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.

中文翻译：

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用于任务感知面部本质图像分析的过滤和传播的偏微分方程学习


过滤和传播是图像分析和渲染中的两个基本操作，它们也广泛应用于计算机图形和机器学习中。然而，过滤和传播模型基于不同的数学公式，尚未得到充分理解。本文旨在从偏微分方程 (PDE) 学习的角度探讨过滤模型和传播模型的属性。我们提出了一个基于非线性反应扩散的统一偏微分方程学习框架，具有引导图、图拉普拉斯算子和反应权重。它表明：1）引导图和反应权重决定偏微分方程是否产生过滤或传播扩散，2）图拉普拉斯核控制扩散模式。基于所提出的偏微分方程框架，我们推导了不同模型之间的数学关系，包括学习扩散（LTD）模型、标签传播、编辑传播和边缘感知过滤器。在实际验证中，我们应用偏微分方程框架来设计具有自适应核的扩散操作，以解决面部内在图像分析（FIIA）的不适定问题。灵活的任务感知 FIIA 系统旨在实现各种面部渲染效果，例如面部图像重新照明和愉悦、艺术照明传输、照明感知面部交换或变形。定性和定量实验展示了任务感知 FIIA 的有效性和灵活性，并为视觉分析和渲染的 PDE 学习提供了新的见解。