Document dewarping is crucial for many applications. However, existing learning-based methods rely heavily on supervised regression with annotated data without fully leveraging the inherent geometric properties of physical documents. Our key insight is that a well-dewarped document is defined by its axis-aligned feature lines. This property aligns with the inherent axis-aligned nature of the discrete grid geometry in planar documents. Harnessing this property, we introduce three synergistic contributions: for the training phase, we propose an axis-aligned geometric constraint to enhance document dewarping; for the inference phase, we propose an axis alignment preprocessing strategy to reduce the dewarping difficulty; and for the evaluation phase, we introduce a new metric, Axis-Aligned Distortion (AAD), that not only incorporates geometric meaning and aligns with human visual perception but also demonstrates greater robustness. As a result, our method achieves state-of-the-art performance on multiple existing benchmarks, improving the AAD metric by 18.2% to 34.5%.

In the context of surface representations, we find a natural structural similarity between grid surface and image data. Motivated by this inspiration, we propose a novel approach: encoding grid surfaces as geometric images and using image processing methods to address surface optimization-related problems. As a result, we have created the first dataset for grid surface optimization and devised a learning-based grid surface optimization network specifically tailored to geometric images, addressing the surface optimization problem through a data-driven learning of geometric constraints paradigm. We conduct extensive experiments on developable surface optimization, surface flattening, and surface denoising tasks using the designed network and datasets. The results demonstrate that our proposed method not only addresses the surface optimization problem better than traditional numerical optimization methods, especially for complex surfaces, but also boosts the optimization speed by multiple orders of magnitude. This pioneering study successfully applies deep learning methods to the field of surface optimization and provides a new solution paradigm for similar tasks, which will provide inspiration and guidance for future developments in the field of discrete surface optimization. The code and dataset are available at https://github.com/chaoyunwang/GSO-Net.