Type-Constrained Direct Fitting of Quadric Surfaces

We present a catalog of type-specific, direct quadric fitting methods: Given a selection of a point cloud or triangle mesh, and a desired quadric type (e.g. cone, ellipsoid, paraboloid, etc), our methods recover a best-fit surface of the given type to the given data. Type-specific quadric fitting methods are scattered throughout the literature; here we present a thorough, practical collection in one place. We add new methods to handle neglected quadric types, such as non-circular cones and general rotationally symmetric quadrics. We improve upon existing methods for ellipsoid- and hyperboloid-specific fitting. Our catalog handles a wide range of quadric types with just two high-level fitting strategies, making it simpler to understand and implement.

[1]  Thomas Randrup.  Approximation by Cylinder Surfaces , 1997 .

[2]  Andrew W. Fitzgibbon,et al.  A Buyer's Guide to Conic Fitting , 1995, BMVC.

[3]  Vaughan R. Pratt,et al.  Direct least-squares fitting of algebraic surfaces , 1987, SIGGRAPH.

[4]  A. Al-Sharadqah,et al.  Error analysis for circle fitting algorithms , 2009, 0907.0421.

[5]  Peter J. Huber,et al.  Robust Statistics , 2005, Wiley Series in Probability and Statistics.

[6]  Carlo H. Séquin,et al.  Generalized, basis-independent kinematic surface fitting , 2013, Comput. Aided Des..

[7]  Ralph R. Martin,et al.  Faithful Least-Squares Fitting of Spheres, Cylinders, Cones and Tori for Reliable Segmentation , 1998, ECCV.

[8]  Dana H. Ballard,et al.  Computer Vision , 1982 .

[9]  D. A. Dunavant.  High degree efficient symmetrical Gaussian quadrature rules for the triangle , 1985 .

[10]  Gene H. Golub,et al.  Matrix computations (3rd ed.) , 1996 .

[11]  John G. Griffiths,et al.  Least squares ellipsoid specific fitting , 2004, Geometric Modeling and Processing, 2004. Proceedings.

[12]  Thomas Randrup,et al.  Approximation of surfaces by cylinders , 1998, Comput. Aided Des..

[13]  Sylvain Petitjean,et al.  A survey of methods for recovering quadrics in triangle meshes , 2002, CSUR.

[14]  N. Chernov,et al.  L EAST SQUARES FITTING OF QUADRATIC CURVES AND SURFACES , 2010 .

[15]  Valérie Burdin,et al.  Type-Constrained Robust Fitting of Quadrics with Application to the 3D Morphological Characterization of Saddle-Shaped Articular Surfaces , 2007, 2007 IEEE 11th International Conference on Computer Vision.

[16]  Matthew Harker,et al.  Direct type-specific conic fitting and eigenvalue bias correction , 2008, Image Vis. Comput..

[17]  Carlo H. Séquin,et al.  Interactive Inverse 3D Modeling , 2012 .

[18]  Andrew W. Fitzgibbon,et al.  Direct Least Square Fitting of Ellipses , 1999, IEEE Trans. Pattern Anal. Mach. Intell..

[19]  Prasanna Rangarajan,et al.  Hyper Least Squares and Its Applications , 2010, 2010 20th International Conference on Pattern Recognition.

[20]  Karl Pearson F.R.S..  LIII. On lines and planes of closest fit to systems of points in space , 1901 .

[21]  Ralph R. Martin,et al.  Reverse engineering of geometric models - an introduction , 1997, Comput. Aided Des..

[22]  Alexandre Boulch,et al.  Fast and Robust Normal Estimation for Point Clouds with Sharp Features , 2012, Comput. Graph. Forum.

[23]  James Demmel,et al.  LAPACK Users' Guide, Third Edition , 1999, Software, Environments and Tools.

[24]  PAUL D. SAMPSON,et al.  Fitting conic sections to "very scattered" data: An iterative refinement of the bookstein algorithm , 1982, Comput. Graph. Image Process..

[25]  Gabriel Taubin,et al.  Estimation of Planar Curves, Surfaces, and Nonplanar Space Curves Defined by Implicit Equations with Applications to Edge and Range Image Segmentation , 1991, IEEE Trans. Pattern Anal. Mach. Intell..

[26]  TaubinGabriel.  Estimation of Planar Curves, Surfaces, and Nonplanar Space Curves Defined by Implicit Equations with Applications to Edge and Range Image Segmentation , 1991 .

[27]  Gene H. Golub,et al.  Matrix computations , 1983 .

[28]  B. Ripley,et al.  Robust Statistics , 2018, Encyclopedia of Mathematical Geosciences.