Nonlinear boundary value inverse cauchy problem formulation for image shadow

Shadow removal is a fundamental challenge in imaging science due to the difficulty of decoupling illumination variations from surface reflectance at shadow boundaries. This paper introduces a novel mathematical framework that reformulates osmosis-based shadow removal as a Cauchy-type inverse problem. A new variational functional is proposed, introducing nonlinear anisotropic diffusion in the shadow transition zone, osmosis-driven transport in the illuminated and shadowed regions, and a convenient boundary fidelity constraint on the accessible interface. The existence of a minimizer is shown under classical regularity hypotheses, and the associated Euler–Lagrange system is derived. To solve the obtained coupled problem, a new hybrid architecture is developed, which consist of finite element methods to handle the well-posed osmosis evolution, while a Physics-Informed Neural Network (PINNs) compute the unknown interface traces through the ill-posed Cauchy reconstruction. Numerical experiments on various shadow images demonstrate superior performance of this new configuration in terms of visual quality and edge preservation compared to competitive osmosis methods.