Extrapolation Convolution for Data Prediction on a 2-D Grid: Bridging Spatial and Frequency Domains With Applications in Image Outpainting and Compressed Sensing

Extrapolation plays a critical role in machine/deep learning (ML/DL), enabling models to predict data points beyond their training constraints, particularly useful in scenarios deviating significantly from training conditions. This article addresses the limitations of current convolutional neural networks (CNNs) in extrapolation tasks within image restoration and compressed sensing (CS). While CNNs show potential in tasks such as image outpainting and CS, traditional convolutions are limited by their reliance on interpolation, failing to fully capture the dependencies needed for predicting values outside the known data. This work proposes an extrapolation convolution (EC) framework that models missing data prediction as an extrapolation problem using linear prediction within DL architectures. The approach is applied in two domains: first, image outpainting, where EC in encoder–decoder (EnDec) networks replaces conventional interpolation methods to reduce artifacts and enhance fine detail representation; second, Fourier-based CS-magnetic resonance imaging (CS-MRI), where it predicts high-frequency signal values from undersampled measurements in the frequency domain, improving reconstruction quality and preserving subtle structural details at high acceleration factors. Comparative experiments demonstrate that the proposed EC-DecNet and FDRN outperform traditional CNN-based models, achieving high-quality image reconstruction with finer details, as shown by improved peak signal-to-noise ratio (PSNR), structural similarity index (SSIM), and kernel inception distance (KID)/Frechet inception distance (FID) scores. Ablation studies and analysis highlight the effectiveness of larger kernel sizes and multilevel semi-supervised learning in FDRN for enhancing extrapolation accuracy in the frequency domain.