Geometry-Aware Interpolation for Inverse Problems

Predicting a complete spatially correlated field from sparse observations is a fundamental challenge in spatial statistics and environmental modelling. Classical interpolation methods such as Kriging rely on Gaussian process assumptions and variography, which can limit their effectiveness in non-stationary settings and require substantial domain expertise. In this work, we leverage an architecture based on convolutional neural networks (CNNs) for spatial interpolation that is trained and applied on a single partially observed field, without access to external data or prior fields. The model is supervised directly on the observed locations and learns to predict values at unobserved points on the user defined grid. Unlike Kriging, our method does not require explicit covariance modelling or variogram estimation, and it can flexibly capture local spatial patterns in a data-driven manner. This work demonstrates the potential of CNNs for single-instance spatial interpolation under sparse supervision, offering a practical alternative to classical geostatistical methods, and extending the use of CNNs to a new problem domain.

Image inpainting aims to recover missing regions while preserving structural consistency. We propose a non-parametric method without network training based on data-guided stochastic dynamics. Starting from a masked image, the missing pixels are evolved through a reverse-time stochastic differential equation with a kernel-weighted correction estimated directly from a reference dataset. This empirical correction guides the reconstruction toward high-density regions of the data distribution without training a neural network or fitting a parametric density model. Experiments on MNIST, Fashion-MNIST, and MVTec show that the proposed method outperforms Mean Fill, Telea, and Navier-Stokes inpainting in PSNR, SSIM, and visual quality. On CelebA, it remains competitive and produces plausible completions for structure-sensitive occlusions. These results demonstrate the effectiveness of empirical reference statistics as a non-parametric prior for image inpainting.

Spatial transformations such as rotation and scale obscure the morphological cues needed for accurate image classification. Careful consideration is required for reliable use in high stakes settings. A model should stay robust under such transformations, expose why a correction was applied, and signal when its input is ambiguous. While geometrically equivariant architectures provide a mathematically grounded solution, they often limit model flexibility through strict symmetry constraints and incur significant computational overhead. Spatial Transformer Networks (STNs) offer a data-driven, flexible alternative for learning pseudo-equivariances to affine transformations. However, STNs have historically been restricted to convolutional architectures and suffer from training instability. To address this, we introduce a novel STN framework. It leverages the global modeling capabilities of transformers to regress the affine transformation acting on the input. For this, we decompose affine transformations into interpretable primitives, regressed under adaptable geometric constraints, thereby preventing the training instability typically caused by degenerate transformations. By sharing weights between the localization network and the classification backbone, the framework requires minimal computational overhead. Extensive experiments on challenging insect biodiversity and medical imaging benchmarks demonstrate that our approach achieves superior predictive performance under diverse spatial transformations while maintaining high efficiency. Code is available at https://github.com/johSchm/TokenSTN.