Fitting curves and surfaces with constrained implicit polynomials

Abstract

Abstract-A problem which often arises while fitting implicit polynomials to 2D and 3D data sets is the following: Although the data set is simple, the fit exhibits undesired phenomena, such as loops, holes, extraneous components, etc. Previous work tackled these problems by optimizing heuristic cost functions, which penalize some of these topological problems in the fit. This paper suggests a different approach-to design parameterized families of polynomials whose zero-sets are guaranteed to satisfy certain topological properties. Namely, we construct families of polynomials with star-shaped zero-sets, as well as polynomials whose zero-sets are guaranteed not to intersect an ellipse circumscribing the data or to be entirely contained in such an ellipse. This is more rigorous than using heuristics which may fail and result in pathological zero-sets. The ability to parameterize these families depends heavily on the ability to parameterize positive polynomials. To achieve this, we use some powerful recent results from real algebraic geometry.

Topologically faithful fitting of simple closed curves

“…Better results are achieved by using cost functions which provide a more accurate approximation to the geometric distance between the zero-set and the points [9], [11], [20], [1], [17], [13], [15], [3], [8], [19].…”

“…For example, spurious components of the zero-set which lie far from the data are not penalized, neither are self-intersections, loops, etc. Various efforts were undertaken to solve this problem including a heuristic which searches for extraneous components and penalizes them [17], methods which seek to approximate not only the data, but also its gradients [6], [2], [12], [18], and restricting the fitting to polynomials which are guaranteed to have a "nice" zero-set [9]. However, all these methods are either restricted in the type of curves they can approximate, or are liable to fail and result in zero-sets with a different topology than that of the data they attempt to fit.…”

“…However, the fit may have a very different topology. For example, see [2], [9]. This paper offers an implicit fitting paradigm which solves this problem, as it is guaranteed to return a simple closed curve.…”

Tractable fitting with convex polynomials via sum-of-squares

“…We can extends the techniques presented in this paper to a different class of polynomials [KG99], that are convex only when restricted to a line passing for a given point x 0 .…”

Surface patch reconstruction via curve sampling

“…Often a subclass of quartic polynomials is chosen and constraints are incorporated into least-squares formulations [13,20]. Multiple level sets may also be used to further constrain the fitting, as in the case of the algorithm [3].…”