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Depth Any Seen: Which Surfaces and How Far?

Xiaohao Xu, Xiaonan Huang

arXiv:2609.32027v1cs.CVcs.AIcs.GRcs.RO

TL;DR

Recovering several visible surfaces from one image requires estimating both their presence and metric depth, which depth maps alone cannot represent. Depth Any Seen uses image-conditioned multi-Bernoulli depth sets and ExactMB to learn both through assignment marginalization, and evaluates depth and support jointly. Against depth stacking, ExactMB reduces overprediction while retaining most ordinal accuracy and comparable conditional metric-depth error.

  • Problem

    A single depth map cannot represent multiple visible surfaces along a ray, and recovery requires determining both which surfaces are present and where they lie.

    Source: §1 · p. 1

  • Method

    Depth Any Seen represents coexisting surfaces as image-conditioned multi-Bernoulli depth sets and uses ExactMB to marginalize one-to-one assignments to complete, distinct targets.

    Source: §1 · p. 2

  • Results

    ExactMB reduces LD-Real OverPred from 98.4% to 11.6% and MD-3K OverPred from 98.5% to 19.2% versus depth stacking, retaining most LD-Real ordinal accuracy and comparable LD-Syn conditional AbsRel.

    Source: §5.1 · p. 6

  • Takeaways & Limitations

    Conditional depth accuracy and correct ordering can conceal incomplete or excessive predictions, so evaluation should jointly measure localization, recovered support, and overprediction.

    Source: §6 · p. 9

    Quantitative benchmarking adapts existing datasets without collecting new data, and the authors identify stronger real-world generalization through mixed synthetic–real supervision as future work.

    Source: §6 · p. 9, ETHICS STATEMENT · p. 10

Abstract

from arXiv · show

When several surfaces are visible along a ray, recovering visible 3D structure from one image requires jointly estimating their presence and metric depth. Depth Any Seen represents these surfaces as image-conditioned multi-Bernoulli depth sets, whose components each contribute one depth or remain absent. Its auxiliary-free Exact Multi-Bernoulli objective (ExactMB) learns depth and presence by marginalizing one-to-one assignments to complete, distinct targets. Our analysis shows that matching expected count can leave component-surface assignment unresolved. We extend real and synthetic layered-depth benchmarks to evaluate depth accuracy, recovered support, and overprediction. Compared to depth stacking, ExactMB reduces overprediction by a relative 88.2% on LD-Real and 80.5% on MD-3K while retaining most ordinal accuracy, with comparable conditional metric-depth error on LD-Syn. Further ablation studies show that ordered assignment improves depth-accurate recall and precision over marginalization, whereas the count-regularized configuration achieves higher deeper-rank precision than ordered assignment at lower recall. Our code will be publicly released.

1 INTRODUCTION

Depth Any Seen targets coexisting visible surfaces by learning their metric depths and presence jointly. Its ExactMB objective and support-aware benchmarks expose recovery trade-offs that depth-only or count-only evaluation can miss.

1 INTRODUCTION

  • ExactMB models coexisting visible surfaces as image-conditioned multi-Bernoulli sets, marginalizing one-to-one assignments to complete, distinct targets so depth and presence are learned jointly.Each component contributes one depth or remains absent, without auxiliary losses.Source: §1 · p. 2
  • Matching expected count can leave component–surface assignment unresolved, and count-regularized and ordered-assignment configurations produce different recovery trade-offs.Count regularization improves aggregate count accuracy and deeper-rank precision; ordered assignment improves depth-accurate recall and precision over marginalization, with higher mean MD-3K overprediction.Source: §1 · p. 2
  • The joint benchmarks combine synthetic metric depth, real-data ordinal relations and membership, and fixed-ray set recovery to assess surface location and recovery.They evaluate depth accuracy, recovered support, and overprediction together.Source: §1 · p. 2
  • 88.2% relative LD-Real overprediction reduction versus depth stacking accompanies retained ordinal accuracy on LD-Real and MD-3K and comparable conditional metric-depth error on LD-Syn.The analysis also reports an 80.5% relative overprediction reduction on MD-3K.Source: §1 · p. 2

2 RELATED WORK

Prior work represents layered geometry, predicts visible or amodal surfaces, and reconstructs transparent scenes using varied inputs and objectives. Depth Any Seen instead targets coexisting visible surfaces and their membership at metric depth from one image.

2 RELATED WORK

  • Depth Any Seen’s target is visible-set membership coupled with metric depth for coexisting surfaces seen along each ray, including through transparent foreground objects.The cited approaches span layered-depth data, unordered components, transparent multilayers, ordinal pairs, and amodal or complete-geometry prediction.Source: §2 · p. 2
  • Feed-forward geometry methods predict depth, pointmaps, cameras, or dense metric reconstruction, while this paper targets coexisting visible depths and their membership from one image.The contrast is stated as complementary to those multi-view and geometry-reconstruction goals.Source: §2 · p. 2
  • Layered geometry and rendering work uses learned layers for view synthesis, masked reconstruction, object-wise decomposition, inpainting, and insertion or shading.Engel et al. (2024) infer depth, color, and opacity intervals from semitransparent volume renderings.Source: §2 · p. 2
  • Figure 2 contrasts recurrent predictions of depth, scale, and existence with ExactMB assignment scoring and annotated visibility that determines target return count.Ordered assignment retains optional emission, and scale describes localization conditional on presence.Source: Figure 2 · p. 3
  • Transparent-scene methods include RGB-D completion, selected-surface depth, and multilayer recovery using active stereo or stereo distance queries; this work uses one monocular image to jointly model visible-layer presence and metric depth.ASGrasp reconstructs two layers from RGB and active stereo, while DepthFocus selects layers from stereo observations.Source: §2 · p. 3

3 DEPTH ANY SEEN VIA RAY-ADAPTIVE DEPTH SETS

Depth Any Seen represents each ray’s visible surfaces as a variable-cardinality set with uncertain depth and presence. ExactMB scores complete distinct-target assignments, while its assignment responsibilities jointly supervise depth and presence but need not resolve component–surface allocation.

3 DEPTH ANY SEEN VIA RAY-ADAPTIVE DEPTH SETS

  • Given one image, the task predicts a set of visible optical-axis depths per ray, with 0 to K targets; opaque-hidden geometry is excluded and cross-pixel surface identities are unspecified.Farther visible layers are assumed to be seen through transparent objects locally modeled as finite-thickness glass.Source: §3 · p. 3

3.1 CONTINUOUS DEPTH, DISCRETE MEMBERSHIP

  • Each image-conditioned component predicts a depth, localization scale, and existence probability, then emits one depth or remains absent; component identities are not depth ranks.Optional emission allows cardinality to vary within capacity K, with selection, sorting, and filtering defining decoded ranks.Source: §3.1 · p. 3

3.2 EXACTMB: ONE LIKELIHOOD FOR DEPTH AND ABSENCE

  • ExactMB sums one-to-one assignments that match each observed depth to a distinct component, scoring unused components as absent and jointly scoring localization and membership without auxiliary penalties.The sum is invariant to target enumeration; absence means visible-set nonmembership, not physical nonexistence.Source: §3.2 · p. 4

3.3 HOW ASSIGNMENT SUPERVISES DEPTH AND PRESENCE

  • Marginalized assignment responsibilities give components presence credit for explaining targets, and those credits sum to the annotated target count.Posterior occupancy also depends on observed depths and competing components, unlike image-predicted existence probability.Source: §3.3 · p. 4
  • For complete distinct targets with 0 < M < K, shifting all logits preserves posterior assignment probabilities, while the set-loss minimum matches expected count to M.The aggregate presence signal still leaves component–surface assignment unresolved; at M = 1 selection is ambiguous, at M = K the occupied subset is fixed, and intermediate counts involve both decisions.Source: §3.3 · pp. 4–5

4 EVALUATING VISIBLE DEPTH LAYERS

The evaluation extends layered-depth benchmarks to jointly assess depth accuracy, recovered support, and overprediction. It combines dense synthetic metric labels with sparse real presence and ordering annotations, using complementary metrics for ordinal correctness, support, depth, and count.

4 EVALUATING VISIBLE DEPTH LAYERS

  • Correct counts can hide missed surfaces and extra predictions, while correct ordering can coexist with incorrect presence, motivating evaluation of depth accuracy, recovered support, and overprediction together.The evaluation distinguishes whether a surface is recovered from how accurately its depth is estimated.Source: §4 · p. 5

4.1 BENCHMARKS AND ANNOTATION CONVENTIONS

  • LD-Real provides 300 real validation images with sparse ordinal queries and 500 synthetic metric-depth images; MD-3K adds 3,161 material-labeled point pairs aligned to LD layer conventions.Dense synthetic labels test metric depth and count, while sparse real labels test presence and ordering.Source: §4.1 · p. 5

4.2 JOINT DEPTH–PRESENCE EVALUATION

  • LD-Real ordinal ACC requires requested depths to be present in strict near-to-far order, while OverPred measures predictions on all-absent quadruplets; MD-3K scores validity patterns, rank recall, overprediction, and joint pattern-plus-ordinal correctness.On LD-Syn, conditional AbsRel is paired with target-support recall and depth-accurate recall and precision; count ACC and MAE are also reported.Source: §4.2 · p. 5
  • Primary ablations use native selection and common geometric filters without alignment or clipping, with means and sample standard deviations reported across seeds.Summary AbsRel equally weights four conditional rank errors per seed; rank-wise scores complement fixed-ray set-recovery diagnostics.Source: §4.2 · p. 5

5 EXPERIMENTS

The experiments compare depth stacking, ExactMB, and supervision variants across depth, presence, and recovery metrics. ExactMB sharply reduces overprediction while retaining most ordinal accuracy, and assignment choices shift deeper-layer precision–recall trade-offs.

5.1 JOINT DEPTH–PRESENCE MODELING REDUCES OVERPREDICTION

  • LD-Real OverPred falls from 98.4% to 11.6% and MD-3K OverPred from 98.5% to 19.2% with ExactMB versus depth stacking.ExactMB retains LD-Real ordinal ACC of 61.7% versus 67.4% and has LD-Syn conditional AbsRel of 19.2% versus 18.3%.Source: §5.1 · p. 6, Table 1 · p. 6
  • Count regularization leads ExactMB on MD pattern ACC (65.9% versus 46.3%) and synthetic count ACC (95.0% versus 88.2%), while ordered assignment reaches 71.4% on four-return rays versus ExactMB’s 52.3%.One-return rays dominate synthetic count ACC; an always-one predictor reaches 87.49% micro count ACC and .2112-layer MAE on occupied rays.Source: §5.1 · p. 7, Figure 3 · p. 7
  • Figure 3(b)’s separate-recipe MAP-to-ExactMB comparison lowers MD OverPred by 5.31 ± 1.49 points and raises pattern ACC by 1.96 ± 1.37, but worsens count MAE by .0069 ± .0030 layers.The primary-study pattern-ACC change is −1.35 ± 6.69 points, so marginalization’s benefits vary by recipe and metric.Source: §5.1 · p. 7, Figure 3 · p. 7

5.2 DEPTH ACCURACY MUST BE READ WITH RECOVERED SUPPORT

  • ExactMB raises second-rank depth-accurate precision from 10.07% to 54.04% versus depth stacking, while recall falls from 80.20% to 75.78%.Count regularization reduces conditional errors relative to marginalization, but second- and third-rank target-support recall falls by 6.71 ± .26 and 10.68 ± .28 points.Source: §5.2 · p. 7, Figure 4 · p. 7
  • Ordered assignment improves fourth-rank depth-accurate recall by 13.95 ± .58 points and precision by 8.76 ± .59 points over ExactMB.It also raises target-support recall at ranks 2–4 in every seed, while MD OverPred increases from 19.23% to 21.65%. Count regularization has higher deeper-rank precision than ordered assignment at lower recall.Source: §5.2 · pp. 7–8, Figure 4 · p. 7
  • Depth stacking’s 85.13% ordinal ACC on MD-3K coincides with 0% Joint ACC, while ExactMB, ordered assignment, and count regularization reach 42.34%, 46.62%, and 58.79% Joint ACC.Joint ACC requires both annotated presence and ordering; ordered assignment’s gain over ExactMB reflects better presence-pattern recovery despite lower ordering accuracy conditional on correct presence.Source: §5.2 · p. 8, Figure 5 · p. 8

5.3 MULTILAYER 3D RECONSTRUCTION AND APPLICATIONS

  • Selected ExactMB predictions show spatially varying retained depth support in real and synthetic examples, with unprojected surfaces used for relighting meshes and local robot-route screening.The route screening tests candidate trajectories against selected first-layer facets; extending it to closed-loop control and whole-scene safety verification remains future work.Source: §5.3 · p. 8, Figure 6 · p. 8, Figure 7 · p. 9

6 CONCLUSION

The conclusion frames visible-surface recovery as jointly estimating presence and depth, and argues that count correction alone may leave surface assignment unresolved. The appendices develop the objective’s properties and distinctions among selection, assignment, and count.

6 CONCLUSION

  • The paper’s central goal is recovering the right surfaces at the right depths, since conditional depth accuracy and correct ordering can conceal incomplete or excessive predictions.The conclusion motivates evaluating localization, recovered support, and overprediction together.Source: §6 · p. 9
  • Synthetic-only task training leaves room for stronger real-world generalization through mixed synthetic–real supervision.Video and streaming reconstruction are proposed as future directions toward temporally consistent recovery.Source: §6 · p. 9
  • The quantitative benchmarks adapt existing datasets rather than collecting new data, and real-world deployment safety and downstream control remain future validation needs.Source: ETHICS STATEMENT · p. 10

A.1.1 WHAT THE COVERAGE OBJECTIVE ENCOURAGES

  • Under the stated coverage-objective assumptions, both losses reach their lower bounds by covering every target and centering every component on a target, so surplus components duplicate target depths.Shared parameters, regularization, finite training, and filtering can alter this idealized behavior.Source: §A.1.1 · p. 15

A.1.2 A POISSON INTENSITY DEFINES A DIFFERENT SET MODEL

  • The Poisson set model allows one component to explain multiple targets and has unbounded count, whereas ExactMB bounds count with optional singletons and injective assignments.Source: §A.1.2 · p. 15

A.1.3 REFERENCE MEASURE, NORMALIZATION, AND PROPRIETY

  • The normalized finite-set logarithmic score is uniquely minimized by the true density under the theorem’s conditions, but this propriety does not establish component identifiability, calibration, or optimization.Source: §A.1.3 · p. 16

A.1.4 COUNT AND CONDITIONAL LOCATION DENSITY; A.2 SELECTING COMPONENTS VERSUS ASSIGNING SURFACES

  • The count–geometry factorization shows that for M=K the component subset is fixed, while as many as K! correspondences may remain plausible.For M=1, within-subset correspondence is unique and only component selection remains ambiguous.Source: §A.1.4 · p. 16, §A.2 · p. 16

A.3 MASS CORRECTION AND SPECIALIZATION ARE DIFFERENT DIRECTIONS

A common shift in existence logits can match expected count without changing component ownership, while relative changes can alter selection; matching count therefore does not settle assignment. The implementation then specifies the shared model, training interface, schedule, and decoding rules used for evaluation.

A.3 MASS CORRECTION AND SPECIALIZATION ARE DIFFERENT DIRECTIONS

  • For 0 < M < K, a common logit shift has a unique finite optimum matching expected count without changing ownership; relative logit changes can alter selection.At fractional symmetry qj = M/K, the existence gradient vanishes but the Hessian has a positive common-mode eigenvalue and negative contrast-direction eigenvalues.Source: §A.3 · p. 17
  • Four .25 gates have mean count one, count-one probability 27/64, and variance 3/4, so expected count alone does not ensure a concentrated pre-decoding count distribution.Thresholding may retain all four gates or none, and merging can remove coincident depths.Source: §A.3 · p. 17

B IMPLEMENTATION, EVALUATION, AND REPRODUCIBILITY

  • The implementation describes auxiliary-free ExactMB, evaluation, and annotation semantics while separating primary ablations from diagnostic and fixed-ray studies.The reported implementation choices connect to their operational roles in Table S1.Source: B IMPLEMENTATION, EVALUATION, AND… · p. 18, Table S1 · p. 19

B.1.1 FROM RGB AND ANNOTATIONS TO SLOT PARAMETERS

  • The image is encoded once, then four recurrent passes predict absolute depth centers, Laplace scales, and existence logits.Depth centers use 20 softplus(u); scales have a positive floor and are clipped to [0.1, 10], while existence is sigmoid-transformed.Source: §B.1.1 · p. 18

B.1.2 PRESENCE-ALIGNED RECURRENT UPDATE

  • The presence-aligned update removes a gated feature contribution subject to a relative norm cap; detaching the gate blocks direct later-slot gradients to earlier existence probabilities.Indirect shared-feature interactions remain, and all K passes execute.Source: §B.1.2 · p. 18

B.1.3 TRAINING LOSS AND THE ANNOTATION INTERFACE

  • Auxiliary-free ExactMB trains on all BHW rays, treating all-false masks as empty targets while element masks do not distinguish empty targets from unannotated rays.Supplied depth ties remain separate valid entries.Source: §B.1.3 · p. 18

B.1.4 TRAINING SCHEDULE AND SAMPLING

  • Training uses four seeds and round-robin cardinality buckets, but this sampling does not equalize full-epoch bucket mass or per-ray exposure.Primary results use terminal/latest checkpoints at 800k updates.Source: §B.1.4 · p. 18

B.1.5 NATIVE SELECTION AND GEOMETRIC DECODING

  • Selection varies by model variant, followed by a shared decoder that filters finite depths and merges nearby candidates in sorted order.Principal thresholds are τ = .01, dmin = .02 m, and Δmerge = .02 m; evaluation does not use target cardinality or test-set threshold optimization.Source: §B.1.5 · pp. 18–19

B.1.6 BATCHED EXACTMB IMPLEMENTATION AND CHECKS

The section details batched ExactMB computation and the benchmark-specific evaluation adapters. These protocols measure distinct combinations of ordering, presence, cardinality, and metric localization, while identifying limits in annotation coverage, replication, and released-system comparability.

B.1.6 BATCHED EXACTMB IMPLEMENTATION AND CHECKS

  • The batched ExactMB implementation scores target-column pair costs, sums over cached permutations, and corrects for permutations of indistinguishable null columns.For K = 4 it differentiates through 24 padded permutations at arithmetic cost O(BHW K!K), with checks against direct injections, target permutations, and finite-difference gradients.Source: §B.1.6 · pp. 19–20

B.2 EVALUATION DEFINITIONS AND ADAPTERS

  • Benchmark ACC refers to ordinal-tuple, presence-pattern, or count accuracy depending on annotations; none provides dense real multilayer metric ground truth or complete physical-interface enumeration.LD OverPred and MD OverPred use different annotation-specific rejection definitions.Source: §B.2 · p. 20

B.2.1 SELECTION AND DIAGNOSTIC CONTEXTS

  • The protocols separate principal from diagnostic settings, with “before fallback” scores excluding shallower-depth substitution and geometric comparisons reported separately.The appendix compares 20 geometric checkpoints and cutoff curves for 12 Bernoulli-slot checkpoints alongside native fixed-count and count-regularized exports.Source: §B.2.1 · p. 20, Table S3 · p. 20

B.2.2 LD-REAL: RECOVERY AND REJECTION OF QUERIED RANKS

  • LD-Real scores valid quadruplet recovery and all-absent rejection separately; its OverPred denominator is invalid quadruplets, while headline ACC is micro-averaged over valid quadruplets.Requested ranks must all be present in strict near-to-far order for valid tuples, and all absent for correct rejection.Source: §B.2.2 · p. 21

B.2.3 MD-3K: PATTERNS AND MATERIAL-CONDITIONED GEOMETRY

  • MD-3K maps foreground and optional transparent-point background to L1 and L3 validity, then evaluates pair-level presence patterns and material-conditioned ordering.Joint correctness combines presence and both foreground/background orderings; headline multilayer ordinal ACC pools TT and TO, whereas Figure 5 uses all pairs.Source: §B.2.3 · p. 21

B.2.4 LD-SYN: CARDINALITY AND METRIC LOCALIZATION

  • LD-Syn evaluates four metric channels on 500 held-out images, with principal cardinality scores on occupied rays and localization errors conditional on each rank’s valid target-prediction intersection.Conditional error must be read alongside support because abstaining on difficult rays changes the evaluated population.Source: §B.2.4 · pp. 21–22

B.2.5 REPLICATION AND RETROSPECTIVE SELECTION

  • Training seed is the replication unit, and four seed pairs cannot attain a two-sided exact sign-flip p < .125; primary outcomes were organized retrospectively without preregistration or score-blind selection.The analysis emphasizes effect sizes and metric trade-offs rather than pixel/image variation as seed replication.Source: §B.2.5 · p. 22

B.2.6 RELEASED-SYSTEM TRANSFER AND ADAPTER BOUNDARIES

  • Released-system scores measure transfer to supplied visible targets, but differing masks, ordering, scale, training, and stochastic generation prevent attributing differences solely to architecture.Released-model AbsRel uses per-image, per-layer affine calibration, so it assesses conditional within-layer localization after alignment rather than native scale or inter-layer geometry.Source: §B.2.6 · p. 22

B.3 ABLATION FACTORS AND CONFIGURATION PROVENANCE

The ablation study compares configurations under shared primary-study conditions while distinguishing controlled assignment comparisons from broader cross-group changes. Configuration records support the reported comparisons but do not fully identify historical training provenance.

B.3 ABLATION FACTORS AND CONFIGURATION PROVENANCE

  • The five primary variants share data, encoder/recurrent trunk, capacity, optimization schedule, four-seed terminal-800k protocol, and disabled geometric auxiliary terms.They test how emitted cardinality is determined and observed depths are assigned to components.Source: §B.3 · p. 22
  • MAP, ExactMB, and ordered assignment share model and training fields within each seed and differ only in assignment criterion, while count regularization versus fixed cardinality changes a complete stopping configuration.Count regularization changes the categorical predictor, loss, and selector; cross-group comparisons also change recurrence gating and selection.Source: §B.3 · pp. 22–23
  • The separate PPP, MAP, and ExactMB study uses a different sampler and schedule with incomplete launch records, and its ExactMB–MAP MD pattern-accuracy difference changes sign across cohorts.It is −1.35 ± 6.69 points in the primary study and +1.96 ± 1.37 in the mechanism study, making interpretation recipe-specific.Source: §B.3 · p. 23
  • Checkpoint and evaluator hashes cover study endpoints, but records do not identify the source version throughout every historical training segment.Primary exports describe resumed states without a source commit, and synthetic geometry records lack a historical evaluator-source hash.Source: §B.3 · p. 23

B.4 ANNOTATION SEMANTICS AND THE TRAINING INTERFACE

The section examines annotation assumptions and how count, assignment, and geometric supervision shape visible-surface recovery. Results show that rankings vary by cardinality, material pattern, and evaluation choice.

B.4 ANNOTATION SEMANTICS AND THE TRAINING INTERFACE

  • The annotations may not satisfy the likelihood’s completeness and distinctness assumptions; audits find quantized ties and close depth pairs, and compact prefixes do not certify sorted, distinct targets.The primary variants differ in emitted-depth selection and target assignment; released-model transfer is a separate comparison without retraining.Source: §B.4 · p. 23, §C.1 · p. 23, §C.1.1 · pp. 23–24

C.1.1 PAIRED EFFECTS OF EXPLICIT COUNT REGULARIZATION

  • Count regularization reduces false emission but loses valid returns, including in LD valid-query accuracy, transparent-point recall, and four-return count accuracy.These are paired complete-configuration comparisons, not an isolated count-loss effect; all four seeds share the displayed directions.Source: §C.1.1 · pp. 23–24, Table S4 · p. 24

C.2 PRIMARY FOUR-SEED GEOMETRIC RECOVERY

  • Ordered assignment improves depth-accurate recall and precision over marginalization at every rank in all four paired seeds.At rank four, gains are 13.95 ± 0.58 and 8.76 ± 0.59 percentage points, respectively; count regularization has higher precision at ranks 2–4 but lower recall than ordered assignment.Source: §C.2 · p. 24

C.3.1 CARDINALITY AND MATERIAL-CONDITIONED REVERSALS

  • Count-regularized rankings reverse at full cardinality: its exactness is 52.5% at c = 4, versus 71.4% for ordered assignment.Count regularization leads at c = 1, 2, and 3; rankings also vary by material pattern, where ordered assignment improves TT and TO pattern accuracy over ExactMB but worsens OO and raises OverPred.Source: §C.3.1 · p. 25, Figure S1 · p. 26

C.3.2 JOINT EXISTENCE AND ORDERING FAILURES

  • From pairs to quadruplets, correct-existence/wrong-order outcomes rise from 13.72% to 31.87%, while mixed-rank existence failures rise from 23.97% to 27.15%.In the separate LD geometry study, quadruplets also have a larger ordering-failure share than pairs even when queried ranks exist.Source: §C.3.2 · p. 26, Figure S2 · p. 27

C.4.1 POINT-PROCESS LAW AND ASSIGNMENT REDUCTION

  • PPP→MAP raises MD pattern ACC by 14.39 ± 2.84 points and synthetic occupied-micro count ACC by 8.79 ± 0.15, while lowering MAE by 0.121 ± 0.003 layers.This contrast bundles changes in point-process law, multiplicity, and assignment; MD point-level OverPred changes by +0.32 ± 1.59 points.Source: §C.4.1 · p. 27, Figure S3 · p. 28
  • MAP→ExactMB lowers LD tuple-level OverPred by 1.56 ± 0.47 points and MD point-level OverPred by 5.31 ± 1.49, while count MAE worsens by 0.0069 ± 0.0030 layers.The change isolates the recorded assignment criterion; its near-zero synthetic occupied-micro exactness change does not establish equivalence.Source: §C.4.1 · p. 27, Figure S3 · p. 28

C.5 MATCHED GEOMETRIC-LOSS INTERVENTIONS

  • Geometric-loss interventions have mixed effects: stronger, faster-ramping SortedGrad improves LD valid-query accuracy and last-visible ordering while reducing LD OverPred, but MD and synthetic-count effects vary across seeds.Intensity supervision reduces mean MD OverPred while lowering MD pattern exactness and raising LD OverPred; AssignGrad’s ordering and MAE effects also depend on SortedGrad strength.Source: §C.5 · p. 29, Table S7 · p. 29

C.6 ENCODER CAPACITY AND DECODER DESIGN

Encoder scaling and decoder design affect recovery differently across metrics and ranks. Archived operating points also show that precision gains can come with lower recall.

C.6 ENCODER CAPACITY AND DECODER DESIGN

  • ViT-S/B/L scaling raises LD ACC from 53.60% to 57.44% and 63.62%, and synthetic count ACC from 81.50% to 85.48% and 88.15%, while MD pattern accuracy peaks at Base.MD pattern accuracy is 28.52%, 46.69%, and 46.24% for Small, Base, and Large, respectively.Source: §C.6 · p. 30, Figure S4 · p. 30
  • Independent heads selectively improve ordinal and presence-pattern accuracy over shared parallel FiLM, but four-return count ACC falls by 4.88 ± 0.39 points and LD last-visible ACC by 3.87 ± 1.30.They use four times as many non-encoder parameters and raise peak memory from 1.94 to 2.50 GiB despite similar operations and throughput.Source: §C.6 · p. 30, Figure S5 · p. 31

C.7 ARCHIVED DECODER OPERATING POINTS

  • ExactMB’s .5 cutoff lowers LD ACC/OverPred by 4.90/6.07 points, raises synthetic occupied-micro count ACC by 5.82 points, and lowers count MAE by .117 layers relative to .01.Count agreement and false emission improve while ordinal recovery falls; the comparisons use separate archived prediction passes.Source: §C.7 · p. 31, Figure S6 · p. 32

C.7.1 GEOMETRIC PRECISION–RECALL ACROSS ARCHIVED OPERATING POINTS

  • ExactMB fourth-rank precision rises from 14.51% to 50.85% as recall falls from 39.34% to 25.70% between archived .01 and .5 passes.At rank four, ordered assignment at .5 has lower recall but higher precision than count regularization: 36.87% versus 40.47% recall and 59.88% versus 52.46% precision.Source: Figure S7 · p. 32, §C.7.1 · p. 33, Table S8 · p. 33
  • The count-regularized model’s archived operating points vary by up to 2.34 recall and 2.21 precision points despite its τ-independent selector; thresholds are not calibrated or independently validated.The rank-wise aggregate records do not provide the joint per-ray residuals needed to assess injective matching or whole-set correctness.Source: Figure S6 · p. 32, Figure S7 · p. 32, §C.7.1 · p. 33

C.8 ADDITIONAL REAL-WORLD QUALITATIVE RESULTS

Additional results show qualitative out-of-distribution examples and examine training dynamics. The dynamics distinguish expected-count fitting from discrete activation, with a held-out deficit at high cardinality.

C.8 ADDITIONAL REAL-WORLD QUALITATIVE RESULTS

  • On fourteen real-world images outside synthetic-only task training, ExactMB’s deeper layers show spatially selective support, while several released-model outputs repeat foreground or background structure across ranks.The comparisons include the auxiliary-regularized ExactMB checkpoint and released SeeGroup, LaRI, and World Tracing models; color indicates relative depth, not comparable distances across methods.Source: §C.8 · p. 33, Figure S8 · p. 34, Figure S9 · p. 35

D.1 COUNT, ACTIVATION, AND HELD-OUT LEARNING

  • Early ExactMB thresholded activation is far below MAP and ordered assignment despite similar expected counts: at c = 1 it averages .037 ± .010 components versus 1.023 ± .007 and 1.029 ± .005.At c = 4, ExactMB instead averages 2.954 ± .119 activations, higher than MAP’s 2.041 ± .136 and ordered assignment’s 2.172 ± .060 in every seed.Source: §D.1 · pp. 48–49, Figure S22 · p. 48
  • At termination, held-out thresholded counts average 2.56–2.60 layers for c = 4, versus four target layers; that stratum is 53 of 64 images but only 1.42% of pixels.The low pixel share dilutes the roughly 1.4-layer conditional deficit in all-pixel averages.Source: Figure S22 · p. 48, §D.1 · p. 49
  • The diagnostic occupied-existence cost falls on training batches but rises on held-out data for all three variants in all four seeds.For ExactMB, paired changes are −.0365 ± .0037 and +.0681 ± .0352 nats/pixel; interpretation is qualified by differences in cropping, model mode, composition, averaging, and posterior weights.Source: §D.1 · p. 49

D.2 FIXED-RAY UNCERTAINTY AND GEOMETRIC RECOVERY

Fixed-ray analyses separate uncertainty about occupied components from uncertainty about their target correspondence. The reported entropy and localization-scale trends differ across cardinalities and aggregation levels.

D.2 FIXED-RAY UNCERTAINTY AND GEOMETRIC RECOVERY

  • The fixed-ray analysis excludes 185 occupied rays with exact ties and evaluates supplied targets of unknown completeness.After exclusions, 4,513 positive, finite, distinct-target rays remain; tie exclusion removes 184 of 1,049 sampled four-target rays.Source: §D.2 · p. 49
  • In the auxiliary-regularized ExactMB trajectory, subset entropy falls from .289 to .178 at M = 2 and .378 to .222 at M = 3, while correspondence entropy changes less.At M = 4, subset entropy is zero but correspondence entropy remains .787 to .719; for M = 4, expected-count MAE falls while count NLL rises.Source: §D.2 · pp. 49–50, Figure S23 · p. 50
  • For M = 4, expected-count MAE falls from 1.350 to 1.224 layers while count NLL rises from 2.627 to 2.845 nats/ray.At M = K, the scores apply different penalties, permitting opposite trends; this is a single training run.Source: §D.2 · p. 50, Figure S23 · p. 50
  • Across epochs 22 to 181, extreme slot probabilities rise from 89.90% to 93.30% overall, versus 39.41% to 60.41% on multilayer rays, so pooling obscures their uncertainty.Only 8.67% of the audited pixels have at least two supplied targets.Source: §D.2 · pp. 50–51
  • The multiplicative localization-scale gap narrows from 1.513 to 1.311 as pair weight near the .1 m floor rises, but pairs with a scale above .101 m show no comparable narrowing.The analysis uses posterior-co-occupied slot pairs from 19 images with multilayer targets.Source: §D.2 · p. 51

D.2.1 CANDIDATE GEOMETRY VERSUS SELECTION AND FILTERING

D.2.1 separates candidate geometry from decoder filtering and shows that missing accurate candidates dominate four-target recovery deficits. D.3 shows that cardinality weighting changes the preferred operating point, while fallback can make depth error appear better as retained support shrinks.

D.2.1 CANDIDATE GEOMETRY VERSUS SELECTION AND FILTERING

  • Across 18 paired four-target images, proposal recall rose 2.95 ± 12.59 points and emitted recall 1.54 ± 10.90 points, while gap-stage loss rose .80 →1.98 points and unmatched emissions fell 2.487 →2.313 per ray.At M = 1, proposal recall fell 38.67 →34.38% while emitted recall changed 25.23 →25.94%.Source: §D.2.1 · p. 51
  • On four-target rays, exact output count fell 64.37 →52.38% while SetOK.05 rose .121 →.686%, showing that count agreement and geometric recovery are distinct evaluation events.At .02 m, emitted recall changed 13.44 →13.88% and complete recovery remained zero; at .10 m they changed 41.14 →42.57% and 4.42 →5.35%.Source: §D.2.1 · p. 51
  • At the final stage, four-target proposal recall was 29.83% and emitted recall 27.63%; all-candidate matching missed 70.17% of returns, versus .224 percentage points lost at the gate and 1.984 at the gap rule.Figure S24 accounts for these losses on the same fixed rays.Source: §D.2.1 · p. 51, Figure S24 · p. 52

D.3 CARDINALITY WEIGHTING AND RETAINED SUPPORT

  • At .5→.3 evaluation passes, micro exact-count accuracy decreased by .343 points while cardinality-macro accuracy increased by 1.730 points and accuracy at c = 4 increased by 7.602 points.All 16 configuration families shared these directions, so the preferred global operating point depends on cardinality weighting.Source: §D.3 · p. 53
  • From τ = .01 to .9, L7 own-slot support fell 69.31 →27.29%, while fallback AbsRel improved 17.15 →14.98% and own-slot AbsRel ended slightly worse at 21.86 →22.05%.Last-visible support remained 100.00 →99.52% because an absent deep slot inherits the last surviving shallower prediction.Source: §D.3 · p. 53

E OUTPUT AND MEMBERSHIP CONVENTIONS

  • Multilayer methods differ in native outputs, supervised geometry, and treatment of unused capacity; variable-count decoding differs from explicit component-wise presence and absence modeling.Fixed-capacity outputs can yield variable counts through validity masks, stopping, or repeated-point removal; ExactMB uses untruncated densities before positive-depth filtering, while ordered assignment fixes correspondence.Source: Table S9 · p. 54, E OUTPUT AND MEMBERSHIP CONVENTIONS · p. 54

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