Pith. sign in

REVIEW 4 major objections 5 minor 78 references

Self-supervised features carry a hidden map of 3D space

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-02 02:43 UTC pith:OQLDCQDV

load-bearing objection Useful probe kit and some honest theory, but the 'emergence' headline rests on three unrepresentative ScanNet scenes and oracle-based selection, so read it for the methodology, not the claim. the 4 major comments →

arxiv 2607.14228 v1 pith:OQLDCQDV submitted 2026-07-15 cs.CV cs.AI

SeeSE3: Emergence of 3D Space in Vision Features

Caroline Chen , Sayna Ebrahimi , Fedor Kitashov , Ming-Hsuan Yang , Leonidas Guibas , Viorica P\u{a}tr\u{a}ucean , Maks Ovsjanikov This is my paper
classification cs.CV cs.AI
keywords vision foundation modelsSE(3) geometryself-supervised learningPoincaré adapterlatent space navigationgeometric probingvisual odometry
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Self-supervised vision models—passive observers that never see depth, pose, or 3D ground truth—can still develop an internal geometry that mirrors three-dimensional Euclidean space, provided the right probe is used. The paper introduces the 'Poincaré task': can a motionless observer discover the structure of SE(3) from visual statistics alone? It answers yes in the affirmative: after a lightweight Siamese adapter unrolls the raw (curved) feature manifold, feature differences become nearly linear in the six-dimensional Lie algebra of rigid motion, yielding held-out R² ≈ 0.65 for DINOv2 on ScanNet. This implies a 'Visual Grid Code'—a latent subspace locally isomorphic to SE(3)—and enables a proof-of-concept latent-space navigation that moves features by vector arithmetic and retrieves physically nearby frames. The practical payoff is that visual odometry and localization may be performed without explicit 3D reconstruction.

Core claim

The paper's central claim is that a motionless, passively trained vision transformer's latent manifold contains a locally Euclidean submanifold isomorphic to SE(3), even though no raw feature space is linearly equivariant. The authors prove that whenever the posed feature map has rank-6 Jacobian, a local linear readout exists; a nonlinear 'Poincaré adapter' removes the curvature obstruction, leaving only a residual from SE(3)'s non-commutativity. They validate this with a Siamese MLP plus linear readout, achieving held-out R² ≈ 0.65 for DINOv2 on ScanNet. They also show rotation is easier to decode than translation because rotational flow is depth-independent while translational flow scales

What carries the argument

The central machinery is the Poincaré adapter: a Siamese two-layer MLP φ applied independently to each frame's feature vector, followed by a bias-free linear readout W, trained so that the adapted-feature difference maps to the 6D Lie-algebra twist: ΔP ≈ W(φ(z_{t+s}) − φ(z_t)). This 'unrolls' the curved feature manifold into a flat coordinate patch, converting nonlinear latent displacements into linear pose increments. The theoretical core is the posed feature map f: SE(3) → R^d, its Jacobian J(g), and the condition rank(J(g)) = 6; success requires the Jacobian to vary little across the pose region (low manifold curvature), and a residual from the Baker–Campbell–Hausdorff bracket remains for

Load-bearing premise

The central argument assumes the feature map from camera pose to feature vector is a smooth six-dimensional surface with full rank-6 Jacobian and no pose collisions—i.e., that distinct viewpoints never land on the same feature—an injectivity that the paper supports only indirectly via intrinsic-dimension estimates and never verifies directly.

What would settle it

Use a static scene with a large textureless wall (e.g., a white-walled room from TUM RGB-D). Collect frames from a dense grid of camera poses that all view the same blank region, and compute the mutual k-nearest-neighbor alignment M1 and intrinsic dimension in that region. If two distinct poses yield nearly identical feature vectors, the local intrinsic dimension will collapse below 6 and an adapter trained on those frames should show sharply lower R²; this would invalidate the embedded-submanifold interpretation in exactly the region where the paper's theory depends on injectivity.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Visual odometry and localization can run directly in adapted latent space: inverting the linear map with the pseudo-inverse W† predicts destination features, and nearest-neighbor retrieval finds physically near frames without any 3D reconstruction.
  • Geometric awareness is an emergent, scale-dependent property: zero-shot pose recovery on unseen rooms improves monotonically with the number of training rooms and does not plateau at 200 rooms, so larger passive observation is predicted to yield even cleaner geometry.
  • Probing design is decisive: Lie-algebra linearization, relative pose targets, and Siamese processing are each necessary; raw difference or absolute-pose formulations fail, indicating latent geometry is relational and only becomes explicit through the right readout.
  • Rotation vs translation magnitude asymmetry is structural: rotation is decodable in 88% of hyperparameter configurations while translation magnitude is positive in only 27%, because translational flow depends on unknown depth; any downstream task should treat metric scale as a prior-restored quantity.
  • The 'Visual Grid Code' gives a concrete interpretation of why self-distillation and large batches (batch size 192 vs 128) are critical for geometry: they allow a pose-aligned manifold to form; the paper's ablation shows topology can survive without them, but flat geometry does not.

Where Pith is reading between the lines

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

  • We infer that because the adapted feature space is homogeneous and transferable across rooms with only a linear refit of the inverse map, it could be treated as a universal 'world grid' for embodied agents; testing this would require closed-loop navigation, which the authors do not run.
  • We infer a testable developmental prediction from the rotation-before-translation difficulty hierarchy: if such geometry is a general property of passive vision, models (and possibly infants) should show reliable orientation recovery before metric-scale recovery.
  • We infer that the gap between train-set R² above 0.9 and test R² 0.65 is likely a generalization gap, not a structural ceiling; a concrete extension is to check whether increasing adapter capacity or training data for the adapter closes this gap.
  • We infer that extending the Poincaré adapter to dynamic scenes would require a temporal disentangling module to separate scene motion from camera motion; the current formulation offers no mechanism for distinguishing 'changes of state' from 'changes of position'.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper asks whether self-supervised vision models, which are never trained with explicit 3D supervision or agency, nevertheless organize their latent spaces according to the geometry of SE(3). It introduces four metrics: M1 (mutual k-NN alignment between feature and pose neighborhoods), M2 (intrinsic dimensionality), M3 (linear readout of pose changes from feature differences), and M4 (a 'Poincaré Adapter', a Siamese MLP followed by a linear readout). The main empirical claim is that the Poincaré Adapter reaches test R2 ≈ 0.65 for DINOv2 on three ScanNet scenes, interpreted as evidence of a locally Euclidean SE(3)-isomorphic submanifold ('Visual Grid Code'). The paper also derives local and global decodability theorems, reports a rotation–translation difficulty asymmetry, presents training ablations, and applies the adapter to latent-space navigation via pseudo-inverse vector arithmetic.

Significance. If the headline result were representative and the protocol free of selection bias, the paper would be significant: it would demonstrate that passive self-supervised encoders contain a decodable, transferable 3D structure, with implications for probing, navigation world models, and theories of spatial representation. The manuscript has genuine strengths: M1 is a clean training-free metric; the cross-room generalization experiment (Fig. 5) is a properly split train/test regression and provides nontrivial evidence of a transferable subspace; and the appendices contain extensive, honest ablations (A.6, A.8) that document fragility rather than hiding it. However, the central 'remarkably strong correlation' claim rests on a small, unspecified subset of scenes and on oracle-based layer/hyperparameter selection, and the theoretical framework is partly tautological. These issues are load-bearing for the paper's interpretation, though they are addressable by re-analysis or by substantially tempering the claims.

major comments (4)
  1. [§5, Fig. 4, Table 2] The headline R2 ≈ 0.65 is 'averaged over 3 scenes from ScanNet', but no sampling or selection procedure for these three scenes is given in §5, App. A.2, or App. A.5. Table 2 reports that, across all 707 ScanNet rooms, DINOv2 achieves only 31% of rooms with R2 > 0 and only 3% with R2 > 0.3. Thus the headline number is not representative of the dataset; it is generated by a tiny, unspecified subset of unusually favorable rooms. This is load-bearing for the claim that self-supervised models possess latent subspaces 'remarkably strongly correlated' with SE(3). Please report aggregate statistics (e.g., median and interquartile range across rooms), specify the exact room selection procedure, and base the abstract's quantitative claim on a representative statistic.
  2. [§4, App. A.6, Table 8] The protocol states that 'for all metrics M1-M4, we perform a layer sweep across the encoder blocks to achieve the best performance' (Sec. 4), and App. A.6 describes a 3,328-configuration hyperparameter sweep. The paper does not state that layer and hyperparameter selection are nested inside the training folds; Table 8 reports results at the 'oracle layer'. If layer or hyperparameter selection uses the test split, the reported R2 is optimistically biased. Please clarify how selection is performed and, if it is not nested, re-run with nested selection or report the unbiased estimate.
  3. [§6, App. A.6, Table 7] The paper's own large-scale sweep (38,929 runs) shows that the positive result is highly fragile: translation magnitude has positive R2 in only 26.7% of configurations, with a mean R2_mag of −3.30 across all runs, and Table 8 shows four identical training runs yielding geometry R2 between 0.08 and 0.35. This contradicts the impression given by the main-text 'well-tuned' numbers (R2 ≈ 0.65; Table 8: 0.63 trans, 0.59 rot) and weakens the claim of a homogeneous coordinate system. The distribution over seeds and hyperparameters should be presented in the main text, with the headline claim revised accordingly.
  4. [A.7, Theorem 5(c), Theorem 3] The theoretical explanation is partially tautological. Theorem 3 is explicitly 'essentially trivial' (any C1 feature map with rank-6 Jacobian admits a local linear readout), and Theorem 5(c) constructs a nonlinear chart that 'unrolls' any injective smooth feature map. Consequently, the existence of a successful adapter does not by itself establish that the feature space 'contains' a pre-existing SE(3) manifold; it shows that a pose-supervised probe can impose one. The only nontrivial empirical evidence for a pre-existing structure is the cross-room generalization in Fig. 5 and Table 10. Please explicitly acknowledge this in the interpretation, and either provide additional evidence that the structure is not merely an artifact of the probe's capacity or weaken the 'Visual Grid Code' language.
minor comments (5)
  1. [Table 5 vs. Table 2] Table 5 reports a 9% R2>0 rate for 7-Scenes, while Table 2 reports 2% for DINOv2 on this dataset. The discrepancy is not explained; please reconcile the numbers or clarify what each table reports.
  2. [§5.1, Table 3] Table 3 is averaged over 3 scenes without specifying which scenes, and the Identity baseline's Hit@0.5 is 0.523 for both backbones, which seems inconsistent with a 'no navigation' baseline; please define how the Identity baseline is computed and report scene-level variances.
  3. [A.6, Table 7] The table reports 'Best R2' for translation magnitude as 0.725 while the text states mean R2_mag is −3.30; this is a useful but easily misread contrast. Please explicitly label that the best is over a large sweep and not a typical result.
  4. [§2, Related Work] Reference [30] is described as 'recent work by Mitchel et al.' but the reference list is incomplete in the provided text; please ensure all citations are complete.
  5. [A.11, MoE adapter] The MoE extension is described only in the appendix and is not evaluated in the main results. If it is not needed for the main claims, consider moving it to future work to reduce scope creep.

Circularity Check

2 steps flagged

Pose-supervised probe and a tautological chart theorem inject part of the SE(3) structure; held-out generalization keeps the core empirical claim partially independent.

specific steps
  1. fitted input called prediction [Section 5, 'Emergence of Linear Structure (Metric M4)'; Eq. (3)]
    "Remarkably, by applying a lightweight Siamese MLP adapter φ, we recover strong linear equivariance. As shown in Figure 4, DINOv2 achieves strong performance, reaching test set R2 ≈0.65, averaged over 3 scenes from ScanNet. This implies that the latent space contains a locally Euclidean sub-manifold isomorphic to SE(3)—effectively a “Visual Grid Code”—which can be accessed via a non-linear projection."

    Eq. (3) is trained with ground-truth pose displacements ΔP as regression targets, so the linearized 'SE(3) sub-manifold' is the label space of the fitted probe. The test R2=0.65 is held-out generalization of that pose-supervised fit, not an unsupervised measurement. The conclusion that the latent space 'contains a locally Euclidean sub-manifold isomorphic to SE(3)' therefore partly asserts what the probe's supervision injected. The qualifier 'when probed correctly' makes the probe part of the claim, but the probe uses the very camera poses defining SE(3). Held-out and zero-shot results mitigate, but do not remove, this by-construction alignment.

  2. self definitional [Section 6, 'Why does the adapter work?'; Appendix A.7.2, Remark 4; Appendix A.7.3, Theorem 5(c)]
    "Theorem 3 is essentially trivial—it follows from Taylor’s theorem and the existence of a left inverse. The content is in recognizing that (C2) is the only condition needed for local decodability: any smooth, locally discriminative encoder automatically admits a Poincaré adapter."

    Theorem 5(c)'s proof defines φ = (log ∘ f^{-1} ∘ π, normal coordinates) and sets W=[I_6|0], so the adapter output is log(g2)-log(g1) up to BCH terms—exactly the supervised target. Thus the claimed theoretical 'explanation' of the unrolling assumes the feature map is already an embedded copy of SE(3) and uses the ground-truth pose chart to build the adapter. It cannot independently establish that the learned φ discovers SE(3); it merely restates the definition of a Poincaré adapter under the embedding assumption. The paper's own remark calls Theorem 3 'essentially trivial'.

full rationale

The main empirical content—DINOv2's held-out adapter R2≈0.65, scaling to unseen rooms, and training-free M1 alignment—is not itself a fabricated equivalence: it is a genuine generalization measurement. However, two steps in the paper's interpretation are circular. First, Eq. (3) trains the Poincaré adapter on camera-pose labels, so the 'SE(3)-isomorphic sub-manifold' is the target of a fitted probe; the abstract's own qualifier 'when probed correctly' concedes that the discovery requires a pose-supervised accessor. Second, the theoretical support (Theorem 3 and Theorem 5(c)) is explicitly tautological: any rank-6 smooth feature map admits a local readout, and any injective immersion can be charted so that differences are Lie-algebra coordinates. The chart theorem constructs the adapter from f^{-1} and the ground-truth logarithm, so it restates the desired unrolling rather than deriving it from the encoder's training objective. These issues weaken the 'Visual Grid Code' and 'emergence' framing, but the held-out and zero-shot results provide independent evidence that the fitted structure transfers, so the circularity is partial, not total. The scene-selection and hyperparameter-sweep concerns noted by the skeptic are correctness risks, not circularity, and are not scored here.

Axiom & Free-Parameter Ledger

3 free parameters · 5 axioms · 1 invented entities

The empirical result is a supervised-probe measurement: the adapter and hyperparameters are fitted to pose labels, so the central finding is conditional on the probe design. The theoretical explanation assumes the feature map is a smooth embedded 6D manifold, which is plausible but not directly verified.

free parameters (3)
  • Poincaré adapter weights (φθ, W) = MLP d→64→20, W ∈ R^(6×20), optimized on training pose targets
    The central M4 result is the held-out R2 of this fitted regressor; the existence of the 'latent subspace' is defined by this fit.
  • Adapter hyperparameters (bottleneck dim, learning rate, loss, etc.) = best of 3,328 configurations (e.g., geo_dim=20)
    Selection over a large sweep could inflate reported R2; only the best configuration is highlighted.
  • Oracle layer selection = varies per model (e.g., DINOv2-B Layer 7/9)
    The paper sweeps encoder layers and reports the layer maximizing each metric, introducing selection bias without correction.
axioms (5)
  • domain assumption Static scene assumption: feature changes are caused only by camera motion.
    Section 3: 'we consider static scenes, where the mapping from camera pose to an image (and thus feature) is stable and well-defined.' All probes and navigation rely on pose being the only source of feature change.
  • domain assumption Smoothness and local discriminability of the posed feature map (C1 and rank-6 Jacobian).
    Theorem 3 (conditions C1-C2) requires df_{g0} rank 6; this is not directly measured, only supported indirectly by intrinsic dimensionality near 6.
  • domain assumption Global injectivity of f on the pose region K.
    Theorem 5c assumes f is globally injective on K so that a nonlinear adapter can unroll the manifold; injectivity is not verified and may fail in textureless or symmetric regions.
  • domain assumption Pose ground truth is accurate enough for regression targets and neighbor graphs.
    All M3/M4 targets and M1 neighbor graphs come from dataset camera poses; noisy single-scan poses (ARKitScenes) coincide with low success rates (Table 2/5).
  • standard math Standard smooth-manifold results: BCH formula, tubular neighborhood theorem, Hausdorff dimension under Lipschitz maps.
    Used in Theorem 5c and Theorem 1; these are standard mathematical facts and not in question.
invented entities (1)
  • Visual Grid Code no independent evidence
    purpose: Name for the hypothesized 6D linearizable subspace in frozen features that mirrors SE(3) tangent structure.
    The evidence for this entity is the fitted Poincaré adapter's held-out R2; it has no falsifiable handle outside the paper's own probe, and the paper's Theorem 5c shows such a chart exists for any injective smooth feature map.

pith-pipeline@v1.3.0-alltime-deepseek · 27044 in / 14663 out tokens · 152161 ms · 2026-08-02T02:43:35.240825+00:00 · methodology

0 comments
read the original abstract

In this paper, we ask whether vision foundation models construct representations that reflect the intrinsic properties of 3D Euclidean space. Unlike previous works that probe 3D awareness of vision features by regressing image-centric quantities such as depth or normals, we investigate the relation between the structure of the space of visual features and the group of Euclidean transformations $SE(3)$. We propose a set of probes to evaluate this relation from both topological and geometric perspectives: a mutual neighborhood metric that measures the alignment between feature neighborhoods and spatial topology, and a Poincar\'e Adapter to test the linear accessibility of the geometry of camera motion from latent displacements in static scenes. We show that self-supervised vision models, which, in principle, have not been trained with direct 3D supervision or active agency, possess latent subspaces that are remarkably strongly correlated with three-dimensional Euclidean space, when probed correctly. Building on this insight we propose a new class of "Latent-Space Navigation" techniques that perform visual odometry and localization purely in the latent space, bypassing the need for explicit 3D reconstruction.

Figures

Figures reproduced from arXiv: 2607.14228 by Caroline Chen, Fedor Kitashov, Leonidas Guibas, Maks Ovsjanikov, Ming-Hsuan Yang, Sayna Ebrahimi, Viorica P\u{a}tr\u{a}ucean.

Figure 1
Figure 1. Figure 1: Overview: we investigate whether vision features are aligned with the geometry of Euclidean transformations, as captured by motion (trajectories) within a static scene (left). While raw features are chaotic and tangled (middle left), we show that a lightweight network that we call the “Poincaré adapter” can identify the subspace of the feature space, associated with spatial motion, and unroll it to closely… view at source ↗
Figure 2
Figure 2. Figure 2: Mutual k-nn Alignment (M1). Alignment between camera poses and vision features for short (x axis) and long (y axis) strides on ScanNet. The former captures pixel-level dependencies, while the latter is more indicative of spatial awareness. Explicit geometric models (DUSt3R, MoGe) achieve high alignment. Among self-supervised models, DINO-family encoders show emergent spatial topology, while video models (4… view at source ↗
Figure 3
Figure 3. Figure 3: Example frame pairs from ScanNet. The frames within each of these two pairs are separated by a stride of s = 40 frames, illustrating the typical viewpoint change that the Poincaré adapter must decode. 2 4 8 16 20 24 32 40 60 Frame stride (s) -0.20 -0.15 -0.10 -0.05 -0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 B est a d a pter R 2 (m e a n) Best adapter R 2 vs frame stride (al… view at source ↗
Figure 4
Figure 4. Figure 4: Poincaré Adapter (M4). Quantitative evaluation across a range of encoders and frame strides s. We evaluate whether changes in the latent features as observed across s frames can be linearly mapped to changes in camera pose. Emergence of Linear Structure (Metric M4). As shown above, Metrics M2 and M3 do not tend to differentiate vision encoders because they are, respectively, too easy (most models have ID c… view at source ↗
Figure 5
Figure 5. Figure 5: Generalization and Scaling (M4). Evaluation of zero-shot pose recovery on 31 unseen test rooms as a function of the number of training rooms (N). Left: The fraction of successfully recovered rooms (Top-R2 > 0) increases monotonically. Right: The total recoverable signal (Clipped Mean R2 ) grows substantially without plateauing. adapter recovers a positive R2 signal grows from roughly 18% at N = 5 to 50% at… view at source ↗
Figure 6
Figure 6. Figure 6: Multi-Step Latent Navigation. A 4-step open-loop trajectory (∆P = 0.85m, 36◦ ). The path drifts slightly due to the open-loop integration, but the retrieved frames remain visually coherent and successfully approach the target view. 2.45m, 37◦ ), the path remains semantically coherent and rotation recovery remains highly precise (< 5 ◦ error), although translation drift accumulates [PITH_FULL_IMAGE:figures… view at source ↗
Figure 7
Figure 7. Figure 7: Overview of the mutual nearest neighbor metric [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Poincaré Adapter architecture: illustration of the Siamese architecture of our “Poincaré [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Multi-Step Navigation Trajectories. Three trajectories of varying difficulty generated via open-loop navigation in the latent space. Top (∆ = 0.68m, 19◦ ): The planner successfully traverses the exact path with zero final NN retrieval error. Middle (∆ = 0.85m, 36◦ ): Moderate drift, but semantically consistent room traversal. Bottom (∆ = 2.45m, 37◦ ): Large translation creates drift, but rotation is recove… view at source ↗
Figure 10
Figure 10. Figure 10: Trajectory 1 (Perfect Recovery). ∆P = 0.68m, 19◦ rotation. The final retrieved frame perfectly matches the target image. (Final error: 0.000m, 0.000rad). 28 [PITH_FULL_IMAGE:figures/full_fig_p028_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Trajectory 2 (Moderate Drift). ∆P = 0.85m, 36◦ rotation. The path drifts slightly due to the open-loop integration, but the retrieved frames remain visually coherent and approach the target view. (Final error: 0.387m, 0.140rad) [PITH_FULL_IMAGE:figures/full_fig_p029_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Trajectory 3 (Extreme Displacement). ∆P = 2.45m, 37◦ rotation. The largest displacement tests the limits of the open-loop linear model. While translation error accumulates, the rotation is extremely well recovered (0.084rad ≈ 5 ◦ ). (Final error: 1.020m, 0.084rad). 29 [PITH_FULL_IMAGE:figures/full_fig_p029_12.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

78 extracted references · 22 linked inside Pith

  1. [1]

    Cognitive maps in rats and men.Psychological review, 55(4):189, 1948

    Edward C Tolman. Cognitive maps in rats and men.Psychological review, 55(4):189, 1948. 1

    1948

  2. [2]

    Psychology press, 2014

    James J Gibson.The ecological approach to visual perception: classic edition. Psychology press, 2014. 1

    2014

  3. [3]

    Mental rotation of three-dimensional objects.Science, 171(3972):701–703, 1971

    Roger N Shepard and Jacqueline Metzler. Mental rotation of three-dimensional objects.Science, 171(3972):701–703, 1971. 1

    1971

  4. [4]

    Routledge, 2013

    Jean Piaget.Child’s Conception of Space: Selected Works vol 4. Routledge, 2013. 1

    2013

  5. [5]

    Allocentric and egocentric spatial representations: Definitions, distinctions, and interconnections

    Roberta L Klatzky. Allocentric and egocentric spatial representations: Definitions, distinctions, and interconnections. InSpatial cognition: An interdisciplinary approach to representing and processing spatial knowledge, pages 1–17. Springer, 1998. 1

    1998

  6. [6]

    Structure-from-motion revisited

    Johannes L Schonberger and Jan-Michael Frahm. Structure-from-motion revisited. InProceed- ings of the IEEE conference on computer vision and pattern recognition, pages 4104–4113,

  7. [7]

    Nerf: Representing scenes as neural radiance fields for view synthesis

    Ben Mildenhall, Pratul P Srinivasan, Matthew Tancik, Jonathan T Barron, Ravi Ramamoor- thi, and Ren Ng. Nerf: Representing scenes as neural radiance fields for view synthesis. Communications of the ACM, 65(1):99–106, 2021. 1, 5

    2021

  8. [8]

    Habitat: A platform for embodied ai research

    Manolis Savva, Abhishek Kadian, Oleksandr Maksymets, Yili Zhao, Erik Wijmans, Bhavana Jain, Julian Straub, Jia Liu, Vladlen Koltun, Jitendra Malik, et al. Habitat: A platform for embodied ai research. InProceedings of the IEEE/CVF international conference on computer vision, pages 9339–9347, 2019. 1

    2019

  9. [9]

    Efficiently reconstructing dynamic scenes one d4rt at a time.arXiv preprint arXiv:2512.08924, 2025

    Chuhan Zhang, Guillaume Le Moing, Skanda Koppula, Ignacio Rocco, Liliane Momeni, Junyu Xie, Shuyang Sun, Rahul Sukthankar, Joëlle K Barral, Raia Hadsell, et al. Efficiently reconstructing dynamic scenes one d4rt at a time.arXiv preprint arXiv:2512.08924, 2025. 1

    arXiv 2025

  10. [10]

    IndyPublish.com, 1709

    George Berkeley.An essay towards a new theory of vision. IndyPublish.com, 1709. 1

  11. [11]

    Movement-produced stimulation in the development of visually guided behavior.Journal of comparative and physiological psychology, 56(5):872, 1963

    Richard Held and Alan Hein. Movement-produced stimulation in the development of visually guided behavior.Journal of comparative and physiological psychology, 56(5):872, 1963. 1

    1963

  12. [12]

    Dover Publications, 2011

    Henri Poincaré.Science and hypothesis. Dover Publications, 2011. 1

    2011

  13. [13]

    Probing the 3D awareness of visual foundation models

    Mohamed El Banani, Amit Raj, Kevis-Kokitsi Maninis, Abhishek Kar, Yuanzhen Li, Michael Rubinstein, Deqing Sun, Leonidas Guibas, Justin Johnson, and Varun Jampani. Probing the 3D awareness of visual foundation models. InProceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 21795–21806, 2024. 2

    2024

  14. [14]

    Z. Teed, L. Lipson, and J. Deng. Deep patch visual odometry. InAdvances in Neural Information Processing Systems (NeurIPS), volume 36, 2023. 2

    2023

  15. [15]

    S. Wang, V . Leroy, Y . Cabon, B. Chidlovskii, and J. Revaud. DUSt3R: Geometric 3D vision made easy. InProc. IEEE/CVF Conf. Comput. Vis. Pattern Recognit. (CVPR), pages 20697– 20709, 2024. 2, 3, 6, 7, 8

    2024

  16. [16]

    J. Wang, M. Chen, N. Karaev, A. Vedaldi, C. Rupprecht, and D. Novotny. VGGT: Visual geometry grounded transformer. InProc. IEEE/CVF Conf. Comput. Vis. Pattern Recognit. (CVPR), pages 5294–5306, 2025. 2, 3, 6

    2025

  17. [17]

    Nomad: Goal masked diffusion policies for navigation and exploration

    Ajay Sridhar, Dhruv Shah, Catherine Glossop, and Sergey Levine. Nomad: Goal masked diffusion policies for navigation and exploration. In2024 IEEE International Conference on Robotics and Automation (ICRA), pages 63–70, 2024. 2, 3

    2024

  18. [18]

    Scaling cross- embodied learning: One policy for manipulation, navigation, locomotion and aviation.arXiv preprint arXiv:2408.11812, 2024

    Ria Doshi, Homer Walke, Oier Mees, Sudeep Dasari, and Sergey Levine. Scaling cross- embodied learning: One policy for manipulation, navigation, locomotion and aviation.arXiv preprint arXiv:2408.11812, 2024. 2, 3 11

    Pith/arXiv arXiv 2024

  19. [19]

    Can vision founda- tion models navigate? zero-shot real-world evaluation and lessons learned.arXiv preprint arXiv:2603.25937, 2026

    Maeva Guerrier, Karthik Soma, Jana Pavlasek, and Giovanni Beltrame. Can vision founda- tion models navigate? zero-shot real-world evaluation and lessons learned.arXiv preprint arXiv:2603.25937, 2026. 2, 3

    Pith/arXiv arXiv 2026

  20. [20]

    Navigation world models

    Amir Bar, Gaoyue Zhou, Danny Tran, Trevor Darrell, and Yann LeCun. Navigation world models. InProceedings of the Computer Vision and Pattern Recognition Conference, pages 15791–15801, 2025. 2, 3, 5

    2025

  21. [21]

    V-JEPA 2.1: Unlocking dense features in video self-supervised learning.arXiv preprint arXiv:2603.14482, 2026

    Lorenzo Mur-Labadia, Matthew Muckley, Amir Bar, Mido Assran, Koustuv Sinha, Mike Rabbat, Yann LeCun, Nicolas Ballas, and Adrien Bardes. V-JEPA 2.1: Unlocking dense features in video self-supervised learning.arXiv preprint arXiv:2603.14482, 2026. 2, 3, 5, 6, 7, 8

    Pith/arXiv arXiv 2026

  22. [22]

    Multiview equivariance improves 3D correspondence understanding with minimal feature finetuning.arXiv preprint arXiv:2411.19458, 2024

    Yang You, Yixin Li, Congyue Deng, Yue Wang, and Leonidas Guibas. Multiview equivariance improves 3D correspondence understanding with minimal feature finetuning.arXiv preprint arXiv:2411.19458, 2024. 2

    Pith/arXiv arXiv 2024

  23. [23]

    Deep ViT features as dense visual descriptors.arXiv preprint arXiv:2112.05814, 2(3):4, 2021

    Shir Amir, Yossi Gandelsman, Shai Bagon, and Tali Dekel. Deep ViT features as dense visual descriptors.arXiv preprint arXiv:2112.05814, 2(3):4, 2021. 2

    Pith/arXiv arXiv 2021

  24. [24]

    How much 3d do video foundation models encode?arXiv preprint arXiv:2512.19949, 2025

    Zixuan Huang, Xiang Li, Zhaoyang Lv, and James M Rehg. How much 3d do video foundation models encode?arXiv preprint arXiv:2512.19949, 2025. 2

    arXiv 2025

  25. [25]

    Lexicon3D: Probing visual foundation models for complex 3D scene understanding

    Yunze Man, Shuhong Zheng, Zhipeng Bao, Martial Hebert, Liang-Yan Gui, and Yu-Xiong Wang. Lexicon3D: Probing visual foundation models for complex 3D scene understanding. Advances in Neural Information Processing Systems, 37:76819–76847, 2024. 3

    2024

  26. [26]

    Bhattad, D

    A. Bhattad, D. McKee, D. Hoiem, and D. A. Forsyth. StyleGAN knows normal, depth, albedo, and more.arXiv preprint arXiv:2306.00987, 2023. 3

    Pith/arXiv arXiv 2023

  27. [27]

    Y . Chen, F. Viégas, and M. Wattenberg. Beyond surface statistics: Scene representations in a latent diffusion model.arXiv preprint arXiv:2306.05720, 2023. 3

    Pith/arXiv arXiv 2023

  28. [28]

    Campos, R

    C. Campos, R. Elvira, J. J. Gómez Rodríguez, J. M. M. Montiel, and J. D. Tardós. ORB- SLAM3: An accurate open-source library for visual, visual–inertial, and multimap SLAM. IEEE Transactions on Robotics, 37(6):1874–1890, 2021. 3

    2021

  29. [29]

    Rust: Latent neural scene representations from unposed imagery

    Mehdi SM Sajjadi, Aravindh Mahendran, Thomas Kipf, Etienne Pot, Daniel Duckworth, Mario Luˇci´c, and Klaus Greff. Rust: Latent neural scene representations from unposed imagery. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 17297–17306, 2023. 3

    2023

  30. [30]

    True self-supervised novel view synthesis is transferable.arXiv preprint arXiv:2510.13063, 2025

    Thomas W Mitchel, Hyunwoo Ryu, and Vincent Sitzmann. True self-supervised novel view synthesis is transferable.arXiv preprint arXiv:2510.13063, 2025. 3

    arXiv 2025

  31. [31]

    Primary visual cortex straightens natural video trajectories.Nature communications, 12(1):5982, 2021

    Olivier J Hénaff, Yoon Bai, Julie A Charlton, Ian Nauhaus, Eero P Simoncelli, and Robbe LT Goris. Primary visual cortex straightens natural video trajectories.Nature communications, 12(1):5982, 2021. 3, 4

    2021

  32. [32]

    LeWorld- Model: Stable end-to-end joint-embedding predictive architecture from pixels.arXiv preprint arXiv:2603.19312, 2026

    Lucas Maes, Quentin Le Lidec, Damien Scieur, Yann LeCun, and Randall Balestriero. LeWorld- Model: Stable end-to-end joint-embedding predictive architecture from pixels.arXiv preprint arXiv:2603.19312, 2026. 3, 4

    Pith/arXiv arXiv 2026

  33. [33]

    Sorscher, G

    B. Sorscher, G. C. Mel, S. A. Ocko, L. M. Giocomo, and S. Ganguli. A unified theory for the computational and mechanistic origins of grid cells.Neuron, 111(1):121–137, 2023. 3

    2023

  34. [34]

    Dorrell and J

    W. Dorrell and J. Whittington. If grid cells are the answer, what is the question? a review of normative grid cell theory.arXiv preprint arXiv:2601.12424, 2026. 3

    arXiv 2026

  35. [35]

    Gnm: A general navigation model to drive any robot

    Dhruv Shah, Ajay Sridhar, Arjun Bhorkar, Noriaki Hirose, and Sergey Levine. Gnm: A general navigation model to drive any robot. In2023 IEEE International Conference on Robotics and Automation (ICRA), pages 7226–7233, 2023. 3

    2023

  36. [36]

    Vint: A foundation model for visual navigation.arXiv preprint arXiv:2306.14846, 2023

    Dhruv Shah, Ajay Sridhar, Nitish Dashora, Kyle Stachowicz, Kevin Black, Noriaki Hi- rose, and Sergey Levine. Vint: A foundation model for visual navigation.arXiv preprint arXiv:2306.14846, 2023. 3 12

    Pith/arXiv arXiv 2023

  37. [37]

    Prior does matter: Visual navigation via denoising diffusion bridge models

    Hao Ren, Yiming Zeng, Zetong Bi, Zhaoliang Wan, Junlong Huang, and Hui Cheng. Prior does matter: Visual navigation via denoising diffusion bridge models. InProceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 12100–12110, 2025. 3

    2025

  38. [38]

    Navmorph: A self-evolving world model for vision-and-language navigation in continuous environments

    Xuan Yao, Junyu Gao, and Changsheng Xu. Navmorph: A self-evolving world model for vision-and-language navigation in continuous environments. InProceedings of the IEEE/CVF International Conference on Computer Vision, pages 5536–5546, 2025. 3

    2025

  39. [39]

    R. M. Murray, Z. Li, and S. S. Sastry.A Mathematical Introduction to Robotic Manipulation. CRC Press, 1994. 4

    1994

  40. [40]

    On the continuity of rotation representations in neural networks

    Yi Zhou, Connelly Barnes, Jingwan Lu, Jimei Yang, and Hao Li. On the continuity of rotation representations in neural networks. InProceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 5745–5753, 2019. 4, 16

    2019

  41. [41]

    Self-supervised video pretraining yields robust and more human-aligned visual representations.Advances in Neural Information Processing Systems, 36:65743–65765, 2023

    Nikhil Parthasarathy, SM Eslami, Joao Carreira, and Olivier Henaff. Self-supervised video pretraining yields robust and more human-aligned visual representations.Advances in Neural Information Processing Systems, 36:65743–65765, 2023. 4

    2023

  42. [42]

    Learning predictable and robust neural representations by straightening image sequences.Advances in Neural Information Processing Systems, 37:40316–40335, 2024

    Xueyan Niu, Cristina Savin, and Eero P Simoncelli. Learning predictable and robust neural representations by straightening image sequences.Advances in Neural Information Processing Systems, 37:40316–40335, 2024. 4

    2024

  43. [43]

    Chirality in action: Time-aware video represen- tation learning by latent straightening.Advances in Neural Information Processing Systems, 38:92695–92726, 2026

    Piyush Nitin Bagad and Andrew Zisserman. Chirality in action: Time-aware video represen- tation learning by latent straightening.Advances in Neural Information Processing Systems, 38:92695–92726, 2026. 4

    2026

  44. [44]

    A. Dai, A. X. Chang, M. Savva, M. Halber, T. Funkhouser, and M. Nießner. ScanNet: Richly- annotated 3D reconstructions of indoor scenes. InProc. IEEE Conf. Comput. Vis. Pattern Recognit. (CVPR), pages 5828–5839, 2017. 4, 6, 7, 8

    2017

  45. [45]

    ARKitScenes: A diverse real- world dataset for 3D indoor scene understanding using mobile RGB-D data.arXiv preprint arXiv:2111.08897, 2021

    Gilad Baruch, Zhuoyuan Chen, Afshin Dehghan, Tal Dimry, Yuri Feigin, Peter Fu, Thomas Gebauer, Brandon Joffe, Daniel Kurz, Arik Schwartz, et al. ARKitScenes: A diverse real- world dataset for 3D indoor scene understanding using mobile RGB-D data.arXiv preprint arXiv:2111.08897, 2021. 4, 8

    Pith/arXiv arXiv 2021

  46. [46]

    A benchmark for the evaluation of rgb-d slam systems

    Jürgen Sturm, Nikolas Engelhard, Felix Endres, Wolfram Burgard, and Daniel Cremers. A benchmark for the evaluation of rgb-d slam systems. In2012 IEEE/RSJ international conference on intelligent robots and systems, pages 573–580, 2012. 4, 8

    2012

  47. [47]

    Learning to navigate the energy landscape

    Julien Valentin, Angela Dai, Matthias Nießner, Pushmeet Kohli, Philip Torr, Shahram Izadi, and Cem Keskin. Learning to navigate the energy landscape. In2016 Fourth International Conference on 3D Vision (3DV), pages 323–332. IEEE, 2016. 4, 8

    2016

  48. [48]

    Scene coordinate regression forests for camera relocalization in rgb-d images

    Jamie Shotton, Ben Glocker, Christopher Zach, Shahram Izadi, Antonio Criminisi, and Andrew Fitzgibbon. Scene coordinate regression forests for camera relocalization in rgb-d images. InProceedings of the IEEE conference on computer vision and pattern recognition, pages 2930–2937, 2013. 4, 8

    2013

  49. [49]

    M. Huh, B. Cheung, T. Wang, and P. Isola. The platonic representation hypothesis.arXiv preprint arXiv:2405.07987, 2024. 4

    Pith/arXiv arXiv 2024

  50. [50]

    Dynamic reflections: Probing video representations with text alignment

    Tyler Zhu, Tengda Han, Leonidas Guibas, Viorica P˘atr˘aucean, and Maks Ovsjanikov. Dynamic reflections: Probing video representations with text alignment. InThe Fourteenth International Conference on Learning Representations, 2026. 4

    2026

  51. [51]

    Levina and P

    E. Levina and P. Bickel. Maximum likelihood estimation of intrinsic dimension. InAdvances in Neural Information Processing Systems (NeurIPS), volume 17, 2004. 4

    2004

  52. [52]

    Facco, M

    E. Facco, M. d’Errico, A. Rodriguez, and A. Laio. Estimating the intrinsic dimension of datasets by a minimal neighborhood information.Scientific Reports, 7(1):12140, 2017. 4 13

    2017

  53. [53]

    Large vision models can solve mental rotation problems

    Sebastian Ray Mason, Anders Gjølbye, Phillip Chavarria Højbjerg, Lenka Tˇetková, and Lars Kai Hansen. Large vision models can solve mental rotation problems. InIEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pages 1571–1575, 2026. 5

    2026

  54. [54]

    Metzer, E

    G. Metzer, E. Richardson, O. Patashnik, R. Giryes, and D. Cohen-Or. Latent-NeRF for shape- guided generation of 3D shapes and textures. InProc. IEEE/CVF Conf. Comput. Vis. Pattern Recognit. (CVPR), pages 12663–12673, 2023. 5

    2023

  55. [55]

    Weinzaepfel, V

    P. Weinzaepfel, V . Leroy, T. Lucas, R. Brégier, Y . Cabon, V . Arora, L. Antsfeld, B. Chidlovskii, G. Csurka, and J. Revaud. CroCo: Self-supervised pre-training for 3D vision tasks by cross-view completion. InAdvances in Neural Information Processing Systems (NeurIPS), volume 35, pages 3502–3516, 2022. 6

    2022

  56. [56]

    Oquab, T

    M. Oquab, T. Darcet, T. Moutakanni, H. V o, M. Szafraniec, V . Khalidov, P. Fernandez, D. Haziza, F. Massa, A. El-Nouby, et al. DINOv2: Learning robust visual features without supervision. arXiv preprint arXiv:2304.07193, 2023. 6, 8

    Pith/arXiv arXiv 2023

  57. [57]

    Oriane Siméoni, Huy V V o, Maximilian Seitzer, Federico Baldassarre, Maxime Oquab, Cijo Jose, Vasil Khalidov, Marc Szafraniec, Seungeun Yi, Michaël Ramamonjisoa, et al. DINOv3. arXiv preprint arXiv:2508.10104, 2025. 6

    Pith/arXiv arXiv 2025

  58. [58]

    Radford, J

    A. Radford, J. W. Kim, C. Hallacy, A. Ramesh, G. Goh, S. Agarwal, G. Sastry, A. Askell, P. Mishkin, J. Clark, et al. Learning transferable visual models from natural language supervision. InInt. Conf. Mach. Learn. (ICML), pages 8748–8763, 2021. 6

    2021

  59. [59]

    R. Wang, S. Xu, C. Dai, J. Xiang, Y . Deng, X. Tong, and J. Yang. MoGe: Unlocking accurate monocular geometry estimation for open-domain images with optimal training supervision. In Proc. IEEE/CVF Conf. Comput. Vis. Pattern Recognit. (CVPR), 2025. 6

    2025

  60. [60]

    H. Lin, S. Chen, J. H. Liew, D. Y . Chen, Z. Li, G. Shi, J. Feng, and B. Kang. Depth Anything 3: Recovering the visual space from any views.arXiv preprint arXiv:2511.10647, 2025. 6

    Pith/arXiv arXiv 2025

  61. [61]

    Bolya, P.-Y

    D. Bolya, P.-Y . Huang, P. Sun, J. H. Cho, A. Madotto, C. Wei, T. Ma, J. Zhi, J. Rajasegaran, H. Rasheed, et al. Perception encoder: The best visual embeddings are not at the output of the network.arXiv preprint arXiv:2504.13181, 2025. 6

    Pith/arXiv arXiv 2025

  62. [62]

    M. Hu, W. Yin, C. Zhang, Z. Cai, X. Long, H. Chen, K. Wang, G. Yu, C. Shen, and S. Shen. Metric3D v2: A versatile monocular geometric foundation model for zero-shot metric depth and surface normal estimation.arXiv preprint arXiv:2404.15506, 2024. 6

    Pith/arXiv arXiv 2024

  63. [63]

    High- resolution image synthesis with latent diffusion models

    Robin Rombach, Andreas Blattmann, Dominik Lorenz, Patrick Esser, and Björn Ommer. High- resolution image synthesis with latent diffusion models. InProceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 10684–10695, 2022. 6

    2022

  64. [64]

    Denoising diffusion implicit models.arXiv preprint arXiv:2010.02502, 2020

    Jiaming Song, Chenlin Meng, and Stefano Ermon. Denoising diffusion implicit models.arXiv preprint arXiv:2010.02502, 2020. 6

    Pith/arXiv arXiv 2010

  65. [65]

    Latent consistency models: Synthesizing high-resolution images with few-step inference.arXiv preprint arXiv:2310.04378,

    Simian Luo, Yiqin Tan, Longbo Huang, Jian Li, and Hang Zhao. Latent consistency models: Synthesizing high-resolution images with few-step inference.arXiv preprint arXiv:2310.04378,

    Pith/arXiv arXiv

  66. [66]

    A tale of two features: Stable diffusion complements dino for zero-shot semantic correspondence.Advances in Neural Information Processing Systems, 36:45533–45547, 2023

    Junyi Zhang, Charles Herrmann, Junhwa Hur, Luisa Polania Cabrera, Varun Jampani, Deqing Sun, and Ming-Hsuan Yang. A tale of two features: Stable diffusion complements dino for zero-shot semantic correspondence.Advances in Neural Information Processing Systems, 36:45533–45547, 2023. 6

    2023

  67. [67]

    Stable video diffusion: Scaling latent video diffusion models to large datasets.arXiv preprint arXiv:2311.15127, 2023

    Andreas Blattmann, Tim Dockhorn, Sumith Kulal, Daniel Mendelevitch, Maciej Kilian, Do- minik Lorenz, Yam Levi, Zion English, Vikram V oleti, Adam Letts, et al. Stable video diffusion: Scaling latent video diffusion models to large datasets.arXiv preprint arXiv:2311.15127, 2023. 6

    Pith/arXiv arXiv 2023

  68. [68]

    L. Wang, B. Huang, Z. Zhao, Z. Tong, Y . He, Y . Wang, Y . Wang, and Y . Qiao. VideoMAE v2: Scaling video masked autoencoders with dual masking. InProc. IEEE/CVF Conf. Comput. Vis. Pattern Recognit. (CVPR), pages 14549–14560, 2023. 6 14

    2023

  69. [69]

    Zoran, N

    D. Zoran, N. Parthasarathy, Y . Yang, D. A. Hudson, J. Carreira, and A. Zisserman. Recurrent video masked autoencoders.arXiv preprint arXiv:2512.13684, 2025. 6

    Pith/arXiv arXiv 2025

  70. [70]

    Carreira, D

    J. Carreira, D. Gokay, M. King, C. Zhang, I. Rocco, A. Mahendran, T. A. Keck, J. Heyward, S. Koppula, E. Pot, et al. Scaling 4D representations.arXiv preprint arXiv:2412.15212, 2024. 6

    Pith/arXiv arXiv 2024

  71. [71]

    Unique lives, shared world: Learning from single-life videos.arXiv preprint arXiv:2512.04085, 2025

    Tengda Han, Sayna Ebrahimi, Dilara Gokay, Li Yang Ku, Maks Ovsjanikov, Iva Babukova, Daniel Zoran, Viorica Patraucean, Joao Carreira, Andrew Zisserman, et al. Unique lives, shared world: Learning from single-life videos.arXiv preprint arXiv:2512.04085, 2025. 10

    Pith/arXiv arXiv 2025

  72. [72]

    Federer.Geometric Measure Theory

    H. Federer.Geometric Measure Theory. Springer, 1969. 21

    1969

  73. [73]

    Poincaré adapter

    J. M. Lee.Introduction to Smooth Manifolds. Springer, 2nd edition, 2012. 21 15 A Appendix Figure 7: Overview of the mutual nearest neighbor metricM1. For each frame in a scene we extract its visual features, as well as its corresponding (ground truth) camera pose. We then compare the nearest neighbor graphs computed in the feature (left) and pose (right) ...

    2012

  74. [74]

    Assuming Φ =F ◦P is a smooth mapping, the intrinsic (Hausdorff) dimensionality of MS is at most6

  75. [75]

    Proof.We address each claim sequentially:

    If Φ =F ◦P is a smoothembedding—meaning it is an injective immersion that maps homeomorphically onto its image—then MS is a regular smooth submanifold of Rd with intrinsic dimensionof exactly6. Proof.We address each claim sequentially:

  76. [76]

    Thus, time t influences the visual observation solely through the trajectory C(t), making P:U → Ia well-defined mapping that assigns a unique, repeatable image to any specific pose

    For a strictly static scene S, the geometry, materials, and illumination of the environment are invariant over time. Thus, time t influences the visual observation solely through the trajectory C(t), making P:U → Ia well-defined mapping that assigns a unique, repeatable image to any specific pose. Moreover, by definition, the camera trajectory is restrict...

  77. [77]

    The domain of valid poses U is an open subset of the Lie group SE(3), which is a smooth manifold of dimension 6

    Let Φ =F ◦P. The domain of valid poses U is an open subset of the Lie group SE(3), which is a smooth manifold of dimension 6. Because the composition Φ :U →R d is assumed to be a smooth mapping, it is continuously differentiable and therefore locally Lipschitz continuous. By standard results in geometric measure theory [72], locally Lipschitz mappings do ...

  78. [78]

    move forward 1 meter

    By definition, a smooth embedding Φ is an immersion (its differential dΦ has a full rank of 6 everywhere) that is injective and maps homeomorphically onto its image. Under these assumptions, standard differential topology [ 73] guarantees that the image of an embedded manifold is a regular smooth submanifold, diffeomorphic to the domain. Because diffeomor...