Observation-Conditioned Latent Energy Priors for Sparse Implicit Neural Shape Completion

• Sparse latent optimization can fit observed SDF samples while drifting toward codes that decode implausible unobserved geometry.Source: §1 · p. 2
• Global L2 and density priors regularize latent plausibility but score codes independently of the particular sparse observation set.L2 provides stable scale control, while stronger global priors capture latent-distribution structure without conditioning on measurements.Source: §1 · p. 2
• The proposed post-hoc conditional energy is trained from frozen-decoder latents paired with subsampled coordinate/SDF observations and acts as an observation-aware residual prior.Its observation encoding is permutation-invariant, allowing unordered sparse sets to condition the latent preference.Source: §1 · p. 2, Figure 1 · p. 3
• The method is evaluated on a controlled cell-nucleus SDF testbed and a public heterogeneous MedShapeNet mesh/SDF completion track.Source: §1 · p. 3
• The study identifies latent drift, introduces the conditional energy alongside tuned L2 for frozen-decoder inference, and compares it with global priors and context ablations.Source: §1 · p. 3

Inference

• The workflow freezes a pretrained INR decoder and optimizes only the latent code using observation loss, an L2 prior, and conditional energy.The energy and context encoder are trained after decoder fitting, while test-time inference keeps both decoder and energy fixed.Source: §2 · p. 3, §2.2 · p. 4

Prior construction

• The base L2 prior standardizes latents using training-set mean and scale, controlling latent magnitude while its validation-selected weight absorbs constant normalization factors.The conditional energy serves as an observation-aware residual expert alongside this global scale-control term.Source: §2.2 · p. 4

Observation-conditioned prior

• The observation context encodes unordered coordinate/SDF pairs with permutation-invariant pooling, and the energy scores standardized latents conditioned on that context.The context encoder maps each coordinate–SDF pair through an embedding, pools the set, and produces a context vector for the scalar energy network.Source: §2.2 · p. 4

Energy training

• The conditional energy is trained post hoc from training latents paired with sparse observation sets subsampled from the same shapes.Training samples observation counts from dataset-specific values and uses only train-split latents and observations.Source: §2.3 · pp. 4–5
• Energy Matching regresses gradients along interpolations from Gaussian noise to matched standardized training latents, conditioned on their observation contexts.Ranking losses distinguish matched from mismatched latent–context pairs, while off-manifold negatives are calibrated to have higher energy than matched pairs.Source: §2.3 · p. 5
• The full energy-training loss combines gradient matching, ranking, negative calibration, and scale terms, while interpolants and Langevin-refined negatives are excluded from test-time inference.At inference, the learned energy is used as a differentiable prior rather than recomputing training-only samples.Source: §2.3 · p. 5

Datasets and sparse observations

• The evaluation covers cell-nucleus and MedShapeNet-derived SDF completion with 16–128 and 16–256 observed samples, respectively.The nucleus split has 2,924/699/492 train/validation/locked-test nuclei; MedShapeNet uses six anatomical categories with a balanced 960/120/120 split.Source: §3.1 · p. 6

Metrics

• Held-out SDF error is primary, with Grid SDF MAE, occupancy Dice/IoU, and nucleus surface Dice as secondary readouts.Surface Dice uses a fixed tolerance of 0.1 normalized coordinate units.Source: §3.1 · p. 6

Baselines and protocol

• All methods share the frozen decoder, sparse observations, latent initialization, optimization budget, and evaluation points within each setting.Baselines include No prior, validation-selected Tuned L2, and a six-component global GMM; the main method is L2 + Conditional Energy.Source: §3.2 · p. 6

Nucleus evaluation

• Table 1 reports nucleus locked-test performance for Conditional Energy alone and the combined L2-plus-Conditional-Energy method.The table caption abbreviates Conditional Energy as CondE and the combined method as Ours.Source: Table 1 · p. 7

MedShapeNet evaluation

• Table 2 reports MedShapeNet locked-test performance for the six-component global mixture and the combined L2-plus-Conditional-Energy method.GMM K6 is the six-component global latent mixture baseline.Source: Table 2 · p. 7

Locked-test performance

• On the locked nucleus test set, L2 + Conditional Energy is best or tied-best across sparse observation settings, while Conditional Energy alone remains below Tuned L2.The ablation improves over No prior, supporting use of the conditional energy as a complement to L2 rather than a replacement.Source: §4.1 · p. 7
• On MedShapeNet, L2 + Conditional Energy outperforms Tuned L2 and GMM K6 across all reported readouts and observed-sample counts.The global GMM often improves over Tuned L2, but the observation-conditioned method remains best across the reported MedShapeNet settings.Source: §4.1 · p. 7, Figure 2 · p. 8

Paired gains over global priors

• At 16–64 nucleus samples, gains over Tuned L2 are significant for all four metrics, whereas at 128 samples only surface Dice remains significant.The largest corrected p-value at 16, 32, and 64 samples is 1.38×10−2; surface Dice at 128 samples has p = 8.0×10−4.Source: §4.2 · p. 8, Figure 2 · p. 8

Context ablation

• On MedShapeNet, matched context improves Grid SDF MAE over shuffled context by 0.0026–0.0072 at every observed-sample count.Shuffled context still beats Tuned L2 on Grid SDF MAE, indicating both a general latent-prior component and an observation-specific contribution.Source: §4.3 · p. 8, Table 3 · p. 9

Qualitative examples

• Qualitative reconstructions compare ground truth, Tuned L2, GMM K6, and L2 + Conditional Energy across anatomical categories.The reported comparison emphasizes preservation of category-compatible shape structure under identical sparse observations.Source: §4.3 · p. 8, Figure 3 · p. 9

Results

• Matched context improves over shuffled context at all tested observation counts, supporting an observation-specific contribution beyond a general latent-prior effect.Tuned L2 provides stable global shrinkage, while conditional energy acts as an observation-conditioned complement.Source: §5 · p. 9

Interpretation

• Observation conditioning is most useful when sparse test-time inference is underconstrained, where a global isotropic L2 preference can be too coarse.This interpretation is consistent with the stronger structure and larger nearest-neighbor distances observed in MedShapeNet latents, but the latent comparison alone does not establish the hypothesis.Source: §5 · p. 9, Figure 4 · p. 10

Results

• Matched conditional energy improves on GMM K6 across the reported MedShapeNet readouts, suggesting information beyond a global multimodal latent prior.The latent-space comparison finds stronger category structure and larger nearest-neighbor distances for MedShapeNet than for nucleus latents.Source: §5 · p. 9, Figure 4 · p. 10

Limitations

• The study covers one controlled cell-nucleus SDF setting and one public MedShapeNet-derived SDF setting, limiting the breadth of its evaluation.Broader context randomization, out-of-distribution tests, and additional structured priors remain future work.Source: §5 · p. 10

Conclusion

• The authors suggest lightweight post-hoc priors can make pretrained INR decoders more observation-aware without retraining.Future work includes stronger conditional energies, realistic partial-surface observations, broader INR tasks, and topology-rich reconstruction.Source: §5 · p. 10