Ellipse-based principal component analysis for self-intersecting curve reconstruction from noisy point sets

Surface reconstruction from cross cuts usually requires curve reconstruction from planar noisy point samples. The output curves must form a possibly disconnected 1-manifold for the surface reconstruction to proceed. This article describes an implemented algorithm for the reconstruction of planar curves (1-manifolds) out of noisy point samples of a self-intersecting or nearly self-intersecting planar curve C. C:a,b]⊂R→R ² is self-intersecting if C(u)=C(v), u≠v, u,v∈(a,b) (C(u) is the self-intersection point). We consider only transversal self-intersections, i.e. those for which the tangents of the intersecting branches at the intersection point do not coincide (C′(u)≠C′(v)). In the presence of noise, curves which self-intersect cannot be distinguished from curves which nearly self-intersect. Existing algorithms for curve reconstruction out of either noisy point samples or pixel data, do not produce a (possibly disconnected) Piecewise Linear 1-manifold approaching the whole point sample. The algorithm implemented in this work uses Principal Component Analysis (PCA) with elliptic support regions near the self-intersections. The algorithm was successful in recovering contours out of noisy slice samples of a surface, for the Hand, Pelvis and Skull data sets. As a test for the correctness of the obtained curves in the slice levels, they were input into an algorithm of surface reconstruction, leading to a reconstructed surface which reproduces the topological and geometrical properties of the original object. The algorithm robustly reacts not only to statistical non-correlation at the self-intersections (non-manifold neighborhoods) but also to occasional high noise at the non-self-intersecting (1-manifold) neighborhoods.