Evolving Implicit Polynomial Interfaces

Although algebraic or so-called “implicit polynomial” curves have been studied rather extensively for several decades, to the best of our knowledge, a dynamic formulation of them, similar to active contours, has not been done yet. This paper develops a dynamic formulation for implicit polynomial curves based on level set formalism. In particular, it is shown that utilization of an implicit polynomial distance function in the level set equation yields an ordinary differential equation (ODE) for the temporal behavior of the polynomial coefficients. Using a control theoretic approach, several problems such as curve morphing, dynamic conic fitting without and with constraint, i.e. dynamic ellipse fit, and dynamic curve fitting can be tackled within this new framework. Results are verified by several examples on real images.