Spatial Interpolation of DEM With Stochastic Differential Equations-Based Diffusion Model

Despite the availability of high-precision surveying technologies, digital elevation models (DEMs) may still contain missing or degraded elevation values due to complex acquisition conditions, terrain occlusions, imaging geometry effects, and subsequent data processing procedures. Spatial interpolation methods provide a technical means to recover complete DEM data from incomplete DEM data. However, traditional interpolation methods often face problems such as insufficient model expressiveness and limited interpolation accuracy. In this study, we innovatively introduce a diffusion model based on the mean-reverting stochastic differential equation (SDE) to the DEM spatial interpolation task and propose a novel data-driven interpolation framework. This method constructs a mean-reverting SDE to map high-quality DEM data to a degraded state with fixed noise, and uses the reverse-time SDE to achieve high-precision reconstruction of missing areas. We also specifically design a unique joint loss function for DEM interpolation and reconstruction, which includes content loss, perceptual loss and terrain feature loss. In addition, we propose a maximum likelihood estimation strategy to optimize the training objective, which significantly improves the training stability of the model and the reliability of the interpolation results. Experimental results show that, compared to traditional interpolation and typical deep learning-based reconstruction models, this method performs excellently in interpolating DEM data for complex terrains (such as steep slopes, gullies and ridges). In particular, it significantly reduces interpolation errors and produces terrain details that are more consistent with the actual terrain.