| `int.angular` | Integration over the solid angle of incidence/emission |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Medical Imaging |
| Sub domain | Water-diffusion-weighted MRI for microstructure/tractography |
| Carrier | radio_wave |
| Problem class | nonlinear_inverse |
| Solution space | 4D_diffusion_tensor_or_ODF |
| Noise model | gaussian |
| Integration axis | angular |
| Difficulty delta | 5 |
| L dag | 3.8 |

## 📡 4. Measurement Model

Existence of the recovered 4D diffusion tensor or ODF 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 ~= 18); eddy_currents dominates the stability cliff; motion and the remaining mismatch parameters contribute higher-order bias terms. Additive gaussian thermal/electronic noise 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:** `dmri` · **Forward operator:** `dmri_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| H | px | 128 |
| W | px | 128 |
| Z | — | 64 |
| Snr db | dB | 20 |
| Motion | — | 0 |
| N directions | — | 32 |
| B0 distortion | — | 0 |
| Eddy currents | — | 0 |
| B max s per mm2 | — | 1000 |
| Partial volume csf | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| H | px | 64 |
| W | px | 64 |
| Z | — | 16 – 256 |
| Snr db | dB | 0.0 – 35.0 |
| Motion | — | 0.0 – 0.5 |
| N directions | — | 6 – 256 |
| B0 distortion | — | 0.0 – 0.3 |
| Eddy currents | — | 0.0 – 0.3 |
| B max s per mm2 | — | 100 – 5000 |
| Partial volume csf | — | 0.0 – 0.3 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 20.0

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

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `360` and δ = `5`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x86eda8fa6529b40c56bdcd45721d0f846b0e76b9d5a8f82f08800d56b1b062c1`
- **Chain tx hash:** `0xae0121d4d4c450e698d264aa07d0cf545572f1c42e70b0f483a4a3c3ea2da7b9`
- **Chain block:** `41552301`

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## File Mapping

This bundle consists of: `L1-045.md`, `L1-045.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-045.md` | Source of truth — edit this | Human or LLM |
| `L1-045.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 `L1-045`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._