Presentation on theme: "Developable Surface Fitting to Point Clouds Martin Peternell Computer Aided Geometric Design 21(2004) 785-803 Reporter: Xingwang Zhang June 19, 2005."— Presentation transcript:
Developable Surface Fitting to Point Clouds Martin Peternell Computer Aided Geometric Design 21(2004) 785-803 Reporter: Xingwang Zhang June 19, 2005
About Martin Peternell Affiliation Institute of Discrete Mathematics and Geometry Vienna University of Technology Web http://www.geometrie.tuwien.ac.at/peternell People Helmut Pottmann Johannes Wallner etc.
Research Interests Classical Geometry Computer Aided Geometric Design Reconstruction of geometric objects from dense 3D data Geometric Computing Industrial Geometry
Overview Problem Developable surfaces Blaschke model Reconstruction of Developable Surfaces Q&A
Problem Given: scattered data points from a developable surface Object: Construct a developable surface which fits best to the given data
Ruled Surface directrix curve a generator A ruled surface Normal vector
Developable Surface Each generator all points have the same tangent plane. Vectors and are linearly dependent Equivalent condition
Developable Surface Three types of developable surfaces
Geometric Properties of Developable Surface Gaussian curvature is zero Envelope of a one-parameter family of planes Dual approach: is a curve in dual projective 3-space.
Singular Point A singular point doesn ’ t possess a tangent plane. Singular curve is determined by the parameter
Three Different Classes Cylinder: singular curve degenerates to a single point at infinity Cone: singular curve degenerates to a single proper point, called vertex Tangent surface: tangent lines of a regular space curve, called singular curve
Literature [Bodduluri, Ravani, 1993] duality between points and planes in 3-D space [Pottmann, Farin, 1995] projective algorithm, dual representation [Chalfant, Maekawa, 1998] optimization techniques [Pottmann, Wallner, 1999] a curve of dual projective 3-D space [Chu, Sequin, 2002] boundary curve, de Casteljau algorithm, equations [Aumann, 2003] affine transformation, de Casteljau algorithm
General Fitting Technique Estimating parameter values Solving a linear problem in the unknown control points fitting unorganized data points Find an developable B-spline surface
Two Difficult Problems Sorting scattered data Estimation of data parameters Estimation of approximated direction of the generating lines Guaranteeing resulting fitted surface is developable Leading a highly non-linear side condition in the control points
Contributions of this Paper Avoid the above two problems Reconstruction of a 1-parameter family of planes close to the estimated tangent planes of the given data points Applicable Nearly developable surfaces Better slightly distorted developable surfaces
Analysis of the Blaschke Image (continued) Check point cloud on fitted well by hyperplane Principal component analysis
Principal Component Analysis (continued) Minimization Eigenvalue problem
Principal Component Analysis (continued) Four small eigenvalues: The Blaschke image is a point-like cluster. The original surface is planar. Two small eigenvalues: The Blaschke image is a planar curve (conic). The original surface is a cone or cylinder of rotation. a cone of rotation. a cylinder of rotation.
Principal Component Analysis (continued) One small eigenvalue and curve-like Blaschke image. The original surface is developable. a general cone a general cylinder a developable of constant slope. One small eigenvalue and surface-like Blaschke-image: The original surface is a sphere.
Example Analysis of the Blaschke image – Sphere
Example Approximation of a developable of constant slope
Example General cylinderTriangulated data points and approximation Original Blaschke image
Example Developable of constant slope Triangulated data points and approximation Spherical image of the approximation with control points.
Reconstruction of Developable Surfaces from Measurements
Reconstruction Find a curve fitting best the tubular region defined by Determine 1-parameter family of tangent planes determined by Compute a point-representation of the corresponding developable approximation of the data points
Parametrizing a Tubular Region Determine relevant cells of carrying points Thinning of the tubular region: Find cells carrying only few points and delete these cells and points Estimate parameter values for a reduced set of points (by moving least squares: marching through the tube) Compute an approximating curve on w.r.t. points