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 of the Poisson Image Equation. In this study, we develop new VFs generated from linear combinations of the above VFs after proving they are not constant multiples of each other. Further, we 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 that we name SP ribbons (SPRs), which are strings of SPs in homogeneous regions. We also prove the conditions for SPR existence and explore SP shape invariance under affine transformations. On the basis of the theory, we developed MATLAB code that embeds the new VFs into images and conducted numerous experiments with public image databases. Further, we showcase convolutional neural network (CNN) experiments that demonstrate that embedding our VFs into a medical image database improves machine learning (ML) image classification accuracy. The paper concludes with a discussion on the advantages of using the new VFs and SPRs and outlines directions for future work.