1. A system for quadrilateral re-meshing of a three-dimensional (3D) input model using progressive implicit approximating levels, comprising: a computer device including: a system for providing a quadrilateral mesh for the 3D input model; a system for building an implicit approximating field for a first approximating level (L) of the 3D input model using an implicit surface modeling technique; a system for extracting an iso-contour of the implicit approximating field; a system for fitting the quadrilateral mesh to the first approximating level (L); a system for estimating a fit between the quadrilateral mesh and the first approximating level (L); a system for determining whether the fit meets a predetermined quality criterion; and a system for refining the initial quadrilateral mesh using at least one of a sequence of topological operations to achieve an improved fit after the determining step if the fit fails to meet the predetermined quality criterion, wherein the quadrilateral mesh is an initial quadrilateral mesh, and wherein the sequence of topological operations comprises snapping vertices, splitting edges, extruding faces, and bridging faces of the quadrilateral mesh.