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

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 → CBCT reconstruction more practical for clinical pipelines.