K-U-KAN: Koopman-Enhanced U-KAN for 3D Dental Reconstruction from a Single Panoramic X-ray Radiograph

Abhijit Sen; Bikram Keshari Parida; Wonsang You

arxiv: 2605.25163 · v1 · pith:TX5I5OO6new · submitted 2026-05-24 · 💻 cs.CV · cs.AI

K-U-KAN: Koopman-Enhanced U-KAN for 3D Dental Reconstruction from a Single Panoramic X-ray Radiograph

Bikram Keshari Parida , Abhijit Sen , Wonsang You This is my paper

Pith reviewed 2026-06-30 11:42 UTC · model grok-4.3

classification 💻 cs.CV cs.AI

keywords 3D reconstructionpanoramic X-rayKoopman operatorKolmogorov-Arnold Networksdental imagingsingle-view reconstructionU-KANfocal trough

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The pith

K-U-KAN reconstructs 3D dental volumes from a single panoramic X-ray by lifting features with KANs then evolving them linearly via Koopman dynamics before focal-trough placement.

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

The paper presents K-U-KAN as a three-stage pipeline to recover 3D jaw anatomy from a single 2D panoramic radiograph. Kolmogorov-Arnold Networks first expand 2D features into depth-aware observables. A Koopman token block then advances those observables through stable phase-aware linear evolution. The results are projected along the horseshoe focal trough and refined by a lightweight 3D attention U-KAN. The method combines Beer-Lambert attenuation physics with known panoramic geometry and learned linear dynamics to aim for sharp boundaries, reduced artifacts, and reliable operation on raw intensities at batch size one. It reaches parity with transformer and implicit baselines on quantitative metrics while improving perceptual quality and cutting training time by roughly half.

Core claim

K-U-KAN is a three-stage pipeline that (i) lifts 2D features into depth-aware observables with Kolmogorov-Arnold Networks, (ii) advances these observables by a stable, phase-aware linear evolution via a Koopman token block, and (iii) places the predicted depth bins onto focal-trough rays before a lightweight 3D attention U-KAN refines the volume. This marriage of physics (Beer-Lambert image formation), geometry (horseshoe focal trough), and learned linear dynamics yields sharp anatomy, fewer artifacts, and robust behavior on native radiographic intensities with batch size one. On held-out data, K-U-KAN matches transformer/implicit baselines on signal and structure metrics, clearly improves p

What carries the argument

Three-stage pipeline of KAN-based lifting to depth-aware observables, Koopman token block for stable phase-aware linear evolution, and projection onto horseshoe focal-trough rays before 3D attention U-KAN refinement.

If this is right

Matches transformer and implicit baselines on signal and structure metrics.
Improves perceptual quality over those baselines.
Trains in roughly half the time of competing methods.
Operates effectively with batch size one on native radiographic intensities.
Yields volumes with sharp anatomy and fewer artifacts through explicit physics and geometry constraints.

Where Pith is reading between the lines

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

The reported efficiency gains could allow deployment on standard clinical hardware without specialized accelerators.
The explicit use of focal-trough geometry may make the reconstructions more consistent with the curved shape of real dental arches than purely learned approaches.
The linear evolution step could support incremental updates when new 2D views become available.
Similar combinations of KAN lifting and Koopman evolution might apply to other single-projection medical reconstruction tasks with known imaging geometry.

Load-bearing premise

The assumption that lifting 2D features into depth-aware observables with KANs, followed by stable phase-aware linear evolution via the Koopman token block and placement onto focal-trough rays, supplies sufficient information for the lightweight 3D attention U-KAN to produce accurate volumes.

What would settle it

An ablation experiment on held-out data showing that removing the Koopman token block or the focal-trough ray placement causes clear drops in perceptual quality or increases in artifacts would falsify the claim that these steps drive the reported gains.

Figures

Figures reproduced from arXiv: 2605.25163 by Abhijit Sen, Bikram Keshari Parida, Wonsang You.

**Figure 1.** Figure 1: Compute–performance trade-off of 3D oral reconstruction models. Bubble plots of PSNR, SSIM, and LPIPS versus wall-clock training time for all compared methods. Each bubble corresponds to one model, with its area and color proportional to the per-forward GFLOPS reported in [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

**Figure 2.** Figure 2: Schematic overview of the proposed K-U-KAN framework for single-view 3D dental reconstruction. (A) K-U-KAN Lifting Module: The input 2D panoramic radiograph is processed by a U-Net-style encoder-decoder where bottleneck features are lifted into a latent observable space. The network utilizes Tokenized Koopman-KAN blocks to predict a flat depth field F consisting of K depth bins. (B) Focal-Trough Inverse Wa… view at source ↗

**Figure 3.** Figure 3: Overview of the data-preparation workflow (Sunilkumar et al., 2025; Parida et al., 2025). Coronal and axial MIPs are first generated to localize the jaw. A. Jaw contour segmentation followed by horizontal tilt correction to standardize alignment. B. Definition of an elliptical focal-trough region that encompasses the dental arch. C. Computation of a dynamic rotation trajectory and simulation of pencil-beam… view at source ↗

**Figure 4.** Figure 4: Qualitative comparison of reconstructed CBCT volumes. From top to bottom: ground truth; K-U-KAN (Ours), ViT-NeBLa (Parida et al., 2025); 3DentAI (P.Sunilkumar et al., 2024); Oral-3D (auto-encoder) (Song et al., 2021); residual CNN (Henzler et al., 2018). Columns show (left) volume renderings, (middle) coronal MIPs, and (right) sagittal MIPs. Our K-U-KAN model yields the most accurate delineation of jaw ana… view at source ↗

**Figure 5.** Figure 5: Boxplots of the (a) PSNR (dB) (↑), (b) SSIM (%) (↑), and (c) LPIPS (VGG) (↓) metrics visually summarize the performance stability of the five reconstruction methods evaluated: Residual CNN (Henzler et al., 2018), Oral-3D (Song et al., 2021), 3DentAI (P.Sunilkumar et al., 2024), ViT-NeBLa (Parida et al., 2025), and K-U-KAN (Ours). The fill color of each box represents the training duration, providing a dual… view at source ↗

**Figure 6.** Figure 6: Qualitative comparison of reconstructed CBCT volumes. From top to bottom: ground truth; K-U-KAN (ours), ViT-NeBLa (Parida et al., 2025); 3DentAI (P.Sunilkumar et al., 2024); Oral-3D (auto-encoder) (Song et al., 2021); and residual CNN (Henzler et al., 2018). Columns show (left) volume renderings, (middle) coronal MIPs, and (right) sagittal MIPs [PITH_FULL_IMAGE:figures/full_fig_p023_6.png] view at source ↗

**Figure 7.** Figure 7: Qualitative comparison of reconstructed CBCT volumes. From top to bottom: ground truth; K-U-KAN (ours), ViT-NeBLa (Parida et al., 2025); 3DentAI (P.Sunilkumar et al., 2024); Oral-3D (auto-encoder) (Song et al., 2021); and residual CNN (Henzler et al., 2018). Columns show (left) volume renderings, (middle) coronal MIPs, and (right) sagittal MIPs. 23 [PITH_FULL_IMAGE:figures/full_fig_p023_7.png] view at source ↗

**Figure 8.** Figure 8: Qualitative comparison of reconstructed CBCT volumes. From top to bottom: ground truth; K-U-KAN (ours), ViT-NeBLa (Parida et al., 2025); 3DentAI (P.Sunilkumar et al., 2024); Oral-3D (auto-encoder) (Song et al., 2021); and residual CNN (Henzler et al., 2018). Columns show (left) volume renderings, (middle) coronal MIPs, and (right) sagittal MIPs [PITH_FULL_IMAGE:figures/full_fig_p024_8.png] view at source ↗

**Figure 9.** Figure 9: Qualitative comparison of reconstructed CBCT volumes. From top to bottom: ground truth; K-U-KAN (ours), ViT-NeBLa (Parida et al., 2025); 3DentAI (P.Sunilkumar et al., 2024); Oral-3D (auto-encoder) (Song et al., 2021); and residual CNN (Henzler et al., 2018). Columns show (left) volume renderings, (middle) coronal MIPs, and (right) sagittal MIPs. 24 [PITH_FULL_IMAGE:figures/full_fig_p024_9.png] view at source ↗

read the original abstract

A panoramic X-ray compresses a 3D jaw into a 2D strip; we aim to recover the missing depth cleanly and fast. Existing implicit neural representations render realistic volumes but are slow to train, sensitive to sampling and positional encodings, and costly in practice. Pure CNN baselines are efficient yet struggle with the dental arch's long-range geometry, blur fine enamel-dentin boundaries, and offer little interpretability. We present K-U-KAN, a three-stage pipeline that (i) lifts 2D features into depth-aware observables with Kolmogorov-Arnold Networks, (ii) advances these observables by a stable, phase-aware linear evolution via a Koopman token block, and (iii) places the predicted depth bins onto focal-trough rays before a lightweight 3D attention U-KAN refines the volume. This marriage of physics (Beer-Lambert image formation), geometry (horseshoe focal trough), and learned linear dynamics yields sharp anatomy, fewer artifacts, and robust behavior on native radiographic intensities with batch size one. On held-out data, K-U-KAN matches transformer/implicit baselines on signal and structure metrics, clearly improves perceptual quality, and trains in roughly half the time-making single-view PX $\to$ CBCT reconstruction more practical for clinical pipelines.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note private letter to a colleague

K-U-KAN puts KAN lifting, Koopman evolution, and U-KAN refinement together with Beer-Lambert and focal-trough priors for single panoramic X-ray to 3D dental volumes, but the abstract supplies no metrics or checks to back the claims.

read the letter

The main thing here is a three-stage pipeline that lifts 2D features into depth-aware observables via KANs, advances them with a stable phase-aware Koopman token block, and places the results onto horseshoe focal-trough rays before a lightweight 3D attention U-KAN refines the output. The abstract ties this to Beer-Lambert image formation and dental arch geometry to handle the ill-posed single-view reconstruction.

What is new is the explicit sequencing of KAN-based lifting, Koopman linear dynamics, and physics-informed ray placement inside one named model. The paper contrasts this with slow implicit representations and CNNs that miss long-range geometry, and it positions the hybrid as more interpretable and faster.

It does well at framing a real clinical need—cutting reliance on higher-dose CBCT—and at naming concrete efficiency targets like batch-size-one training on native intensities and roughly halved training time. The perceptual quality improvement claim is the kind of outcome that could matter if it holds.

The soft spots are that the abstract gives no numbers, no error bars, no ablation results, and no verification that the Koopman operator preserves invertibility or that the observables stay stable. The stress-test point about whether KAN lifting plus Koopman evolution plus ray placement actually supplies enough depth information stands, because nothing in the provided text shows a derivation or check on those steps. Without that, the U-KAN could be learning a generic mapping rather than using the priors.

This paper is for people working on medical image reconstruction in dentistry or on hybrid physics-ML methods. A reader who wants to see domain knowledge injected into reconstruction pipelines could pick up ideas, but it needs the full experiments to be useful.

I would send it for peer review because the pipeline is specific and the efficiency angle is worth testing, even if the current description leaves the central claims open.

Referee Report

3 major / 1 minor

Summary. The manuscript presents K-U-KAN, a three-stage pipeline for 3D dental volume reconstruction from a single panoramic X-ray. Stage (i) lifts 2D features to depth-aware observables via Kolmogorov-Arnold Networks; stage (ii) evolves them with a stable phase-aware linear Koopman token block; stage (iii) places the depth bins onto horseshoe focal-trough rays before refinement by a lightweight 3D attention U-KAN. The approach integrates Beer-Lambert image formation and focal-trough geometry priors. The authors claim the method produces sharp anatomy with fewer artifacts, matches transformer/implicit baselines on signal and structure metrics, improves perceptual quality, and trains in roughly half the time on native intensities with batch size one.

Significance. If the central claims hold with supporting quantitative evidence, the work would be significant for clinical dental pipelines by offering a faster, physics-informed alternative to implicit representations or standard CNNs for single-view PX-to-CBCT reconstruction. The explicit use of geometric and dynamic priors could improve interpretability and robustness on long-range dental arch geometry.

major comments (3)

[Abstract] Abstract: performance claims (matching baselines on signal/structure metrics, halved training time, improved perceptual quality) are stated without any numerical values, tables, figures, error bars, or ablation results, rendering the central claim of practical superiority impossible to evaluate from the supplied information.
[Method (Koopman token block)] Method description of Koopman token block (stage ii): no derivation of the Koopman operator, no eigenvalue-magnitude analysis for stability, and no verification that the resulting observables remain invertible to accurate depth bins consistent with the Beer-Lambert and focal-trough model; this is load-bearing because the lightweight 3D U-KAN may simply learn a generic mapping rather than benefiting from the claimed priors.
[Experiments] Experiments / results section: the assumption that KAN lifting + Koopman evolution + focal-trough ray placement supplies sufficient depth information is presented without explicit checks on operator stability, ray-placement consistency, or depth-bin accuracy on held-out data, leaving the robustness claims untested.

minor comments (1)

[Abstract] Abstract: the final sentence contains a missing space ('time-making').

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive feedback. We address each major comment below and indicate the revisions made to the manuscript.

read point-by-point responses

Referee: [Abstract] Abstract: performance claims (matching baselines on signal/structure metrics, halved training time, improved perceptual quality) are stated without any numerical values, tables, figures, error bars, or ablation results, rendering the central claim of practical superiority impossible to evaluate from the supplied information.

Authors: We agree that the abstract would be strengthened by explicit numerical support for the claims. In the revised version we have incorporated key quantitative results drawn from the experiments (e.g., matching SSIM/PSNR within reported margins, perceptual metric gains, and training-time reduction) together with pointers to the relevant tables and figures. revision: yes
Referee: [Method (Koopman token block)] Method description of Koopman token block (stage ii): no derivation of the Koopman operator, no eigenvalue-magnitude analysis for stability, and no verification that the resulting observables remain invertible to accurate depth bins consistent with the Beer-Lambert and focal-trough model; this is load-bearing because the lightweight 3D U-KAN may simply learn a generic mapping rather than benefiting from the claimed priors.

Authors: We accept this observation. The revised method section now contains an explicit derivation of the phase-aware Koopman operator, an eigenvalue-magnitude analysis confirming stability, and verification that the observables invert to depth bins consistent with the physical models. These additions clarify that the block contributes structured dynamics beyond a generic mapping. revision: yes
Referee: [Experiments] Experiments / results section: the assumption that KAN lifting + Koopman evolution + focal-trough ray placement supplies sufficient depth information is presented without explicit checks on operator stability, ray-placement consistency, or depth-bin accuracy on held-out data, leaving the robustness claims untested.

Authors: We agree that explicit validation is required. The updated experiments section adds analyses of operator stability (eigenvalue spectra), ray-placement consistency, and depth-bin accuracy on held-out data; these results are reported in a new subsection and the supplementary material. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper's three-stage pipeline combines external physical priors (Beer-Lambert image formation) and geometric constraints (horseshoe focal trough) with KAN lifting, Koopman evolution, and U-KAN refinement. These elements are presented as an integration of independent known models rather than any quantity being defined in terms of itself or a fitted parameter being relabeled as a prediction. No self-citations, uniqueness theorems from the authors, or ansatzes smuggled via prior work appear in the provided abstract or method description, and the claims are evaluated against held-out data using standard metrics. The derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on the effectiveness of applying KANs and Koopman operators to depth observables and on the accuracy of focal-trough ray placement; these modeling choices are domain assumptions whose validity is not independently demonstrated in the abstract.

axioms (2)

domain assumption Koopman operators can provide stable, phase-aware linear evolution of depth-aware observables extracted from 2D radiographic features
Invoked for the second stage of the pipeline.
domain assumption Horseshoe-shaped focal trough rays accurately represent the geometry for placing predicted depth bins
Used when mapping observables to 3D volume in the third stage.

invented entities (1)

Koopman token block no independent evidence
purpose: To advance depth-aware observables via stable phase-aware linear evolution
Introduced as a core component of the three-stage pipeline.

pith-pipeline@v0.9.1-grok · 5779 in / 1822 out tokens · 49580 ms · 2026-06-30T11:42:41.351738+00:00 · methodology

discussion (0)

Reference graph

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