Lie-Group-Aware and Geometry-Admissibility-Constrained ECG Signal Estimation from Scanned Records Using Projected Bansal-Kalman Algorithm

Electrocardiogram (ECG) analysis in real-world clinical, archival, and telemedicine environments is frequently performed on scanned images, printed charts, or degraded digital recordings rather than on clean, directly sampled signals. In such settings, conventional ECG processing methods-largely designed for unconstrained, probabilistic signal estimation-are prone to temporal incoherence, non-physical waveform evolution, and instability under scanning artifacts and segmentation errors. This paper presents a deterministic, geometry-aware signal processing framework based on the Bansal-Kalman formulation, which redefines ECG analysis as constrained waveform state estimation on an admissible manifold rather than unconstrained statistical filtering. The proposed approach operates strictly at the waveform level and deliberately avoids modeling cardiac electrophysiology or diagnostic physiology. Instead, ECG signals are treated as structured geometric objects characterized by amplitude, phase progression, temporal rate, and orientation descriptors. The core contribution is a projected Kalman estimation architecture in which standard linear Kalman prediction and update equations are preserved, while admissibility is enforced through an explicit deterministic projection onto a physiological waveform manifold. This projection guarantees invariance, boundedness, and temporal coherence of the estimated waveform state, even in the presence of severe noise, discontinuities, or scanning-induced distortions. Waveform phase is modeled on the Lie group S 1 , yielding a toroidal manifold structure that naturally encodes cyclic P-QRS-T morphology. Admissibility enforcement is shown to admit equivalent interpretations in terms of Riemannian projection, Lyapunov invariance, control barrier functions, and viability kernels. From a theoretical perspective, the Bansal-Kalman framework is demonstrated to be fundamentally non-equivalent to linear, extended, or unscented Kalman filters. Nonlinearity is not embedded in the state dynamics or measurement model but is introduced exclusively through geometric admissibility projection, yielding a hybrid projected stochastic system with guaranteed manifold invariance. Category-theoretic and differentialgeometric formulations are provided to formalize the signalstate-manifold-observable pipeline. The proposed framework is implemented in the fully free BVidAl Heart and ECG Analyzer and is designed for explainable, auditable, and deterministic ECG waveform analysis suitable for scanned records, legacy archives, and software-as-a-medical