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Geodesic-based manifold learning for parameterization of triangular meshes

  • Diego A. Acosta
  • Oscar E. Ruiz
  • Santiago Arroyave
  • Roberto Ebratt
  • Carlos CadavidEmail author
  • Juan J. Londono
Original Paper

Abstract

Reverse Engineering (RE) requires representing with free forms (NURBS, Spline, Bézier) a real surface \(S_0\) which has been point-sampled. To serve this purpose, we have implemented an algorithm that minimizes the accumulated distance between the free form and the (noisy) point sample. We use a dual-distance calculation point to / from surfaces, which discourages the forming of outliers and artifacts. This algorithm seeks a minimum in a function \(f\) that represents the fitting error, by using as tuning variable the control polyhedron for the free form. The topology (rows, columns) and geometry of the control polyhedron are determined by alternative geodesic-based dimensionality reduction methods: (a) graph-approximated geodesics (Isomap), or (b) PL orthogonal geodesic grids. We assume the existence of a triangular mesh of the point sample (a reasonable expectation in current RE). A bijective composition mapping \(S_0 \subset \mathbb {R}^3 \longleftrightarrow \mathbb {R}^2\) allows to estimate a size of the control polyhedrons favorable to uniform-speed parameterizations. Our results show that orthogonal geodesic grids is a direct and intuitive parameterization method, which requires more exploration for irregular triangle meshes. Isomap gives a usable initial parameterization whenever the graph approximation of geodesics on \(S_0\) be faithful. These initial guesses, in turn, produce efficient free form optimization processes with minimal errors. Future work is required in further exploiting the usual triangular mesh underlying the point sample for (a) enhancing the segmentation of the point set into faces, and (b) using a more accurate approximation of the geodesic distances within \(S_0\), which would benefit its dimensionality reduction.

Keywords

Computational geometry Parametric surfaces Surface reconstruction Reverse engineering 

Abbreviations

\(PL\)

Piecewise linear

\(B\)

Solid object in \(\mathbb {R}^3\). \(B \subset \mathbb {R}^3\) is the closure of a bounded and connected open set, whose border \(\partial B\) is a 2-dimensional manifold.

\(S_{0}\)

Freeform parametric surface on which a Face of \(\partial B\) is mounted

\(\mathbf P \)

\(\{p_0,p_1,...\}\) Unordered point sample of \(S_{0}\)

\(S(u,v)\)

Parametric surface, which fits the set \(\mathbf P \),so \(S \approx S_{0}\)

\(u,v\)

Surface parameters

\(N_{i,p},N_{j,q}\)

B-spline base functions \(\mathbb {R} \rightarrow \mathbb {R}\),

\(n, m\)

Number of control points of \(S\) in \(u, v\) directions respectively

\(\mathbf {Cp} \)

Control polyhedron for \(S\)

\(k\)

Norm degree. \(|(x_{1},x_{2},...,x_{n})|_{k}=\root k \of {\sum ^{i=n}_{i=1}|x_{i}|^k}\)

\(f\)

Function minimized when fitting \(S\) to \(\mathbf P \)

\(d_{i}\)

Minimum distance between the i-th point \(p_{i}\) of \(\mathbf P \) and \(S\)

LM

Levenberg-Marquardt

RE

Reverse engineering

\(Gr\)

Regular, axis-aligned vertex grid in \(\mathbb {R}^2\)

\(G\)

Graph \((\mathbf P ,E)\) with vertex set \(\mathbf P \) and edge set \(E\), nearly embedded in \(S_{0}\)

\(D\)

Square matrix in which \(D(i,j)=dist(p_{i},p_{j})\), with dist() approaching the geodesic distance on \(S_{0}\) between sample points \(p_{i}\) and \(p_{j}\)

\(T\)

\(\{t_1, t_2,\ldots \}\) Triangular mesh of triangles \(t_{i}\) with vertices in \(\mathbf P \)

\(B_{UV}\)

Parametric rectangular connected subset of \(\mathbb {R}^{2}\)

\(c_{G}\)

PL geodesic curve on \(T\)

Notes

Acknowledgments

The authors wish to thank undergraduate U. EAFIT students Juan Pablo Velasquez for the testing of discrete geodesics MATLAB (TM) code and Daniel Burgos for the end-user segmentation of data sets for this article.

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Copyright information

© Springer-Verlag France 2014

Authors and Affiliations

  • Diego A. Acosta
    • 1
  • Oscar E. Ruiz
    • 2
  • Santiago Arroyave
    • 2
  • Roberto Ebratt
    • 2
  • Carlos Cadavid
    • 2
    Email author
  • Juan J. Londono
    • 2
  1. 1.DDP Research GroupUniversidad EAFITMedellinColombia
  2. 2.Laboratory of CAD CAM CAEUniversidad EAFITMedellinColombia

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