Directional Total Variation-Regularized Implicit Neural Representations (DTV-INR) for Continuous Super-Resolution in Degraded Imaging Domains

In this paper, we introduce the Directional Total Variation-Regularized Implicit Neural Representation (DTV-INR), an advanced variational paradigm that synergistically integrates coordinate-driven implicit neural networks with an anisotropic, structure-tensor-informed total variation regularizer tailored for resolution-agnostic image super-resolution. Casting the continuous-to-discrete acquisition process into an ill-posed inverse problem framework, our formulation equips a SIREN-architected coordinate network with a dynamic Riemannian metric tensor field D(x). By leveraging its spectral decomposition, the proposed regularizer preferentially directs diffusion parallel to dominant structural contours while penalizing cross-edge dissipation, successfully circumventing the classical staircasing artifacts inherent to scalar total variation schemes. We rigorously prove the well-posedness of this formulation in H^1(Omega) by establishing the existence, uniqueness, and metric stability of the variational minimizer, and realize this via an alternating projected optimization algorithm that decouples network parameter tuning from adaptive tensor field updates. Comprehensive experiments conducted on clinical brain magnetic resonance imaging (MRI) and biomedical transmission electron microscopy confirm substantial quantitative and qualitative improvements, yielding PSNR enhancements reaching +5.05 dB over baseline unregularized INRs and +1.71-2.85 dB over isotropic TV-INR across continuous (non-integer) upsampling factors, alongside remarkable noise robustness up to sigma_eta = 0.10 and monotonic preconditioned convergence behavior.