On the String Shaped Singularities of Linear Combinations of Vector Fields on the Solution of the Poisson Image Equation

This research is a continuation of work that developed the vector fields (VFs) \(\nabla\hat{\phi}\),\(\overline{\nabla}\hat{\phi}\),\(\nabla\hat{\psi}\), and \(\overline{\nabla}\hat{\psi}\) generated from the solution to the Poisson Image Equation. In this study, we prove that these VFs are not constant multiples of each other, develop new VFs generated from linear combinations of the above VFs, classify their singular points (SPs), determine their locations, and formulate mappings between critical points (CPs) and corresponding VF SPs. From the new VFs, we discover and describe a novel feature we name SP ribbons (SPRs) which are strings of SPs in homogeneous regions. We also prove the conditions for SPR existence. On the basis of the theory, we developed MATLAB code that embeds the new VFs into images and conducted numerous experiments with images from public databases. The paper concludes with a discussion on the advantages of using the new VFs and SPRs and outlines directions for future work.