L1 ⚛ L1-390 ⊙ Testnet

Phase Unwrapping (2D modulo-2pi inversion)

Signal Processing · Wrapped-to-absolute phase reconstruction

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⬇ L1-390.md

# ⚛ L1 Principle — Phase Unwrapping (2D modulo-2pi inversion) **ID:** `L1-390` · **Status:** ⊙ Testnet (genesis catalog) > **🌐 Domain:** Signal Processing — *Wrapped-to-absolute phase reconstruction* > **🎯 Problem class:** integer ambiguity inverse · **🧮 Solution space:** 2D scalar phase > **📡 Carrier:** electromagnetic · **🌫 Noise:** wrapped gaussian > **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41555198 --- ## 🧠 1. Introduction **Phase Unwrapping (2D modulo-2pi inversion)** is a **integer ambiguity inverse** whose unknown lives in **2D scalar phase** space, within the **Wrapped-to-absolute phase reconstruction** sub-domain of **Signal Processing**. Measurements consist of electromagnetic field measurements via a **phase wrapping** sensing mechanism. The forward operator applies, in order: L · modulo · 2pi operator; pixel-level spatial averaging on the detector. Observations are corrupted by wrapped gaussian. Well-posed when Itoh condition holds everywhere; ill-posed around phase residues (where gradient circulation is non-zero). LS unwrap smooths but biases; branch-cut correct when cuts routed through low-coherence regions. ## ⚙ 2. Forward Model Physical chain: **x** → L · modulo · 2pi → Spatial integration → **y** (detector). ``` y = ∫_A dA `L.modulo.2pi` x ``` **Measurement DAG:** | Primitive | What it does | |---|---| | `L.modulo.2pi` | L · modulo · 2pi operator | | `int.spatial` | Pixel-level spatial averaging on the detector | ## 🔬 3. Physics Fingerprint | Property | Value | |---|---| | Domain | Signal Processing | | Sub domain | Wrapped-to-absolute phase reconstruction | | Carrier | electromagnetic | | Problem class | integer_ambiguity_inverse | | Solution space | 2D_scalar_phase | | Noise model | wrapped_gaussian | | Integration axis | spatial | | Difficulty delta | 3 | | L dag | 2 | ## 📡 4. Measurement Model Well-posed when Itoh condition holds everywhere; ill-posed around phase residues (where gradient circulation is non-zero). LS unwrap smooths but biases; branch-cut correct when cuts routed through low-coherence regions. | Metric | Value | |---|---| | Metric | RMSE_rad | | Secondary | cycle_slip_count | ## 📏 5. Operating Range (Ω) **Center problem class:** `2d_phase_unwrap` · **Forward operator:** `modulo_2pi` **Center point:** | Parameter | Unit | Value | |---|---|---| | H | px | 512 | | W | px | 512 | | Sigma phi | — | 0.3 | | Coherence floor | — | 0.8 | | Undersampling ratio | — | 1 | | Discontinuity fraction | — | 0 | **Allowed bounds:** | Parameter | Unit | Range | |---|---|---| | H | px | 128 – 4096 | | W | px | 128 – 4096 | | Sigma phi | — | 0.05 – 2.0 | | Coherence floor | — | 0.1 – 1.0 | | Undersampling ratio | — | 0.5 – 1.0 | | Local fringe gradient | — | 0.1 – 3.0 | | Discontinuity fraction | — | 0.0 – 0.3 | ## 🎯 6. Tolerance (ε) **Center tolerance:** 0.1 rad RMSE | Metric | Range | |---|---| | Rmse rad | 0.01 – 2.0 | ## ⚖ 7. Hardness Function Hardness scales as **`epsilon_fn`** on **RMSE_rad**, with κ = `1000` and δ = `3`. ## 💾 8. Reference Dataset - **primary** · weight 1.0 · IPFS _(not pinned yet)_ ## 9. On-chain Registration - **Chain hash:** `0x5e389706d22b79eedc948cfe29ad82ea682ec67a463242fe55e02f2c12bfc889` - **Chain tx hash:** `0x41ae3fa547a068a3780388a3e2a4c3b0c1033c41ffea15b6576d12766d209a6e` - **Chain block:** `41555198` --- ## File Mapping This bundle consists of: `L1-390.md`, `L1-390.json`. | File | Role | How to regenerate | |------|------|-------------------| | `L1-390.md` | Source of truth — edit this | Human or LLM | | `L1-390.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 `L1-390`._ _Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._

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Read the attached Markdown (L1-390.md).
Regenerate L1-390.json so every field matches the Markdown.
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⌥ Hierarchy

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L2-390-001 Phase Unwrapping Noisy-Coherence Spec (InSAR family)

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📋 JSON metadata

{
  "artifact_id": "L1-390",
  "chain_block": 41555198,
  "chain_hash": "0x5e389706d22b79eedc948cfe29ad82ea682ec67a463242fe55e02f2c12bfc889",
  "chain_tx_hash": "0x41ae3fa547a068a3780388a3e2a4c3b0c1033c41ffea15b6576d12766d209a6e",
  "domain": "Signal Processing",
  "hardness_fn": {
    "delta": 3,
    "kappa": 1000,
    "metric": "RMSE_rad",
    "type": "epsilon_fn"
  },
  "initiator_dataset": [
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      "ipfs_cid": null,
      "license_hash": null,
      "name": "primary",
      "weight": 1.0
    }
  ],
  "layer": "L1",
  "observable_profile": {
    "metric": "RMSE_rad",
    "regime": "Well-posed when Itoh condition holds everywhere; ill-posed around phase residues (where gradient circulation is non-zero). LS unwrap smooths but biases; branch-cut correct when cuts routed through low-coherence regions.",
    "secondary": "cycle_slip_count"
  },
  "physics_fingerprint": {
    "L_DAG": 2.0,
    "carrier": "electromagnetic",
    "difficulty_delta": 3,
    "domain": "Signal Processing",
    "integration_axis": "spatial",
    "noise_model": "wrapped_gaussian",
    "primitives": [
      "L.modulo.2pi",
      "int.spatial"
    ],
    "problem_class": "integer_ambiguity_inverse",
    "sensing_mechanism": "phase_wrapping",
    "solution_space": "2D_scalar_phase",
    "sub_domain": "Wrapped-to-absolute phase reconstruction",
    "title": "Phase Unwrapping (2D modulo-2pi inversion)"
  },
  "size_tiers": {
    "allowed_forward_operators": [
      "modulo_2pi",
      "modulo_2pi_with_coherence"
    ],
    "allowed_omega_dimensions": [
      "H",
      "W",
      "sigma_phi",
      "discontinuity_fraction",
      "coherence_floor",
      "local_fringe_gradient",
      "undersampling_ratio"
    ],
    "allowed_problem_classes": [
      "2d_phase_unwrap",
      "3d_phase_unwrap",
      "temporal_unwrap_insar"
    ],
    "center_spec": {
      "epsilon_fn_center": "0.1 rad RMSE",
      "forward_operator": "modulo_2pi",
      "input_format": "wrapped_only",
      "omega": {
        "H": 512,
        "W": 512,
        "coherence_floor": 0.8,
        "discontinuity_fraction": 0.0,
        "sigma_phi": 0.3,
        "undersampling_ratio": 1.0
      },
      "problem_class": "2d_phase_unwrap"
    },
    "epsilon_bounds": {
      "rmse_rad": [
        0.01,
        2.0
      ]
    },
    "omega_bounds": {
      "H": [
        128,
        4096
      ],
      "W": [
        128,
        4096
      ],
      "coherence_floor": [
        0.1,
        1.0
      ],
      "discontinuity_fraction": [
        0.0,
        0.3
      ],
      "local_fringe_gradient": [
        0.1,
        3.0
      ],
      "sigma_phi": [
        0.05,
        2.0
      ],
      "undersampling_ratio": [
        0.5,
        1.0
      ]
    }
  },
  "staked_pwm": 0.0,
  "status": "testnet",
  "sub_domain": "Wrapped-to-absolute phase reconstruction",
  "title": "Phase Unwrapping (2D modulo-2pi inversion)"
}

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