Martin Peternell, Geometric Modeling and Industrial ...Analysis of the Blaschke image 2 Let H i be...
Transcript of Martin Peternell, Geometric Modeling and Industrial ...Analysis of the Blaschke image 2 Let H i be...
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Reconstruction of Developable Surfaces
Martin Peternell,
Geometric Modeling and Industrial Geometry Group,
TU Wien,
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Overview
1. The Problem
2. Geometric Properties of Developable Surfaces
3. The Basic Concept
4. The Algorithm
5. Examples
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ProblemGiven: Triangulated datapoints from a developablesurface
Object: Construct a devel-opable surface which fits bestto the given data
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Literature
Hoschek, Schneider, Pottmann, Bodduluri, Ravani, Leopoldseder,Wallner, Redont, Farin, Aumann, Clements, Leon, ...
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Geometric Properties of Developable Surfaces
There are three types of developables:
Cylinder Cone Surface formed by
tangent lines of space curves
• The Gauss curvature of a developable surfaces is zero.
• A developable surface S is envelope of its one-parameter familyof tangent planes
T (t) : d(t) + a(t)x + b(t)y + c(t)z = 0.
• Dual approach: T (t) is a curve in dual projective 3-space.
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The Basic Concept
• The surface S is given by a data pointspi (measurements).
• Estimate tangent planes Ti at points pi.
• The estimated tangent planes Ti will lieclose to the true tangent planes T (t) ofthe surface S.
Idea: We compute a 1-parameter family of planes T (t) best fittingthe estimated tangent planes Ti.
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Tangent planes of a Developable
• Tangent planes T (t) : d(t) + a(t)x + b(t)y + c(t)z = 0 of adevelopable surface S.
• Normalizationa2 + b2 + c2 = 1
of the normal vectors n(t) = (a(t), b(t), c(t)) of S and T (t).• The image curve
C(t) = (a, b, c, d)(t) ∈ R4
is a curve on the
• Blaschke-Cylinder
B : x21 + x2
2 + x23 = 1 ⊂ R4.
• The normal vectors n(t) of S form acurve on the unit sphere S2.
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The Algorithm
• Compute estimates of the tangent planes
Ti : di + aix + biy + ciz = 0
at the data points pi. The points Ci = (ai, bi, ci, di) ∈ B ⊂ R4
should be arranged in a tubular region.
• Find a curve C(t) ⊂ B fitting best the tubular region definedby Ci.
• Determine the one-parameter family of tangent planes T (t)determined by C(t).
• Compute a point-representation of the correspondingdevelopable approximation S∗ of the data points.
• Avoid occurring singularities in the point representation.
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Estimating tangent planes
There are different methods:
• Compute a plane of regression w.r.t. adja-cent points of each data point.
• Compute a linear function z = ax + by + c
w.r.t. adjacent points in the l2-sense (or inthe l1-sense).
• Compute a quadratic function w.r.t. adja-cent points and find the tangent plane atthe considered data point.
• . . .
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Curve Fitting on the Blaschke-Cylinder
• Given: Estimated tangent planes
Ti : di + aix + biy + ciz = 0,
with a2i + b2
i + c2i = 1
and corresponding points Ci = (ai, bi, ci, di)on B (Blaschke cylinder).
• Object: Curve C(t) approximating the setof points Ci.
C(t) =∑
i
Bi(t)bi
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Curve Fitting on the Blaschke-Cylinder 2
• Parametrization problem: How to estimate parameter values ofthe given points Ci?
• Data access on the Blasche cylinder?
• How to handle outliers?
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Cell Structure on the Blaschke-Cylinder
• Tesselation of S2 by subdividing an ikosahedral net.
• Cell structure on the Blaschke-Cylinder B.
1. 20 triangles, 12 vertices, 2 intervals
2. 80 triangles, 42 vertices, 4 intervals
3. 320 triangles, 162 vertices, 8 intervals
4. 1280 triangles, 642 vertices, 16 intervals
5. 5120 triangles, 2562 vert., 32 intervals
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Parametrizing a tubular region
• Determine relevant cells of B carryingpoints Ci = (ai, bi, ci, di).
• Thinning of the tubular region: Find cellscarrying only few points and delete thesecells and points.
• Estimate parameter values for a reducedset of points Ck (by moving least squares:marching through the tube).
• Compute an approximating curve C(t) onB w.r.t. points Ck.
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Parametrizing a tubular region 2
Spherical image n = (a, b, c)(t) Support function (fourth coord. d(t))
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Point Representation of the Approximation
• Approximating curve C(t) = (a, b, c, d)(t) on B determines theplanes
E(t) : d(t) + a(t)x + b(t)y + c(t)z = 0,
which envelope the approximation S∗.
• Generating lines L(t) of S∗ are
L(t) = E(t) ∩ E(t), with E(t) =d
dtE(t).
• Compute planar boundary curveski(t) in planes Hi (bounding box):
ki(t) = E(t) ∩ E(t) ∩Hi.
• Point representation of S∗:
x(t, u) = (1− u)k1(t) + uk2(t).
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Example
Left: Triangulated data points of a developable. Middle & Right:Spherical image.
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Example, cont.
Spherical image n = (a, b, c)(t)and support function d(t) of theapproximation S∗.
Approximating developable S∗
and data points.
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Singular Points of a Developable
• Singular points
s(t) = E(t) ∩ E(t) ∩ E(t).
are corresponding to osculating planes
σ : c(t) + λc(t) + µc(t)
of the Blaschke image C = c(t).
• Let N = c ∨ c ∨ c = (n1, . . . , n4). ⇒ s(t) = 1n4
(n1, n2, n3).
• Data points pi satisfy ‖pi‖ < 1 (or |pi1|, |pi2|, |pi3| < 1)
• Singular curve s(t) has to satisfy
‖s(t)‖ > 1, ( or |s1|, |s2|, |s3| > 1).
Singular curve s is in the outer region of the bounding box.
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Singular Points of a Developable
• The condition ‖s‖ > 1 is (highly) non-linear in thecontrol-points.
• C is a biarc, interpolating Hermiteelements Pj , Vj and Pj+1Vj+1. ⇒‖s‖ > 1 is a quadratic inequality inthe free parameter of the 1-par. so-lution.
• Pottmann and Wallner: Considerspecial parametrizations of devel-opable surfaces which make a con-trol of the singular curve easier.
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Blaschke Images of Special Shapes
Commonly used geometric objects, like cones, cylinders and spherescan be characterized by their images of the estimated tangentplanes on the Blaschke cylinder B:
Objects in R3 Images in the Blaschke-Cylinder
as set of tangent planes as point sets
cones and cylinders of revolution planar intersections (conics on B)
general cones and cylinders curves on B lying in a 3-space
developables of constant slope
spheres surfaces on B lying in a 3-space
(ellipsoids)
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Analysis of the Blaschke image
• Let Ti be reliable estimated tangent planes of a region R on theobject S.
• Let Rb be the Blaschke image ⊂ B of the considered region.
• Compute the ellipsoid of inertia (PCA) of Rb.(axes-hyperplanes and corresponding eigenvalues)
• Analyze the eigenvalues and eigenvectors
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Analysis of the Blaschke image 2
Let Hi be the 3-spaces with respect to eigenvalues λi,
Hi : hi0 + hi1x1 + hi2x2 + hi3x3 + hi4x4 = 0.
• Four small eigenvalues: The Blaschke image is a point-likecluster. All estimated tangent planes are nearly identical.
⇒ The original surface R is planar.
• Two small eigenvalues: The Blaschke image is a planar curve(conic).
⇒ The original surface R is a cone or cylinder of rotation.
– |h10 − h20| > ε: R is a cone of rotation.
– |h10 − h20| ≤ ε: R is a cylinder of rotation.
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Analysis of the Blaschke image 3
• One small eigenvalue and curve-like Blaschke-image:
⇒ The original surface R is developable.
– |h10| < ε: R is a general cone.
– |h10| < ε and |h14| < ε : R is a general cylinder. All planesare orthogonal to a fixed direction v.
– |h14| < ε : R is a developable of constant slope. All planesform a constant angle with a fixed direction v.
• One small eigenvalue and surface-like Blaschke-image:
⇒ The original surface R is a sphere.
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Analysis of the Blaschke image–Sphere
Let Hi be the 3-spaces with respect to eigenvalues λi,
Hi : hi0 + hi1x1 + hi2x2 + hi3x3 + hi4x4 = 0, h14 � ε.
• S is a sphere (λ1 < ε): Its center and radiusare determined by
m =1
h14(h11, h12, h13) , r =
−h10
h14
Region 1 contains 4166 data points.
Eigenvalues 0.00004 0.18598 0.36039 0.36837
Eigenvector -0.0024 0.0116 -0.3500 0.9367
-0.2972 0.9547 -0.0041 -0.0141
0.0015 -0.0087 -0.9368 -0.3499
0.9548 0.2972 -0.0006 -0.0015
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Cylinder and Cone of Rotation
Let Hi be the 3-spaces with respect to eigenvalues λi,
Hi : hi0 + hi1x1 + hi2x2 + hi3x3 + hi4x4 = 0, h14, h24 � ε.
If h14 < ε, then (hi1, hi2, hi3) = a is the axis of R.
• λ1, λ2 < ε ⇒ centers and radii of two spheres
m1 = 1h14
(h11, h12, h13) , r1 = −h10h14
m2 = 1h24
(h21, h22, h23) , r2 = −h20h24
, with
• |r1 − r2| < ε: R is a cylinder of rot. with axis and radius
a = m2 −m1, r = mean(r1, r2).
• |r1 − r2| � ε: R is a cone of rot. with vertex, axis and angle
v =1
r2 − r1(r2m1− r1m2), a = m2−m1, sinφ =
r2 − r1
‖m2 −m1‖.
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Example: Cylinder of rotation
Region 1 contains 3553 data points.
Eigenvalues 0.00013 0.00024 0.48129 0.51860
Eigenvector 0.0245 0.0017 -0.9958 0.0881
0.1074 -0.9942 0.0014 0.0050
-0.0239 0.0026 0.0875 0.9959
0.9936 0.1075 0.0265 0.0212
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Developable of Constant Slope
Let Hi be the 3-spaces with respect to eigenvalues λi,
Hi : hi0 + hi1x1 + hi2x2 + hi3x3 + hi4x4 = 0.
• λ1 < ε and curve-like Blaschke-image:
⇒ The original surface R is developable.
– |h14| < ε : R is a developable of constant slope. All planesform a constant angle with a fixed direction v.
Eigenvalues 0.0000 0.0028 0.1473 0.3175
Eigenvector 0.0013 0.9815 0.1849 0.0498
0.0006 0.1407 -0.8727 0.4675
1.0000 -0.0014 0.0004 -0.0002
-0.0002 -0.1299 0.4518 0.8826
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Segmentation of Geometric Objects
• Estimate tangent planes Ti with respect to given triangulateddata points pi of the object S.
• Edge detection to perform a pre-segmentation of the object.
• Compute connected components of the object (w.r.ttriangulation).
• Compute Blaschke images T bi of the estimated tangent planes
Ti.
• Segmentation and analysis of the Blaschke image Sb.
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Example: Simple composed object
Region 1 contains 1933 data points.
Eigenvalues: 0.00000 0.28204 0.37189 0.47503
Eigenvector 0.5705 -0.0035 0.0927 -0.8161
-0.2269 0.6355 0.7339 -0.0779
0.1618 -0.6957 0.6729 0.1926
0.7726 0.3349 0.0061 0.5393
Region 2 contains 1933 data points.
Eigenvalues: 0.00000 0.27870 0.36828 0.42823
Eigenvector: -0.4899 0.1305 0.1641 0.8462
-0.2408 0.5691 -0.7826 -0.0754
0.1718 -0.7100 -0.6005 0.3254
0.8201 0.3938 -0.0060 0.4152
Region 3 contains 811 data points.
Eigenvalues: 0.00000 0.00000 0.00001 0.00001
Eigenvector: .0052 -0.2686 0.9624 0.0410
0.0003 0.2839 0.1198 -0.9513
0.9992 -0.0361 -0.0149 -0.0124
0.0406 0.9198 0.2435 0.3051
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Example: Simple composed object 2
Region 4 contains 822 data points.
Eigenvalues: 0.00000 0.00000 0.06475 0.46085
Eigenvector 0.1916 0.1297 -0.1339 0.9636
-0.5603 0.8283 0.0001 -0.0000
0.3355 0.2269 0.9138 0.0297
-0.7326 -0.4956 0.3834 0.2657
Region 5 contains 333 data points.
Eigenvalues: 0.00000 0.00000 0.00002 0.00006
Eigenvector: 0.0394 0.4624 0.5194 0.7175
0.9980 -0.0234 -0.0592 0.0032
0.0497 0.2202 0.7146 -0.6621
0.0067 -0.8586 0.4647 0.2166
Region 6 contains 153 data points.
Eigenvalues: 0.00000 0.00000 0.00001 0.00036
Eigenvector: -0.0473 0.5308 0.8455 -0.0327
0.0367 0.2322 -0.1807 -0.9550
0.9807 0.1684 -0.0475 0.0876
-0.1863 0.7975 -0.5002 0.2813
. . .
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Future research
• Continuous transformations of developable surfaces.
• Treatment of nearly developable surfaces.
• Segmentation using both points pi and tangent planes Ti.
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Thank You for Your attention!
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