A summary of geometric level-set analogues for a general class of parametric active contour and surface models

Geometric active contours (GAC) and surfaces (GAS) implemented via level set techniques enjoy many advantages over parametric active contours (PAC) and surfaces (PAS), such as computational stability and the ability to change topology during deformation. While many capabilities of earlier PAC and PAS have been reproduced by various GAC and GAS, and while relationships have been discussed for a variety of specific cases, a comprehensive accounting of the connections between these two worlds (particularly regarding rigid forces) has not been consolidated thus far. We present the precise mathematical relationships between the two for an extensive family of both active contour and surface models, encompassing spatially varying coefficients, both tension and rigidity, and both conservative and non-conservative external forces. The result is a very general geometric formulation for which the intuitive design principles of PAC and PAS can be applied. We also point out which type of PAC and PAS methodologies cannot be adapted to the geometric level set framework. We conclude by demonstrating several geometric adaptations of specific PAC and PAS in several simulations.