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The Visual Computer

, Volume 27, Issue 3, pp 211–226 | Cite as

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

  • O. Ruiz
  • C. Vanegas
  • C. Cadavid
Original Article

Abstract

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]⊂RR 2 is self-intersecting if C(u)=C(v), uv, 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.

Keywords

Self-intersecting curve reconstruction Elliptic support region Principal component analysis Noisy samples 

Glossary

PL:

Piecewise Linear.

C:

Planar open or closed, possibly self-intersecting or nearly self-intersecting, curve.

S={p0,p1,…,pn}:

An unorganized noisy point sample of C.

ε:

Stochastic component of the point sample.

B(p,r):

The disk of radius r centered at point p.

\(L(\lambda)=p+\lambda*\hat{v}\):

Parametric form of the straight line passing through p, directed by the unit vector \(\hat{v}\) with signed distance parameter λ.

f1,f2:

Foci of an ellipse in R 2.

E(f1,f2,α):

Ellipse {pR 2:d(p,f 1)+d(p,f 2)=2α}.

ρX,Y:

Linear regression correlation coefficient between variables Y and X.

\({[}\rho,p,\hat{v}{]}=pca(S_{E})\):

Principal Component Analysis of the point set S E , rendering as a result the linear trend \(L(\lambda)=p+\lambda*\hat{v}\) with correlation coefficient ρ.

Q:

Queue whose elements are pairs [p,v] formed by a vector v anchored at point p.

PL_Curve_Set={c1,c2,…,cm}:

Set of PL pairwise disjoint curves c 1,c 2,…,c m .

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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Department of Computer SciencePurdue UniversityWest LafayetteUSA
  2. 2.Laboratory of CAD CAM CAEEAFIT UniversityMedellinColombia

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