Poisson Equation Solution and Its Gradient Vector Field to Geometric Features Detection

In this paper we solve the Poisson partial differential equation (PDE) with a free right side, which is a function of the image and its gradient. We call such a PDE Poisson Image (PI) equation. Further, we define the function \(\phi ={u+\left\| \nabla u\right\| }^2\), where u is the PI’s solution. Then, we generate the Poisson gradient vector fields (PGVFs) \(\nabla u\) and \(\nabla \phi \) and study the patterns of their trajectories in the vicinity of the singular points (SPs). Next, we use the critical points (CPs) of u and \(\phi \), the SPs of \(\nabla u\) and \( \nabla \phi \), and their relations, to determine the image objects’ concavities and convexities, and use them for automatic objects partitioning. We validated the theoretical concepts with experiments on above 80 synthetic and real-life images, and show some of them in the paper. At the end we compare the new method with contemporary methods in the field and list its contributions, advantages and bottlenecks.