Decoupled Global-Local Implicit Representation for Continuous Nonlocal Reconstruction

Recovering signals from incomplete or noisy observations requires structural priors that remain effective across diverse sampling patterns. Nonlocal self-similarity (NSS) provides a powerful source of such prior knowledge by exploiting recurring local structures across spatially distant regions. Conventional NSS methods, however, typically rely on explicit neighborhood construction and grouping, making them less suitable for irregularly sampled observations and continuous coordinate queries. Implicit neural representations (INRs) offer a flexible alternative by modeling signals as continuous coordinate functions, but conventional INR formulations generally treat the signal as a single global implicit function and do not explicitly exploit recurring nonlocal structures. To bridge this gap, we propose a Decoupled Global-Local Implicit Representation (D-GLIR) for continuous nonlocal reconstruction. D-GLIR decomposes the implicit representation into shared local implicit bases and a coordinate-conditioned coefficient field. The local branch parameterizes reusable structural patterns over relative coordinates, while the global branch continuously maps spatial coordinates to the coefficients that combine these patterns at arbitrary query locations. Unlike conventional INRs that model the signal through a single global coordinate-to-value mapping, D-GLIR allows spatially distant regions to reuse the same local structural bases through different coefficient combinations. This introduces NSS as an intrinsic inductive bias of the representation while preserving its continuous coordinate-based nature, without explicit block matching, precomputed patch groups, or an additional NSS regularization term. Experiments on image inpainting, image denoising, off-grid function regression, point-cloud recovery, and weather data reconstruction demonstrate the effectiveness and flexibility of D-GLIR across diverse signal types, sampling patterns, and degradation conditions.