L1 ⚛ L1-098 ⊙ Testnet

Photometric Stereo (shape-from-shading with multiple light directions)

Depth Imaging · Surface-normal recovery from shading

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⬇ L1-098.md

# ⚛ L1 Principle — Photometric Stereo (shape-from-shading with multiple light directions) **ID:** `L1-098` · **Status:** ⊙ Testnet (genesis catalog) > **🌐 Domain:** Depth Imaging — *Surface-normal recovery from shading* > **🎯 Problem class:** linear inverse · **🧮 Solution space:** 2D normal and depth > **📡 Carrier:** photon · **🌫 Noise:** gaussian > **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41547812 --- ## 🧠 1. Introduction **Photometric Stereo (shape-from-shading with multiple light directions)** is a **linear inverse problem** whose unknown lives in **2D normal and depth** space, within the **Surface-normal recovery from shading** sub-domain of **Depth Imaging**. Measurements consist of photons collected by an optical detector via a **multi light shading** sensing mechanism. The forward operator applies, in order: S · scan · light direction operator; L · reflect lambertian operator; L · normal ls operator; pixel-level spatial averaging on the detector. Observations are corrupted by additive Gaussian noise. With K >= 3 non-coplanar light directions and Lambertian BRDF, the linear system has rank 3 and is well-conditioned. Degrades for specular highlights (outlier fraction), cast and attached shadows (reduced effective K), non-Lambertian materials; robust methods (RANSAC, rank-3 SVD, learned PS) handle outliers up to ~30% specular fraction. ## ⚙ 2. Forward Model Physical chain: **x** → S · scan · light direction → L · reflect lambertian → L · normal ls → Spatial integration → **y** (detector). ``` y = ∫_A dA `L.normal_ls` `L.reflect_lambertian` `S.scan.light_direction` x + n, n ~ 𝒩(0, σ²) ``` **Measurement DAG:** | Primitive | What it does | |---|---| | `S.scan.light_direction` | S · scan · light direction operator | | `L.reflect_lambertian` | L · reflect lambertian operator | | `L.normal_ls` | L · normal ls operator | | `int.spatial` | Pixel-level spatial averaging on the detector | ## 🔬 3. Physics Fingerprint | Property | Value | |---|---| | Domain | Depth Imaging | | Sub domain | Surface-normal recovery from shading | | Carrier | photon | | Problem class | linear_inverse | | Solution space | 2D_normal_and_depth | | Noise model | gaussian | | Integration axis | angular | | Difficulty delta | 3 | | L dag | 3.2 | ## 📡 4. Measurement Model With K >= 3 non-coplanar light directions and Lambertian BRDF, the linear system has rank 3 and is well-conditioned. Degrades for specular highlights (outlier fraction), cast and attached shadows (reduced effective K), non-Lambertian materials; robust methods (RANSAC, rank-3 SVD, learned PS) handle outliers up to ~30% specular fraction. | Metric | Value | |---|---| | Metric | normal_MAE_deg | | Secondary | depth_RMSE_mm | ## 📏 5. Operating Range (Ω) **Center problem class:** `photometric_stereo_normal` · **Forward operator:** `lambertian_multi_light` **Center point:** | Parameter | Unit | Value | |---|---|---| | H | px | 512 | | W | px | 512 | | K lights | — | 10 | | Photon count | — | 500 | | Ambient residual | — | 0 | | Specular fraction | — | 0 | | Cast shadow fraction | — | 0 | | Light direction error | — | 0 | **Allowed bounds:** | Parameter | Unit | Range | |---|---|---| | H | px | 128 – 2048 | | W | px | 128 – 2048 | | K lights | — | 3 – 96 | | Photon count | — | 50 – 10000 | | Ambient residual | — | 0.0 – 0.3 | | Inter reflections | — | 0.0 – 0.3 | | Specular fraction | — | 0.0 – 0.5 | | Cast shadow fraction | — | 0.0 – 0.4 | | Light direction error | — | 0.0 – 0.05 | ## 🎯 6. Tolerance (ε) **Center tolerance:** 8.0 deg normal MAE | Metric | Range | |---|---| | Normal mae deg | 2.0 – 35.0 | ## ⚖ 7. Hardness Function Hardness scales as **`epsilon_fn`** on **normal_MAE_deg**, with κ = `400` and δ = `3`. ## 💾 8. Reference Dataset - **primary** · weight 1.0 · IPFS _(not pinned yet)_ ## 9. On-chain Registration - **Chain hash:** `0x9dd147d10f81e507e5d5aaabda58501a20e80830aee9eb85908e5564f18491f6` - **Chain tx hash:** `0x75b422eecd7831cc562bfdad63c58077a1029471d3e78f3712c030979d9dbb7b` - **Chain block:** `41547812` --- ## File Mapping This bundle consists of: `L1-098.md`, `L1-098.json`. | File | Role | How to regenerate | |------|------|-------------------| | `L1-098.md` | Source of truth — edit this | Human or LLM | | `L1-098.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 `L1-098`._ _Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._

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Read the attached Markdown (L1-098.md).
Regenerate L1-098.json so every field matches the Markdown.
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L2-098-001 Photometric Stereo Mismatch-Only Spec

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📋 JSON metadata

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  "chain_block": 41547812,
  "chain_hash": "0x9dd147d10f81e507e5d5aaabda58501a20e80830aee9eb85908e5564f18491f6",
  "chain_tx_hash": "0x75b422eecd7831cc562bfdad63c58077a1029471d3e78f3712c030979d9dbb7b",
  "domain": "Depth Imaging",
  "hardness_fn": {
    "delta": 3,
    "kappa": 400,
    "metric": "normal_MAE_deg",
    "type": "epsilon_fn"
  },
  "initiator_dataset": [
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      "license_hash": null,
      "name": "primary",
      "weight": 1.0
    }
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  "layer": "L1",
  "observable_profile": {
    "metric": "normal_MAE_deg",
    "regime": "With K \u003e= 3 non-coplanar light directions and Lambertian BRDF, the linear system has rank 3 and is well-conditioned. Degrades for specular highlights (outlier fraction), cast and attached shadows (reduced effective K), non-Lambertian materials; robust methods (RANSAC, rank-3 SVD, learned PS) handle outliers up to ~30% specular fraction.",
    "secondary": "depth_RMSE_mm"
  },
  "physics_fingerprint": {
    "L_DAG": 3.2,
    "carrier": "photon",
    "difficulty_delta": 3,
    "domain": "Depth Imaging",
    "integration_axis": "angular",
    "noise_model": "gaussian",
    "primitives": [
      "S.scan.light_direction",
      "L.reflect_lambertian",
      "L.normal_ls",
      "int.spatial"
    ],
    "problem_class": "linear_inverse",
    "sensing_mechanism": "multi_light_shading",
    "solution_space": "2D_normal_and_depth",
    "sub_domain": "Surface-normal recovery from shading",
    "title": "Photometric Stereo (shape-from-shading with multiple light directions)"
  },
  "size_tiers": {
    "allowed_forward_operators": [
      "lambertian_multi_light",
      "general_brdf_ps",
      "near_point_light_ps"
    ],
    "allowed_omega_dimensions": [
      "H",
      "W",
      "K_lights",
      "photon_count",
      "specular_fraction",
      "cast_shadow_fraction",
      "ambient_residual",
      "light_direction_error",
      "inter_reflections"
    ],
    "allowed_problem_classes": [
      "photometric_stereo_normal",
      "uncalibrated_ps",
      "shape_from_shading"
    ],
    "center_spec": {
      "epsilon_fn_center": "8.0 deg normal MAE",
      "forward_operator": "lambertian_multi_light",
      "input_format": "K_images_plus_light_directions",
      "omega": {
        "H": 512,
        "K_lights": 10,
        "W": 512,
        "ambient_residual": 0.0,
        "cast_shadow_fraction": 0.0,
        "light_direction_error": 0.0,
        "photon_count": 500,
        "specular_fraction": 0.0
      },
      "problem_class": "photometric_stereo_normal"
    },
    "epsilon_bounds": {
      "normal_mae_deg": [
        2.0,
        35.0
      ]
    },
    "omega_bounds": {
      "H": [
        128,
        2048
      ],
      "K_lights": [
        3,
        96
      ],
      "W": [
        128,
        2048
      ],
      "ambient_residual": [
        0.0,
        0.3
      ],
      "cast_shadow_fraction": [
        0.0,
        0.4
      ],
      "inter_reflections": [
        0.0,
        0.3
      ],
      "light_direction_error": [
        0.0,
        0.05
      ],
      "photon_count": [
        50,
        10000
      ],
      "specular_fraction": [
        0.0,
        0.5
      ]
    }
  },
  "staked_pwm": 0.0,
  "status": "testnet",
  "sub_domain": "Surface-normal recovery from shading",
  "title": "Photometric Stereo (shape-from-shading with multiple light directions)"
}

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