Reconstruction Quality Objectives

Definition

A reconstruction-quality objective is a scalar (or scalar field) that scores how good the reconstructed model of the target is, or how good it would become under a hypothetical future measurement — computed cheaply enough to be evaluated over many candidate sensor poses, so that it can serve as the objective function of a view or trajectory planner. It is the objective half of the inspection planning problem. The constraint half — remaining outside a keep-out zone while the state is uncertain — is a separate object, and is where the risk layer enters (see Related Topics).

The top of the tree is scott2003view, which separates model-quality-driven from coverage-driven view planning, and argues for performance-oriented view planning: plan to a specified sampling precision and density, not merely to full-surface coverage. Under that split, three families of objective recur in the literature.

1. Information-theoretic map uncertainty. Score a candidate view by the expected reduction in the uncertainty of a map representation (occupancy grid, octree, truncated signed-distance field). The ancestor is connolly1985determination, which scores a candidate by the projected area of unseen voxels — a volumetric information-gain proxy predating the entropy formulation. vasquezgomez2017view (in the corpus .bib, no source page yet) carries the family title-explicitly: view/state planning for 3D object reconstruction under uncertainty.

2. Optimal-experimental-design accuracy. Score a candidate by a scalar summary of the estimator’s error covariance or Fisher information — the A/D/E/T-optimality family. This is the photogrammetric network-accuracy criterion in modern dress: it predicts the accuracy the reconstruction will attain, rather than the information a measurement will supply. papachristos2019localization (in the .bib, no source page yet) uses the determinant member as the BeliefGain of a belief-propagated path; the criterion and its normalized form are developed on D Optimality and its dual object on Fisher Information Matrix.

3. Geometric coverage-completeness. Score by the fraction of target surface actually observed — no probabilistic model at all. galceran2013survey is the coverage-path-planning canon; our own project metric is the area-ratio form on Inspection Objective Metrics.

The three are not interchangeable: coverage says whether a surface patch was seen, information gain says how much belief a look would buy, and an optimal-design criterion says how accurate the final model will be. A patch can be fully covered and still poorly reconstructed.

A quality objective is also never maximized in isolation: it trades against motion cost. liu2022coverage (in the .bib, no source page yet) is title-explicit that coverage inspection paths are planned with control on measurement uncertainty; for a free-flying inspector the cost side is a propellant/impulse budget, not a shortest-path proxy.

Key Equations

Symbols per notation.md. The symbols below ($\mathcal{V}$, $U$, $C$, $\lambda$, $M$, $Z_v$) are defined inline and are not in the central registry — see Open Questions.

Selection rule. With $\mathcal{V}$ a discretized candidate viewpoint set (the generate-and-test pattern of scott2003view §9.2, as opposed to synthesis over a continuous viewpoint space), $U$ the reconstruction-quality utility and $C$ the motion cost:

$$v^{\star }=\arg \underset{v\in \mathcal{V}}{\max }[U(v)-\lambda C(v)]$$

The three families differ only in $U$:

• Coverage-completeness — $U$ is the observed surface-area fraction. Defined once as eq 7.1 on Inspection Objective Metrics; not duplicated here.
• Optimal design — $U=-D_{\mathrm{opt}}(\mathbf{\Sigma })$ on the propagated error covariance $\mathbf{\Sigma }$. Defined once on D Optimality; not duplicated here.
• Information gain — $U$ is the mutual information between the map $M$ and the measurement $Z_v$ a candidate view would return:

$$U(v)=I(M;Z_v)=H(M)-H(M\mid Z_v)$$

Unsourced form

The mutual-information objective above has no in-corpus source page yet: it is written here as the family’s defining shape, not attributed to any paper we hold. The reading queue that would ground it (notes/reconstruction_quality_planning_objectives.md — Saulnier et al 2020 ICRA on TSDF information gain, Xu et al 2021 RA-L “CRMI” on confidence-rich mutual information) is not yet ingested and has no bibkey. Do not cite this equation until one of those lands.

Source Support

Verified against the corpus .bib:

• scott2003view — the taxonomy anchor: model-quality-driven vs coverage-driven planning; performance-oriented view planning to a precision/density spec; the set-cover / measurability-matrix reduction; generate-and-test vs synthesis.
• connolly1985determination — origin of “next-best-view”; scores candidates by projected unseen area (the volumetric information-gain ancestor). Held as a historical anchor; the primary text is not yet converted, so nothing beyond the term and the two named algorithms should be attributed to it.
• galceran2013survey — coverage-path-planning survey; grounds the coverage-completeness family.
• vasquezgomez2017view · papachristos2019localization · liu2022coverage — bibkeys verified present in Docs/raw/references.bib, but no source page exists yet; cited above only at the level of what their titles assert. Ingest before attributing any method or equation to them.

Not yet in the corpus. The ranked reading queue for this topic (notes/reconstruction_quality_planning_objectives.md) lists the modern objectives — TSDF information gain (Saulnier et al 2020, DOI 10.1109/ICRA40945.2020.9196882), confidence-rich mutual information (Xu et al 2021, DOI 10.1109/LRA.2021.3093023), optimal-experimental-design NBV on a Gaussian-splat scene (Wilson et al 2025 “POp-GS”, arXiv 2503.07819) — none of which has a bibkey. They are named here by title/DOI deliberately; do not invent keys for them.

Related Topics

• Inspection Objective Metrics — our project’s own instantiation: area coverage, the ANCHOR viewpoint score, and versine pointing error. This page is the literature frame that metric sits inside.
• D Optimality · Fisher Information Matrix — the optimal-design family (2), in full.
• Keep Out Zone — the constraint half of the same planning problem: the objective says where it is worth looking from, the KOZ says where it is permitted to fly.
• Chance Constraints · Conditional Value at Risk — how the constraint half is made probabilistic once the state is uncertain. Note the settled position: under a prescribed Gaussian covariance an individual chance constraint reduces exactly to a second-order cone constraint, so CVaR cannot be justified by tractability or by distributional ignorance. Its defensible justification is tail-severity semantics — a coherent risk measure distinguishes how badly a constraint is violated, where a chance constraint sees only whether it was (hakobyan2019risk) — together with robustness to covariance misspecification under the coherent-risk dual / ambiguity-set reading.
• Motion Planning — the layer that turns a chosen viewpoint into an executable trajectory for the free-flying base and arm.

Open Questions

• No manipulator-specific reconstruction objective exists. Every entry in the reading queue is a mobile robot, a UAV, or a turntable. Our free-flying inspector has a fully-actuated 6-DOF base carrying a redundant arm, which introduces two effects none of these objectives model: arm self-occlusion of the camera, and the base attitude’s effect on view quality. This is a research gap, not a citation gap — no ingestion closes it.
• Which family do we adopt? An accuracy-prediction (optimal-design) objective and an information-gain objective answer different questions, and coverage-completeness answers a third. The choice fixes what the inspection deliverable is.
• How does the objective couple to the risk constraint? Two architectures are open: maximize quality subject to a probabilistic keep-out constraint, or fold both into a single risk-sensitive cost. The former keeps the objective interpretable; the latter is what a risk-sensitive planner would naturally do.
• What is $C(v)$ for a free-flyer? The cost term is a propellant/impulse budget over a base-plus-arm maneuver, not a path length. It is not yet defined anywhere in the wiki.
• Registry. $\mathcal{V}$, $U$, $C$, $\lambda$, $M$, $Z_v$ need central registration in notation.md before this page’s equations are cited elsewhere; note $\lambda$ here is a cost-trade weight, not an eigenvalue.