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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
         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         Object: Construct a
    data points from a        developable surface
    developable surface       which fits best to
                              the given data
     Ruled Surface
   A ruled surface x(u, v)  c(u)  ve(u)

    c (u ) : directrix curve
    e (u ) :   a generator
   Normal vector
     n(u, v)  c(u)  e(u)  ve(u)  e(u)
     Developable Surface
   Each generator all points have the same
    tangent plane.
   Vectors c  e and e  e are linearly
    dependent
   Equivalent condition
            det(c, e, e)  0
Developable Surface
Three types of developable surfaces
    Geometric Properties of
    Developable Surface
   Gaussian curvature is zero
   Envelope of a one-parameter family of
    planes
      T (u ) : n4 (u )  n1 (u ) x  n2 (u ) y  n3 (u ) z  0
   Dual approach: T (u) is a curve in dual
    projective 3-space.
       Singular Point
   A singular point doesn’t possess a tangent
    plane.
   Singular curve s(u )  x (u, vs ) is determined by
    the parameter
                      ( c  e )  (e  e )
               vs                        .
                           (e  e ) 2
     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
Find an n1 developable B-spline surface
           b(u, v)   Ni (u)N j (v)bij
fitting unorganized data points pk
   Estimating parameter values (ui , v j )
   Solving a linear problem in the unknown
    control points bij
    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
Blaschke Model
     Blaschke Model
   An oriented plane in Hesse normal form:
      E : n1 x  n2 y  n3 z  d  0, n12  n2  n3  1
                                             2    2


   Defining Blaschke mapping:
       b:E     b( E )  (n1 , n2 , n3 , d )  ( n, d )
   Blaschke cylinder:
               B : u12  u2  u3  1
                          2    2
       Incidence of Point and Plane
   A fixed point p  ( p1 , p2 , p3 ) , planes    E : n x  d  0
    passing through this point
             p1n1  p2 n2  p3n3  d  p  n  d  0
   Image points b( E )  (n1 , n2 , n3 , d ) lie in the
    three space
                 H : p1u1  p2u2  p3u3  u4  0

   The intersection of H B is an ellipsoid.
Blaschke Images of a Pencil of Lines
and of Lines Tangent to a Circle




              Back
       Tangency of sphere and plane
    S oriented sphere with center m and signed
    radius r
                S : ( x  m)  r  0
                         2    2



   Tangent planes:
      TS : n1m1  n2 m2  n3m3  d  n  m  d  r

   Blaschke image of tangent planes:
         H : m1u1  m2u2  m3u3  u4  r  0
     Offset operation
 Maps a surface F  3 (as set of tangent
  planes) to its offset Fr at distance r
  S is the offset surface of m at distance r
 Appearing in the Blaschke image B as

  translation by the vector (0,0,0, r )
                     See Figure
         Laguerre Geometry
   q  (q1 , q2 , q3 , q4 )  B   satisfy :
               H : a0  u1a1  u2 a2  u3a3  u4 a4  0
    T = b 1 (q )inverse Blaschke image
        a4  0. T tangent to a sphere
                        1                         a0
                     m    (a1 , a2 , a3 ), r  
                        a4                        a4
        a4  0. T form a constant angle with the
         direction vector a = (a , a , a )
                                        1   2   3
     The Tangent Planes of a
     Developable Surface
   T (u) be a 1-parameter family of planes

      T (u ) : n4 (u )  n1 (u ) x  n2 (u ) y  n3 (u ) z  0

   Generating lines: L(u )  T (u ) T (u )
   Singular curve: L(u )  T (u ) T (u ) T (u )
   Blaschke image b T (u)   b(D) is a curve on
    the Blaschke cylinder B
Classification
       Classification
   Cylinder: H : a1u1  a2u2  a3u3  0
   Cone: H : p1u1  p2u2  p3u3  u4  0
   Developable of constant slope: normal n(u)
    form a constant angle with a fixed direction
           H :   a1u1  a2u2  a3u3  0
   Tangent to a sphere:
        H : r  u1m1  u2 m2  u3m3  u4  0
Recognition of Developable
Surfaces from Point Clouds
      Estimation of Tangent Planes
    p , triangles   t j , adjacent points q k
   Estimating tangent plane T at p
   Best fitting data points q k , MIN dist (qk , T )
   Original surface with measurement point pi
    developable, b(Ti ) form a curve-like region
    on B
      A Euclidean Metric in the Set
      of Planes
   Distance dist ( E , F ) between E and F
     E : e1 x  e2 y  e3 z  e4  0, F : f1 x  f 2 y  f 3 z  f 4  0
             dist ( E , F )  i 1 (ei  fi )2
                            2        4



   Geometric meaning:
     F : x  m  0 b(F )  (1, 0, 0, m) dist ( E , F )  r
    b( E ) : intersection of B with sphere
    (u1  1) 2  u2  u3  (u4  m) 2  r 2
                  2    2
Boundary Curves of Tolerance
Regions of Center Lines
   A Cell Decomposition of the
   Blaschke Cylinder
Tesselation of S 2 by subdividing an
icosahedral net
        A Cell Decomposition of the
        Blaschke Cylinder (continued)
   Cell structure on the Blaschke cylinder B
       20 triangles, 12 vertices, 2 intervals
       80 triangles, 42 vertices, 4 intervals
       320 triangles, 162 vertices, 8 intervals
       1280 triangles, 642 vertices, 16 intervals
Analysis of the Blaschke
Image
     Analysis of the Blaschke
     Image (continued)
   Check point cloud bi  b(Ti ) on            B      fitted well by
    hyperplane H
    H : h0  h1u1           h4u4  0, h12             h4  1
                                                           2

   Principal component analysis
      c =   bi  / M            qi  bi  c

     d (qi , H )  h  qi , h  (h1 , h2 , h3 , h4 )
      Principal Component
      Analysis (continued)
   Minimization                        M                     M
                                    1                   1
           F (h1 , h2 , h3 , h4 ) 
                                    M
                                         d (qi , H )  M
                                        i 1
                                            2
                                                             ( h  qi ) 2
                                                              i 1

   Eigenvalue problem
                                                 M
                                             1
          F (h)  hT  C  h,           C :
                                             M
                                                  qi  qiT
                                                 i 1


        Eigenvalues:1  2  3  4
        Deviations: 1   2   3   4
        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.
       | h10  h20 |  a cone of rotation.
       | h10  h20 |  a cylinder of rotation.
      Principal Component
      Analysis (continued)
   One small eigenvalue and curve-like
    Blaschke image. The original surface is
    developable.
      |h 10 |     a general cone
      | h |  ,| h |  a general cylinder
          10        14
      | h |  a developable of constant slope.
          14

   One small eigenvalue and surface-like
    Blaschke-image: The original surface is a
    sphere.
Example




  Analysis of the Blaschke image–Sphere
Example




    Cylinder of rotation
Example




Approximation of a developable of constant slope
       Example




General cylinder   Triangulated data points   Original Blaschke image
                   and approximation
      Example




                 Triangulated data   Spherical image of the
Developable of   points and          approximation with
constant slope   approximation       control points.
Reconstruction of Developable
Surfaces from Measurements
        Reconstruction
   Find a curve c (t )  B fitting best the tubular
    region defined by b(Ti )
   Determine 1-parameter family of tangent
    planes E (t ) determined by c (t )
   Compute a point-representation of the
    corresponding developable approximation D*
    of the data points
         Parametrizing a Tubular
         Region
   Determine relevant cells of B carrying points
     Ci  (ai , bi , ci , di )
   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 C k (by moving least squares: marching
    through the tube)
   Compute an approximating curve c (t ) on B
    w.r.t. points C k
Parametrizing a Tubular
Region (continued)
Curve Fitting




Blaschke image   approximating curve to thinned point cloud
Curve Fitting (continued)




   support function (fourth coordinate)
        A Parameterization of the
        Developable Surface
   Approximating curve c (t )  (c1 , c2 , c3 , c4 )(t ) on B
    determines the planes
         E (t ) : c4 (t )  c1 (t ) x  c2 (t ) y  c3 (t ) z  0
   Compute planar boundary curves ki (t ) in H i
    planes (bounding box):
              ki (t )  E (t ) E (t ) H i
   Point representation of D :
                     x (t , u )  (1  u ) k1 (t )  uk2 (t )
Boundary Curves
       Example




                         Projection of the   Approximating
                         Blaschke image      curve with control
                                             polygon
Developable surface
approximating the data
points
    Deviation
   Distance between estimated planes Ti , i  1, , N
    and the approximation D *
                      1
         d ( D, D*) 
           2

                      N
                          i
                            dist 2 (Ti , E (ti ))
   Distance between measurements pi and the
    approximation D *
                      1
         d ( D, D*) 
           2

                      N
                          i
                            dist 2 ( pi , E (ti ))
  Nearly Developable




Nearly developable surface   Projection of the original Blaschke image
        Nearly Developable
        Approximation




developable approximation Thinned Blaschke image with approximating curve
       Singular Points
   Singular points s (t )  E (t )                          E (t )      E (t )
                                                      1
    n  c  c  c = (n1 , n2 , n3 , n4 )  s (t )          (n1 (t ), n2 (t ), n3 (t ))
                                                    n4 (t )

   Data Points pi satisfy pi  1
   Singular points have to satisfy s(t )  1
    Singular curve s (t ) is in the outer region
    of the bounding box.
        Conclusions
   Advantages
       Avoiding estimation of parameter values
       Avoiding estimation of direction of generators
       Guaranteeing approximation is developable
       Improving avoidance of singular points
       etc.
Q&A
      Questions?
  Thanks all!
Especial thanks
to Dr Liu’s help

				
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