Iterative Optimization Coordinates

In order to ensure the non-negativity of generalized barycentric coordinates and improve computational efficiency, an optimization model minimizing the l_p-norm was established based on the properties of barycentric coordinates. Firstly, projected the polygon vertices onto a unit circle centered at the point, generated unit vectors. Then, the bisectors of adjacent vectors and their intersections with the unit circle were calculated. This step of finding the bisectors was repeated k times, converting the model into an l_p-minimization problem, which was solved through three iterations using the generalized soft thresholding(GST) algorithm GST(y,λ,p). Finally, the iterative optimization coordinates of any point inside the polygon were obtained through back-substitution. The paper provides comparisons of mean coordinates, iterative optimization coordinates, and iterative coordinates in contour maps for five different polygons, as well as image deformation comparisons. Experimental results show that the iterative optimization coordinates have good non-negativity and smoothness, and for most complex polygons, only two iterations are typically needed, significantly reducing the iteration count compared to the iterative coordinates.