ToothForge: Automatic Dental Shape Generation using Synchronized Spectral Embeddings

Fran\c{c}ois Guibault; Herv\'e Lombaert; Michal \v{S}pan\v{e}l; Tibor Kub\'ik

REVIEW 3 major objections 4 minor 1 cited by

ToothForge: Automatic Dental Shape Generation using Synchronized Spectral Embeddings

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read ToothForge claims that synchronizing spectral embeddings before training a beta-VAE produces more accurate tooth-shape reconstructions than unaligned spectra, while generating a 10,000-vertex tooth in under a millisecond.

desk verdict A plausible spectral generative model for teeth, but the key comparison is confounded and the synchronization step is under-specified. read the letter →

arxiv 2506.02702 v1 pith:IPSSWGR4 submitted 2025-06-03 cs.CV

Tibor Kubík , François Guibault , Michal Španěl , Hervé Lombaert This is my paper

classification cs.CV

keywords 3Dtoothshapegenerationspectrallearningsynchronizedembeddingsbeta-VAEdigitaldentistrygeometricdeepcompressioninterpolation

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper argues that a generative model for 3D tooth shapes should be trained on spectral coefficients that have been aligned to a common reference basis before training, rather than on each tooth's raw spectrum. On a private dataset of 430 dental crowns, this synchronized training gives lower reconstruction error than training on unaligned spectra and allows meshes with different triangle connectivities to be mixed in one model. The aim is to let dental labs synthesize realistic high-resolution tooth meshes in under a millisecond, so small or imbalanced datasets can be augmented cheaply. If the central claim holds, the same recipe should transfer to other anatomical shapes collected from different scanners, where a shared mesh connectivity cannot be assumed.

What carries the argument

The load-bearing object is the synchronized spectral embedding. For each mesh $M_i$ with truncated eigenbasis $\Phi_{k,i}$ and coefficient array $C_{k,i}$, a reference mesh supplies a common basis $\Phi_{k,\text{ref}}$, and an assumed spectral transformation $R_i$ built from a correspondence map $c_i$ is optimized by minimizing $\|\Phi_{k,i}c_i C_i - \Phi_{k,\text{ref}}R_i C_i\|_2$, with symmetry enforced by the inverse mapping. All aligned coefficient arrays $R C_i$ are then used to train a β-VAE with cyclical annealing, and at synthesis the decoder's output $\hat{C}_k$ is multiplied by $\Phi_{k,\text{ref}}$ and paired with the reference mesh's edges to produce a new tooth mesh.

What would settle it

Run the pipeline on meshes with deliberately scrambled vertex orders and no known correspondence, or on synthetic tooth pairs with ground-truth correspondence; if reconstruction error trained on synchronized spectra is no better than trained on unaligned spectra, or the alignment optimization fails to converge, the central claim fails.

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Extended reading notes

Core claim

ToothForge's central claim is that the instability of Laplacian harmonic decomposition biases generative models unless all shape spectra are first synchronized to a common eigenbasis. The authors report that a β-VAE trained on aligned spectral coefficients of 256 harmonics reconstructs unseen teeth with lower mean squared error and synthesizes shapes with lower Minimum Matching Distance (MMD) than the same model trained on unaligned coefficients, and that a generated tooth with more than 10,000 vertices costs roughly 0.7–0.8 ms for decoder inference plus projection into spatial coordinates. They further claim this removes the fixed-connectivity restriction of earlier spectral autoencoders, because the alignment supplies a common reference basis and a vertex-wise correspondence, and that 256 spectral coefficients capture realistic tooth morphology where the same number of spatial points produces noisy meshes missing molar cusps.

Load-bearing premise

The method rests on reliably matching every tooth mesh to a single reference mesh; if that matching is wrong or fails to converge, the synchronized spectra lose their advantage and the claimed benefit collapses.

Editorial extensions

If this is right

Training on synchronized embeddings yields lower spectral and spatial reconstruction error and lower Minimum Matching Distance for synthesized teeth than training on unaligned embeddings.
A new tooth mesh with more than 10,000 vertices can be generated in about 0.7–0.8 ms on a single GPU, including decoder inference and spatial reconstruction.
The fixed-connectivity constraint of earlier spectral generative models is removed, so datasets assembled from different intra-oral scanners with different mesh connectivity can be used together.
The learned alignment gives vertex-wise correspondence among all generated shapes, which can support downstream shape analysis.
256 spectral coefficients are sufficient for realistic tooth reconstructions, while the same number of spatial point coordinates produces noisy meshes that miss features such as molar cusps.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the correspondence optimization is robust in practice, ToothForge effectively turns spectral generation into a template-based method: every synthesized tooth shares the reference mesh's edges, so downstream tasks receive a consistent mesh structure and correspondence for free.
The same synchronization idea could be attached to other generative backbones, such as diffusion models trained on latent codes or on aligned spectra, if larger dental datasets become available.
A direct test of the mechanism is to measure how reconstruction error degrades as the alignment optimization is perturbed, or to compare ToothForge against a spatial autoencoder with identical latent size and vertex count.
The paper motivates tooth synthesis as data augmentation but does not itself measure improved downstream task accuracy; that is a testable consequence rather than a demonstrated result.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

To be fair, the idea is neat: train a β-VAE on synchronized spectral coefficients so that meshes with different connectivity can share a common basis, and generate a 10k-vertex tooth in under a millisecond. That's a real practical contribution for digital dentistry, and the paper is clearly written. The code is public, and the reported reconstruction errors and generation times are concrete. I believe the authors when they say this works.

The soft spot is exactly the one the stress-test hits. Figure 5 compares ToothForge (k=256) against SAE-LP-128, so the 'without synchronization' row uses half the coefficients. Increasing k alone improves shape recovery (their own Fig. 1), so the visual superiority of ToothForge could be due to the extra coefficients, not the synchronization. Table 1 gives no unaligned baseline. The text claims 'reconstruction accuracy is consistently higher' but the only displayed evidence is not controlled. That is a fixable flaw—add a k=256 unaligned baseline and report dMSE for both—but as it stands, the central claim is not established.

The synchronization step itself is also under-specified. Equation (5) assumes a correspondence map c_i and says it is optimized by minimizing (6), but no algorithm, initialization, or evidence that the optimization succeeds is given. For a method whose selling point is handling variable connectivity, this matters. It could be that they use some off-the-shelf functional map routine; the paper doesn't say. That's a reproducibility gap.

Minor points: the spectral synchronization idea is standard in the shape correspondence literature and should be cited; and the lack of any downstream augmentation experiment means the stated motivation ('enhance downstream tasks') is untested. Neither is fatal.

Overall, this is a genuine extension of [14,15] to a useful domain, with a public implementation and plausible numbers. The gaps are addressable in revision. I'd send it to review; the reviewers should ask for a controlled ablation and a precise description of the alignment optimization. I wouldn't cite it in my own work until those are in the paper.

Referee Report

3 major / 4 minor

Summary. The paper presents ToothForge, a generative model for dental crown meshes based on a β-VAE trained on spectral (Laplace-Beltrami) coefficients. The key methodological novelty is 'spectral synchronization': before training, the eigenvectors of each mesh are aligned to a common reference eigenbasis using an estimated correspondence map c_i, and the autoencoder is trained on the aligned coefficients. After training, new shapes are obtained by decoding sampled latents and projecting them through the reference basis. The authors report reconstruction errors of order 0.002 (spatial MSE), MMD values below 0.008, and generation times under 1 ms for 10,000+ vertex meshes. They also qualitatively compare against an unaligned, unregularized baseline (SAE-LP-128) and a spatial-coefficient baseline.

Significance. If substantiated, the approach would be a meaningful step for shape generation in medical imaging, particularly because it attempts to remove the fixed-connectivity assumption of prior spectral autoencoders and demonstrates near-interactive generation speed. The authors provide code, and the idea of synchronizing spectra across a training set before fitting a generative model is an interesting and largely unexplored direction. However, the evidence for the central claim—that synchronization improves generation—is currently qualitative and confounded; the paper's contribution would be significantly strengthened by a quantitative, controlled ablation.

major comments (3)

[§4 / Fig. 5 vs. Table 1 and §3 Implementation details] The claim that synchronized embeddings improve reconstruction over unaligned ones is not supported by a controlled comparison. Fig. 5 contrasts ToothForge with 'SAE-LP-128 (ToothForge−sync.−reg.)'; since SAE-LP-k denotes k spectral coefficients (Sec. 2.2) and ToothForge uses k=256 (Sec. 3), the baseline uses only 128 coefficients. Fig. 1 shows that increasing k alone substantially improves shape recovery, so any qualitative improvement in Fig. 5 may result from the extra 128 coefficients rather than from synchronization or regularization. Table 1 reports no dMSE-spectral, dMSE-spatial, or MMD numbers for models trained on unaligned spectra. The Conclusion statement that 'Reconstruction accuracy is consistently higher when training on aligned embeddings, supported by qualitative analysis' is therefore not backed by quantitative evidence. Please add same-k ablations (e.g., 256 vs 256) with and without synchronization and with and without KL regularization, reporting all three metrics.
[§2.3, Eq. (5)-(6)] The spectral synchronization step is under-specified. Eq. (5) defines R_i in terms of the unknown correspondence map c_i, and Eq. (6) minimizes an L2 difference between aligned coefficients, but the manuscript never states how c_i is represented (e.g., hard permutation, soft assignment), what constraints it satisfies, how it is initialized, which optimization algorithm is used, or how 'symmetry is enforced by adding the inverse mapping.' As written, R_i and c_i are mutually dependent, so the objective is not well-defined until a concrete alternating or joint estimation procedure is given. Without such a specification, the method cannot be reproduced or judged. Moreover, no evidence is provided that solving Eq. (6) reliably recovers meaningful correspondences on dental meshes with variable connectivity, which is the very assumption on which the method's benefit rests. Please provide the full algorithm and a quantitative validation of alignment (e.g., spectral coefficient distance before/after synchronization, or correspondence accuracy on meshes with known ground-truth correspondence).
[§3 Dataset and §1 Introduction] The central motivation for spectral synchronization is that dental meshes from different clinics have different connectivity, but the experimental section does not report any connectivity information about the 430 crowns used. It is never stated whether the dataset contains meshes with varying vertex counts, triangulations, or topology, nor how many distinct connectivities are present. Without this information, the paper does not actually demonstrate that the method handles variable connectivity. Please include dataset statistics (vertex count ranges, edge counts, number of distinct triangulations) or run an experiment in which connectivity is deliberately varied while geometry is held fixed.

minor comments (4)

[§3, Evaluation metrics] Table 1 reports MMD without specifying whether dMSE or dCD was used as the structural distance; please state it in the table or caption.
[§1, Introduction] There is a typo 'potentionally' in the contributions paragraph; also the phrase 'comparative benchmarks highlight the superiority of our method over spatial approaches' is not backed by a table comparing with spatial methods, though Fig. 7 does provide some evidence for coefficient efficiency.
[Fig. 1 caption] The colorbar label 'relative Shape Recovery error 0%1%2+%' is difficult to read; please format it as a standard colorbar.
[Fig. 7 caption] The statement 'Values are scaled by 10' is ambiguous; please clarify the scaling and whether the axes are linear or logarithmic.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: ToothForge's synchronized-spectral derivation is self-contained; the main weakness is an experimental confound, not circular reasoning.

full rationale

The derivation chain is not circular. The spectral decomposition (Eqs. 1-3) is standard linear algebra on the cotangent Laplacian, the baseline spectral autoencoder is adopted from the external work of Lemeunier et al. [14], and the beta-VAE objective (Eq. 7) is a standard reconstruction-plus-KL loss. The spectral synchronization step (Eqs. 5-6) defines an alignment transformation and an optimization objective for the unknown correspondence map c_i; this is an algorithmic proposal, not a fitted parameter renamed as a prediction. The central empirical claim that aligned embeddings improve reconstruction is evaluated on a private dataset, and although the comparison in Fig. 5 is confounded (ToothForge uses k=256 while the unaligned SAE-LP baseline uses k=128) and Table 1 reports no quantitative unaligned baseline, that is a weakness in experimental support, not a case where the conclusion is equivalent to the input by construction. Self-citations to Lombaert's earlier spectral work [8] and to Guibault's dental-crown paper [11] appear only as background or downstream-task context; neither is load-bearing for the claimed result. The under-specified optimization for c_i is a reproducibility risk but does not make the derivation circular. No prediction reduces to a fitted constant or to a self-citation chain, so the circularity score is 0.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central claim rests on standard spectral geometry math plus a load-bearing ad hoc assumption that a spectral transformation and correspondence exist and are recoverable by the unspecified alignment optimization. The chosen hyperparameters (k=256, latent=16, beta schedule) influence the results but are not fitted constants in the derivation.

free parameters (3)

Number of spectral coefficients k = 256
Chosen by hand; limits spectral input dimension and reconstruction detail. Central to the method's compactness claim.
Latent dimension = 16
Chosen by hand; controls smoothness and capacity of the generative manifold.
Beta (KL weight) schedule = 0 to 0.05
Cyclical annealing schedule; chosen by hand to balance reconstruction and smoothness.

assumptions (4)

standard math Cotangent-weight discretization of the Laplace-Beltrami operator correctly approximates the continuous operator on tooth meshes.
Equation (1); standard FEM discretization from Meyer et al. [17].
domain assumption Dental crown shapes can be represented well by the first 256 spectral coefficients.
Supported by Fig. 1 but not by a formal error bound; depends on tooth geometry.
ad hoc to paper A meaningful spectral transformation R_i exists between each mesh's eigenbasis and the reference eigenbasis (Eq. 5), and minimizing Eq. 6 recovers it.
The paper says 'We assume the existence of a spectral transformation' and gives no convergence or correctness analysis, making this the load-bearing assumption of the synchronization step.
domain assumption The stochastic latent space of the beta-VAE is smooth enough that samples from N(0,I) produce plausible teeth.
Evaluated indirectly via MMD, but no explicit verification of latent space smoothness.

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Cite this review

Pith. "Pith review of ToothForge: Automatic Dental Shape Generation using Synchronized Spectral Embeddings." pith.science (2026). https://pith.science/paper/IPSSWGR4

@misc{pith2026250602702,
  author       = {Pith},
  title        = {Pith review of: ToothForge: Automatic Dental Shape Generation using Synchronized Spectral Embeddings},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IPSSWGR4}},
  note         = {Machine review of arXiv:2506.02702}
}

read the original abstract

We introduce ToothForge, a spectral approach for automatically generating novel 3D teeth, effectively addressing the sparsity of dental shape datasets. By operating in the spectral domain, our method enables compact machine learning modeling, allowing the generation of high-resolution tooth meshes in milliseconds. However, generating shape spectra comes with the instability of the decomposed harmonics. To address this, we propose modeling the latent manifold on synchronized frequential embeddings. Spectra of all data samples are aligned to a common basis prior to the training procedure, effectively eliminating biases introduced by the decomposition instability. Furthermore, synchronized modeling removes the limiting factor imposed by previous methods, which require all shapes to share a common fixed connectivity. Using a private dataset of real dental crowns, we observe a greater reconstruction quality of the synthetized shapes, exceeding those of models trained on unaligned embeddings. We also explore additional applications of spectral analysis in digital dentistry, such as shape compression and interpolation. ToothForge facilitates a range of approaches at the intersection of spectral analysis and machine learning, with fewer restrictions on mesh structure. This makes it applicable for shape analysis not only in dentistry, but also in broader medical applications, where guaranteeing consistent connectivity across shapes from various clinics is unrealistic. The code is available at https://github.com/tiborkubik/toothForge.

Figures

Figures reproduced from arXiv: 2506.02702 by the authors.

**Figure 2.** Impact of spectral synchronization. We perform spectral synchronization across all shapes in the dataset as depicted in (a). This process standardizes the harmonics, ensuring they all speak a common language. As illustrated in (b), ommiting this step leads to incorrect reconstructions, introducing significant noise into the network when trained on unaligned data. ensures fidelity to the original representation with… view at source ↗

**Figure 3.** Effect on regularization of latent space modeled on synchronized [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: Method outline. Our framework for generating synthetic shapes of teeth via sampling on a latent manifold. Such manifold is modeled using spectral synchronized (spectrally aligned to common basis) coefficients of tooth shapes, denoted as C. For novel data sampling using…

**Figure 5.** Figure 5: Synthetic data generation via interpolation in latent space. [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

**Figure 6.** Figure 6: We use the network to encode and decode the spectral coefficients of the [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]

**Figure 6.** Figure 6: Reconstructions of unseen tooth shapes. The predictions accurately capture the overall tooth shape. The reconstructions may appear smoothed at times (rightmost premolar). This is due to inaccuracies in predicting high-frequency components. structions occasionally app…

**Figure 7.** Figure 7: Spectral vs. spatial coordinates in tooth analysis. [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

SCALP: Semi-Supervised Statistical Shape Modeling from Imperfect 3D Photogrammetry via Landmark-Anchored Spectral Warp
cs.CV 2026-07 conditional novelty 6.0 of 10

A landmark-guided, band-limited spectral template warp, preceded by semi-supervised landmark detection, produces dense and anatomy-confined correspondences directly from imperfect 3D photogrammetry.

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