Mathematical representation of 2D image boundary contour using fractional implicit polynomial

Implicit polynomial(IP)fitting is an effective method to quickly represent two-dimensional(2D)image boundary contour in the form of mathematical function.Under the same maximum degree,the fractional implicit polynomial(FIP)can express more curve details than IP and has obvious advantages for the representation of complex boundary contours.In existing studies,algebraic distance is mainly used as the fitting objective of the polynomial.Although the time cost is reduced,there are problems of low fitting accuracy and spurious zero set.In this paper,we propose a two-stage neural network with differentiable geometric distance,which uses FIP to achieve mathematical representation,called TSEncoder.In the first stage,the continuity constraint is used to obtain a rough outline of the fitting target.In the second stage,differentiable geometric distance is gradually added to fine-tune the polynomial coefficients to obtain a contour representation with higher accuracy.Experimental results show that TSEncoder can achieve mathematical representation of 2D image boundary contour with high accuracy.