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arxiv: 2605.02667 · v1 · submitted 2026-05-04 · 💻 cs.RO · cs.CV

Recognition: 3 theorem links

· Lean Theorem

AnchorD: Metric Grounding of Monocular Depth Using Factor Graphs

Simon Dorer , Martin B\"uchner , Nick Heppert , Abhinav Valada
Authors on Pith no claims yet

Pith reviewed 2026-05-08 18:14 UTC · model grok-4.3

classification 💻 cs.RO cs.CV
keywords monocular depth estimationfactor graph optimizationdepth groundingnon-Lambertian surfacesrobotic depth sensingaffine alignmenttraining-free methodsbenchmark dataset
0
0 comments X

The pith

A training-free factor graph method aligns monocular depth priors patch-wise to raw sensor readings to recover accurate metric depth.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proposes a framework that takes depth predictions from monocular foundation models, which capture good local structure but lack correct scale and offset, and anchors them to actual sensor depth measurements. It achieves this through factor graph optimization that computes an affine transform separately for each image patch. This grounds the predictions in real-world metric units while keeping fine geometric details and sharp discontinuities intact. The approach matters for robotics because current depth sensors produce errors on transparent, shiny, and other non-Lambertian surfaces, and the method lets existing models be used directly without any retraining or fine-tuning. Evaluations on a new benchmark with dense ground truth across challenging scenes confirm consistent accuracy gains over either source alone.

Core claim

The central claim is that monocular depth estimation priors from foundation models contain sufficiently accurate local geometric structure to be grounded in metric real-world depth via patch-wise affine alignment performed through factor graph optimization. This process anchors the predictions to raw sensor depth without requiring any training, preserves fine-grained structure and discontinuities, and yields improved depth maps suitable for robotic tasks. The paper supports the claim with evaluations across diverse sensors and domains plus a new benchmark dataset that provides dense scene-wide ground truth depth even in the presence of non-Lambertian objects, obtained via matte reflection sp

What carries the argument

Factor graph optimization performing patch-wise affine alignment between monocular depth priors and raw sensor measurements

If this is right

  • Robotic manipulation and navigation systems gain usable metric depth from monocular models without any model retraining.
  • Depth accuracy improves specifically on transparent, specular, and other surfaces where sensors fail but monocular priors retain structure.
  • The new benchmark enables direct comparison of depth methods under realistic non-Lambertian conditions using dense ground truth.
  • The alignment step works with a variety of existing depth sensors and foundation models across indoor and outdoor domains.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same patch-wise grounding idea could be applied to correct other visual priors such as surface normals or semantic labels when paired with sparse sensor cues.
  • Inserting the factor graph step into existing SLAM or visual odometry pipelines would likely increase robustness to sensor failures on reflective objects.
  • Because the method is training-free and modular, it offers a lightweight way to adapt future depth foundation models to new sensor hardware without re-optimization.

Load-bearing premise

Monocular depth priors from foundation models contain local geometric structures accurate enough that a simple affine transform per patch can align them to sensor data without introducing new distortions or erasing discontinuities.

What would settle it

Applying the patch-wise alignment on the introduced benchmark and measuring that the resulting depth maps match the matte-sprayed multi-camera ground truth no better than the unaligned monocular predictions or the raw sensor data alone.

Figures

Figures reproduced from arXiv: 2605.02667 by Abhinav Valada, Martin B\"uchner, Nick Heppert, Simon Dorer.

Figure 1
Figure 1. Figure 1: Approach overview. We present AnchorD as a training-free method for grounding monocular depth predictions (MDE) using real-world sensor depth using a factor graph approach, which yields grounded, highly-accurate depth predictions on typical non-Lambertian objects. corresponding RGB image typically retains informative cues, such as edges, textures, and semantic structure. Therefore, RGB-guided depth groundi… view at source ↗
Figure 2
Figure 2. Figure 2: Dataset camera setup. We employ three cameras: two RGB-D sensors, an Azure Kinect DK (green), and an Intel RealSense D415 (yellow), as well as a ZED2 stereo camera (blue). The point clouds captured by all cameras are registered in a shared reference frame and fused to generate ground truth depth view at source ↗
Figure 3
Figure 3. Figure 3: Dataset collection. We visualize one of the recorded scenes with the industrial-grade matte diffuser spray applied. masks using Grounding DINO [35] and SAM2 [36], [37], using the text prompt foreground objects to obtain class-agnostic object segments. IV. TECHNICAL APPROACH In this section, we introduce AnchorD, a novel factor-graph￾based optimization method for zero-shot monocular depth completion in scen… view at source ↗
Figure 4
Figure 4. Figure 4: Ground truth depth generation process. Depth obtained from the raw scene (left), after applying a diffuse spray to non-Lambertian objects in the scene (middle), and after fusing the depth maps from all three cameras in a common reference frame (right). (MDE) foundation model f mde, e.g. DepthAnything3 [27]: Dmde = f mde(I) ∈ R H×W >0 (1) As it is common in monocular depth estimation, MDE models only provid… view at source ↗
Figure 5
Figure 5. Figure 5: Factor graph formulation. Dense per-pixel depth variables Dij and patch-wise affine parameters (si, bi) are jointly optimized. Ternary MDE factors ϕmde align depth to the monocular prediction within each patch, unary sensor factors ϕ sen enforce metric consistency, and binary logarithmic slope factors ϕ slp preserve relative depth structure across pixels and patch boundaries. For simplicity, only four pixe… view at source ↗
Figure 6
Figure 6. Figure 6: Qualitative comparison of ablations. We compare the outputs of the ablated variants (w/ and w/o patches) to AnchorD where we additionally apply Gaussian smoothing on the patch-wise affine alignment parameters. p ∈ Ω′ i we define the factor as ϕ mde(Dp, si , bi) = ρδ1 view at source ↗
Figure 7
Figure 7. Figure 7: Qualitative results on SprayD. We compare AnchorD (ours) with metric Depth Anything3 [27] as well as affine scaling in the three rightmost columns. For reference, we display the raw sensor depth and ground truth depth in the two leftmost columns. All depth images along each row feature a consistent color scaling. Metrics: As standard for depth completion, we evaluate quan￾titative performance using mean ab… view at source ↗
Figure 8
Figure 8. Figure 8: Qualitative results on ScanNet++. We compare the point cloud aggregation quality when aggregating the depth over multiple camera poses stemming from the raw iPhone LiDAR observations (left), a metric foundation model (DepthAnything3) taking RGB as input (middle), and AnchorD (ours) (right). MAE [m] 0.125 0.25 0.5 1 2 0.625 1.25 2.5 5 10 0.011 0.013 0.015 0.017 λsen λmde Uncertainty 0.00 0.02 0.04 0.06 0.08 view at source ↗
Figure 9
Figure 9. Figure 9: Model Sensitivity and Uncertainty. We display a parameter sensitivity matrix delineating the influence of various combinations of λ mde and λ sen on the resulting global MAE (left) and a normalized residual map representing our model’s uncertainty across image regions for the sample from view at source ↗
read the original abstract

Dense and accurate depth estimation is essential for robotic manipulation, grasping, and navigation, yet currently available depth sensors are prone to errors on transparent, specular, and general non-Lambertian surfaces. To mitigate these errors, large-scale monocular depth estimation approaches provide strong structural priors, but their predictions can be potentially skewed or mis-scaled in metric units, limiting their direct use in robotics. Thus, in this work, we propose a training-free depth grounding framework that anchors monocular depth estimation priors from a depth foundation model in raw sensor depth through factor graph optimization. Our method performs a patch-wise affine alignment, locally grounding monocular predictions in metric real-world depth while preserving fine-grained geometric structure and discontinuities. To facilitate evaluation in challenging real-world conditions, we introduce a benchmark dataset with dense scene-wide ground truth depth in the presence of non-Lambertian objects. Ground truth is obtained via matte reflection spray and multi-camera fusion, overcoming the reliance on object-only CAD-based annotations used in prior datasets. Extensive evaluations across diverse sensors and domains demonstrate consistent improvements in depth performance without any (re-)training. We make our implementation publicly available at https://anchord.cs.uni-freiburg.de.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper proposes AnchorD, a training-free framework for metric grounding of monocular depth estimates. It anchors priors from depth foundation models to raw sensor depth via factor graph optimization using patch-wise affine alignment, with the goal of preserving fine-grained geometric structure and discontinuities. The authors introduce a new benchmark dataset providing dense scene-wide ground truth depth for non-Lambertian objects, constructed via matte reflection spray and multi-camera fusion. They report consistent depth performance improvements across diverse sensors and domains without retraining and release the implementation publicly.

Significance. If the central claims hold, the work addresses a practical robotics challenge by fusing metric sensor data with structural priors from foundation models without requiring retraining or fine-tuning. This could enable more reliable depth for manipulation and navigation on challenging surfaces. The new benchmark with dense GT is a notable contribution over prior object-only CAD annotations, and the public code supports reproducibility. The approach's training-free nature is a strength for rapid deployment.

major comments (3)
  1. [§3.2] §3.2 (Factor Graph Optimization): The patch-wise affine alignment formulation does not specify the exact factors or regularization terms used to couple overlapping patches while respecting depth discontinuities. This is load-bearing for the central claim, as non-affine local errors common in foundation models on specular surfaces could introduce distortions unless edge-aware constraints are explicitly enforced.
  2. [§4.3] §4.3 (Quantitative Evaluation): The reported consistent improvements lack region-specific error analysis (e.g., on discontinuities or non-Lambertian patches) and convergence diagnostics for the factor graph optimization. Without these, it is unclear whether the alignment preserves structure or merely averages errors, undermining the guarantee of no new artifacts.
  3. [§5] §5 (Benchmark Construction): The matte spray + multi-camera fusion method for dense GT may smooth fine discontinuities, as the spray alters surface properties. Validation against unsprayed high-resolution references or ablation on discontinuity preservation is needed to confirm the benchmark's superiority for evaluating the method's claims.
minor comments (2)
  1. [Figure 4] Figure 4: Add zoomed insets on discontinuity regions to visually support the preservation claim.
  2. [Related Work] Related Work: Include recent comparisons to other sensor-fusion depth methods (e.g., those using similar optimization frameworks) to better contextualize novelty.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed feedback. We address each major comment below, clarifying the existing formulation where possible and committing to revisions that strengthen the manuscript without altering its core claims.

read point-by-point responses

  1. Referee: [§3.2] §3.2 (Factor Graph Optimization): The patch-wise affine alignment formulation does not specify the exact factors or regularization terms used to couple overlapping patches while respecting depth discontinuities. This is load-bearing for the central claim, as non-affine local errors common in foundation models on specular surfaces could introduce distortions unless edge-aware constraints are explicitly enforced.

    Authors: Section 3.2 presents the factor graph with unary factors that minimize the affine alignment error between each monocular patch and the corresponding sensor depth measurements, and binary factors that enforce consistency of the affine parameters across overlapping patches. An additional regularization term penalizes deviations from the identity affine transform while being modulated by image gradients to respect discontinuities. We acknowledge that the exact factor definitions and the gradient-based weighting were described at a high level rather than with full equations. In the revised manuscript we will insert the precise mathematical expressions for all factors, the overlap consistency term, and the edge-aware regularization, ensuring the formulation is fully specified and reproducible. revision: yes

  2. Referee: [§4.3] §4.3 (Quantitative Evaluation): The reported consistent improvements lack region-specific error analysis (e.g., on discontinuities or non-Lambertian patches) and convergence diagnostics for the factor graph optimization. Without these, it is unclear whether the alignment preserves structure or merely averages errors, undermining the guarantee of no new artifacts.

    Authors: We agree that aggregate metrics alone leave open the question of whether structure is preserved. The revised evaluation section will report separate error statistics on non-Lambertian regions (identified via intensity variance and sensor confidence) and on depth discontinuity boundaries (extracted via Canny edges on the reference depth). We will also add convergence plots showing the factor-graph cost and per-iteration depth change, together with a qualitative comparison of edge sharpness before and after optimization. These additions will demonstrate that the method reduces error without smoothing or introducing new artifacts. revision: yes

  3. Referee: [§5] §5 (Benchmark Construction): The matte spray + multi-camera fusion method for dense GT may smooth fine discontinuities, as the spray alters surface properties. Validation against unsprayed high-resolution references or ablation on discontinuity preservation is needed to confirm the benchmark's superiority for evaluating the method's claims.

    Authors: The spray is applied in a minimal, uniform layer specifically chosen to reduce specular reflection while preserving macroscopic geometry; the multi-camera fusion further recovers fine detail through photometric consistency across views. Direct unsprayed high-resolution ground truth for the identical real-world scenes is not available, as the non-Lambertian surfaces prevent reliable capture without the spray. In the revision we will expand the benchmark description with a limitations paragraph acknowledging this trade-off and will add a synthetic ablation that applies an analogous surface modification to rendered scenes, quantifying discontinuity preservation before and after the simulated spray. This will support the claim that the benchmark remains superior to prior CAD-only annotations for evaluating metric grounding on challenging surfaces. revision: partial

Circularity Check

0 steps flagged

No circularity: independent optimization on external priors and sensor data

full rationale

The derivation consists of a factor-graph optimization that takes two independent inputs—monocular depth priors from a foundation model and raw metric sensor depths—and produces an aligned output via patch-wise affine transforms. No equation or claim reduces to a tautology, a fitted parameter renamed as prediction, or a self-citation chain. The benchmark construction (matte spray + multi-camera fusion) is a data-acquisition procedure external to the algorithm. All load-bearing steps remain falsifiable against held-out sensor measurements and are not self-referential by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on abstract only; no explicit free parameters, invented entities, or ad-hoc axioms are detailed. The approach assumes standard properties of factor graphs and affine transformations in depth alignment.

axioms (1)
  • domain assumption Factor graph optimization can locally align monocular depth structure to sensor measurements via affine transforms without global distortion.
    Invoked in the description of the patch-wise grounding process.

pith-pipeline@v0.9.0 · 5519 in / 1233 out tokens · 57200 ms · 2026-05-08T18:14:16.438771+00:00 · methodology

discussion (0)

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Cost (Jcost = ½(x+x⁻¹)−1) washburn_uniqueness_aczel echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    ϕ_slp(D_p, D_q) = ρ_δ2( [log D_p − log D_q] − [log D'^mde_p − log D'^mde_q] )

  • RS framework as a whole — zero adjustable parameters reality_from_one_distinction unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    λ_mde = 2.5, λ_sen = 0.5, λ_slp = 1.0, δ_1 = 0.002, δ_2 = 0.01, k=64; hyperparameter sensitivity matrix in Fig. 9

What do these tags mean?
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supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

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