Inspection Objective Metrics

Definition

The inspection task is judged by three project-original metrics: area coverage (how much of the target surface the camera has seen), a per-viewpoint quality score (used only to choose aim points in ANCHOR mode), and pointing error (how well the camera axis tracks its desired direction).

All three read off a single geometric ground truth. The target surface is a triangular mesh, and a per-triangle hit counter $h_{\mathrm{tri}}$ is incremented whenever that triangle is struck by a ray cast through the camera frustum within the working depth band. Coverage and the novelty term of the score share this same counter, so novelty and coverage cannot disagree about what has been seen.

The two camera modes matter here (see coordinated_control and terminology): in POSE mode the camera looks directly at the scheduled surface point, achieving complete coverage, and the viewpoint score is inert; the scorer runs only in ANCHOR mode, which is retained for risk-aware planning.

Key Equations

Symbols per notation.md. The metric-specific symbols below ($A_i$, $\mathcal{S}$, $h$, ${\sigma }_i$, $w_i$, ${\sigma }_n$) are defined inline and are not yet in the central registry — see the inconsistency note and Open Questions.

Area coverage (eq 7.1) — the true surface-area fraction seen at least once. With $A_i$ the area of triangle $i$ and $\mathcal{S}=\{i:h_{\mathrm{tri}}\ge 1\}$ the set of triangles struck at least once, coverage is an area ratio and is therefore independent of how candidate viewpoints were sampled:

$$\text{coverage}=\frac{{\sum }_{i\in \mathcal{S}}{A}_i}{{\sum }_j{A}_j}$$

Viewpoint score (eq 7.2, ANCHOR mode only — inert in POSE) — normalized quality terms ${\sigma }_i\in [0,1]$ combined by weighted product with weights $w_i$ (all $1.0$ currently):

$$\Omega =\prod _i{\sigma }_i^{w_i}$$

The novelty term rewards unseen surface, ${\sigma }_n=1/(1+h)$, where $h$ is the same per-triangle hit count $h_{\mathrm{tri}}$ that feeds coverage.

Inconsistency — symbol collision on $\Omega$ and ${\sigma }_i$

The canonical equation sheet reuses $\Omega$ for both the ANCHOR viewpoint score (eq 7.2, above) and the singularity-free stability region $\Omega =\{\mathbf{z}:{\sigma }_{\min }({\mathbf{J}}_{{\nu }_e}^{\oplus })>0\}$ of the Lyapunov analysis (see coordinated_control_lyapunov_stability). Likewise ${\sigma }_i$ is a normalized score term here but an SVD singular value in the singularity sections. Reproduced source-faithfully; disambiguate by context until a central rename lands.

Pointing error (eq 7.3) — reported as the versine angle $1-\cos \theta$ between the camera axis and its desired direction, never a raw vector difference. Defined once in terminology (“Versine pointing error”); not duplicated here.

Source Support

• galceran2013survey — coverage path planning survey; grounds the area-coverage objective.
• scott2003view — view planning for automated 3D inspection/reconstruction; grounds the viewpoint-scoring / next-best-view context.

The specific forms above are project-original (final.tex §4.2.2 for the per-triangle marking and area coverage; final.tex §4.1.1 for the weighted-product score; the novelty form ${\sigma }_n=1/(1+h)$ from the project’s sota_Jun16 §3–4), derived here from the canonical equation sheet (eqs 7.1–7.3).

Related Topics

• d_optimality — the belief-gain NBV metric; the risk-aware sibling of the ANCHOR viewpoint score.
• coordinated_control — POSE mode of this controller achieves the complete coverage the metric measures; pointing error quantifies its EE-pointing accuracy.
• manipulability_measure — another scalar quality measure, and part of the $\sigma$-symbol family flagged above.

Open Questions

• The ANCHOR viewpoint score is inert in the current POSE-mode pipeline. Which normalized terms ${\sigma }_i$ beyond novelty enter it, and how are the weights $w_i$ to be set for risk-aware planning?
• Coverage and novelty share one hit counter $h_{\mathrm{tri}}$. How do the working depth band and frustum ray density bias the surface-area estimate?
• The metric symbols ($\Omega$-score, ${\sigma }_n$, $A_i$, $h$, $\mathcal{S}$, $w_i$) are not yet in notation.md; they need central registration and a rename to clear the $\Omega$/${\sigma }_i$ collision.