Developable Surface Fitting to Point Clouds

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Developable Surface Fitting to Point Clouds

• Martin Peternell Computer Aided Geometric
  Design 21(2004) 785-803

Reporter Xingwang Zhang June 19, 2005
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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.

3
Research Interests

• Classical Geometry
• Computer Aided Geometric Design
• Reconstruction of geometric objects from dense 3D
  data
• Geometric Computing
• Industrial Geometry

4
Overview

• Problem
• Developable surfaces
• Blaschke model
• Reconstruction of Developable Surfaces
• QA

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Problem

• Given scattered data points from a developable
  surface

• Object Construct a developable surface which
  fits best to the given data

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Ruled Surface

• A ruled surface

directrix curve
a generator

• Normal vector

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Developable Surface

• Each generator all points have the same tangent
  plane.
• Vectors and are linearly
  dependent
• Equivalent condition

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Developable Surface
Three types of developable surfaces
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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.

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Singular Point

• A singular point doesnt possess a tangent plane.
• Singular curve is determined by
  the parameter

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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

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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

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General Fitting Technique
Find an developable B-spline surface
fitting unorganized data points

• Estimating parameter values
• Solving a linear problem in the unknown control
  points

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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

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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

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Blaschke Model
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Blaschke Model

• An oriented plane in Hesse normal form
• Defining Blaschke mapping
• Blaschke cylinder

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Incidence of Point and Plane

• A fixed point , planes
  passing
  through this point
• Image points lie in the
  three space
• The intersection of is an ellipsoid.

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Blaschke Images of a Pencil of Lines and of
Lines Tangent to a Circle
Back
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Tangency of sphere and plane

• oriented sphere with center and signed
  radius
• Tangent planes
• Blaschke image of tangent planes

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Offset operation

• Maps a surface (as set of tangent
  planes) to its offset at distance
• is the offset surface of at distance
• Appearing in the Blaschke image as
  translation by the vector
• See Figure

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Laguerre Geometry

• satisfy
• inverse Blaschke image
• tangent to a sphere
• form a constant angle with the
  direction vector

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The Tangent Planes of a Developable Surface

• be a 1-parameter family of planes
• Generating lines
• Singular curve
• Blaschke image is a curve on
  the Blaschke cylinder

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Classification
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Classification

• Cylinder
• Cone
• Developable of constant slope normal
• form a constant angle with a fixed direction
• Tangent to a sphere

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Recognition of Developable Surfaces from Point
Clouds
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Estimation of Tangent Planes

• , triangles , adjacent points
• Estimating tangent plane at
• Best fitting data points , MIN
• Original surface with measurement point
• developable, form a curve-like region
  on

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A Euclidean Metric in the Set of Planes

• Distance between and
• Geometric meaning
• intersection of with sphere
  

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Boundary Curves of Tolerance Regions of Center
Lines
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A Cell Decomposition of the Blaschke Cylinder
Tesselation of by subdividing an
icosahedral net
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A Cell Decomposition of the Blaschke Cylinder
(continued)

• Cell structure on the Blaschke cylinder
• 20 triangles, 12 vertices, 2 intervals
• 80 triangles, 42 vertices, 4 intervals
• 320 triangles, 162 vertices, 8 intervals
• 1280 triangles, 642 vertices, 16 intervals

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Analysis of the Blaschke Image
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Analysis of the Blaschke Image (continued)

• Check point cloud on fitted
  well by hyperplane
• Principal component analysis

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Principal Component Analysis (continued)

• Minimization
• Eigenvalue problem

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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.

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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.

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Example
Analysis of the Blaschke imageSphere
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Example
Cylinder of rotation
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Example
Approximation of a developable of constant slope
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Example
General cylinder
Triangulated data points and approximation
Original Blaschke image
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Example
Spherical image of the approximation with control
points.
Triangulated data points and approximation
Developable of constant slope
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Reconstruction of Developable Surfaces from
Measurements
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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

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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

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Parametrizing a Tubular Region (continued)
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Curve Fitting
Blaschke image
approximating curve to thinned point cloud
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Curve Fitting (continued)
support function (fourth coordinate)
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A Parameterization of the Developable Surface

• Approximating curve on
  determines the planes
• Compute planar boundary curves in planes
  (bounding box)
• Point representation of

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Boundary Curves
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Example
Approximating curve with control polygon
Projection of the Blaschke image
Developable surface approximating the data points
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Deviation

• Distance between estimated planes and
  the approximation
• Distance between measurements and the
  approximation

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Nearly Developable
Nearly developable surface
Projection of the original Blaschke image
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Nearly Developable Approximation
developable approximation
Thinned Blaschke image with approximating curve
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Singular Points

• Singular points
• Data Points satisfy
• Singular points have to satisfy
• Singular curve is in the outer region
  of the bounding box.

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Conclusions

• Advantages
• Avoiding estimation of parameter values
• Avoiding estimation of direction of generators
• Guaranteeing approximation is developable
• Improving avoidance of singular points
• etc.

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QA

• Questions?

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Thanks all!Especial thanks to Dr Lius help