L1 ⚛ L1-043 ⊙ Testnet

Functional MRI (BOLD)

Medical Imaging · BOLD-contrast brain functional imaging

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⬇ L1-043.md

# ⚛ L1 Principle — Functional MRI (BOLD) **ID:** `L1-043` · **Status:** ⊙ Testnet (genesis catalog) > **🌐 Domain:** Medical Imaging — *BOLD-contrast brain functional imaging* > **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** 4D BOLD signal > **📡 Carrier:** radio_wave · **🌫 Noise:** gaussian > **⚖ Difficulty (δ):** 10 · **⛓ Block:** 41553372 --- ## 🧠 1. Introduction **Functional MRI (BOLD)** is a **nonlinear inverse problem** whose unknown lives in **4D BOLD signal** space, within the **BOLD-contrast brain functional imaging** 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 · gradient encoding operator; L · T2 star weighting operator; detector accumulates flux over the exposure window. Observations are corrupted by additive Gaussian noise. Existence of the recovered 4D BOLD signal 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 ~= 25); physiological_noise 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 · gradient encoding → L · T2 star weighting → Temporal integration → **y** (detector). ``` y = ∫_t dt `L.T2_star_weighting` `L.gradient_encoding` 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.gradient_encoding` | L · gradient encoding operator | | `L.T2_star_weighting` | L · t2 star weighting operator | | `int.temporal` | Detector accumulates flux over the exposure window | ## 🔬 3. Physics Fingerprint | Property | Value | |---|---| | Domain | Medical Imaging | | Sub domain | BOLD-contrast brain functional imaging | | Carrier | radio_wave | | Problem class | nonlinear_inverse | | Solution space | 4D_BOLD_signal | | Noise model | gaussian | | Integration axis | temporal | | Difficulty delta | 10 | | L dag | 4 | ## 📡 4. Measurement Model Existence of the recovered 4D BOLD signal 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 ~= 25); physiological_noise 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:** `fmri` · **Forward operator:** `fmri_forward` **Center point:** | Parameter | Unit | Value | |---|---|---| | H | px | 64 | | W | px | 64 | | Z | — | 40 | | Tr s | s | 2 | | Te ms | ms | 30 | | Snr db | dB | 15 | | Motion | — | 0 | | N frames | — | 300 | | Physiological noise | — | 0.02 | | Susceptibility artifact | — | 0 | **Allowed bounds:** | Parameter | Unit | Range | |---|---|---| | H | px | 32 | | W | px | 32 | | Z | — | 16 – 100 | | Tr s | s | 0.5 – 5.0 | | Te ms | ms | 15 – 80 | | Snr db | dB | 0.0 – 30.0 | | Motion | — | 0.0 – 0.5 | | N frames | — | 50 – 1000 | | Multi echo bias | — | 0.0 – 0.2 | | Acquisition jitter | — | 0.0 – 0.1 | | Physiological noise | — | 0.0 – 0.2 | | Susceptibility artifact | — | 0.0 – 0.3 | ## 🎯 6. Tolerance (ε) **Center tolerance:** 12.0 | Metric | Range | |---|---| | Psnr db | 5.0 – 45.0 | ## ⚖ 7. Hardness Function Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `500` and δ = `10`. ## 💾 8. Reference Dataset - **primary** · weight 1.0 · IPFS _(not pinned yet)_ ## 9. On-chain Registration - **Chain hash:** `0x04d1d395d338d40f815b99d087248bcfd8137135379330a62c970ebe3addb5ac` - **Chain tx hash:** `0x7108982bdb302508fca241c067411799873d84e30a153268af2d480ce8a17d7f` - **Chain block:** `41553372` --- ## File Mapping This bundle consists of: `L1-043.md`, `L1-043.json`. | File | Role | How to regenerate | |------|------|-------------------| | `L1-043.md` | Source of truth — edit this | Human or LLM | | `L1-043.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 `L1-043`._ _Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._

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Read the attached Markdown (L1-043.md).
Regenerate L1-043.json so every field matches the Markdown.
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L2-043-001 Functional MRI (BOLD) Mismatch-Only Spec

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📋 JSON metadata

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  "status": "testnet",
  "sub_domain": "BOLD-contrast brain functional imaging",
  "title": "Functional MRI (BOLD)"
}

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