PDE Learning of Filtering and Propagation for Task-Aware Facial Intrinsic Image Analysis

Filtering and propagation are two basic operations in image analysis and rendering, and they are also widely used in computer graphics and machine learning. However, the models of filtering and propagation were based on diverse mathematical formulations, which have not been fully understood. This article aims to explore the properties of both filtering and propagation models from a partial differential equation (PDE) learning perspective. We propose a unified PDE learning framework based on nonlinear reaction–diffusion with a guided map, graph Laplacian, and reaction weight. It reveals that: 1) the guided map and reaction weight determines whether the PDE produces filtering or propagation diffusion and 2) the kernel of graph Laplacian controls the diffusion pattern. Based on the proposed PDE framework, we derive the mathematical relations between different models, including learning to diffusion (LTD) model, label propagation, edit propagation, and edge-aware filter. In practical verification, we apply the PDE framework to design diffusion operations with the adaptive kernel to tackle the ill-posed problem of facial intrinsic image analysis (FIIA). A flexible task-aware FIIA system is built to achieve various facial rendering effects, such as face image relighting and delighting, artistic illumination transfer, illumination-aware face swapping, or transfiguring. Qualitative and quantitative experiments show the effectiveness and flexibility of task-aware FIIA and provide new insights on PDE learning for visual analysis and rendering.