Geometry-Aware Interpolation for Inverse Problems

Recovering missing visual information from incomplete or degraded images remains a challenging problem, particularly when spatial variations and complex image structures affect reconstruction accuracy. Although Spatial Transformer Networks (STNs) facilitate spatial feature alignment, their reliance on conventional linear or bilinear interpolation limits their ability to capture the underlying geometric characteristics of image data. Furthermore, interpolation based on a restricted set of neighbouring pixels can introduce inconsistencies in gradient propagation, potentially affecting optimisation stability. To address these limitations, we propose a Geometry-Aware Interpolation Network (GAIN), a variational autoencoder-based framework that incorporates manifold geometry into spatial transformation and image reconstruction. Rather than estimating pixel intensities solely from fixed spatial neighbours, GAIN exploits the local tangent structure of the data manifold to construct geometry-informed intensity approximations. An entropy-based regularisation mechanism further guides the estimation of spatial transformation parameters, promoting more adaptive interpolation. In addition, a convolutional normalisation strategy is introduced to regulate the magnitude of non-zero activations and improve gradient stability during training. Extensive experiments on challenging image completion and classification benchmarks demonstrate the effectiveness of GAIN in improving predictive accuracy while reducing computational cost compared with existing methods.