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Transcript of Www.geometrie.tuwien.ac.at IMA Tutorial, 20. 4. 2001 Instantaneous Motions - Applications to...
IMA Tutorial, 20. 4. 2001 www.geometrie.tuwien.ac.at
Instantaneous Motions -
Applications to Problems in
Reverse Engineering
and 3D Inspection
H. PottmannInstitut für Geometrie, TU Wien
IMA Tutorial, 20. 4. 2001 www.geometrie.tuwien.ac.at
One parameter motions
moving system
fixed system 0
( )t0x
( )tu
x
( ) = ( ) + ( )t t t0 Ax u x
with .TA A E
Trajectory of a point x:
Velocity vector field of x:
( ) = ( ) + ( )t t t0 Ax u x
Representation in the same system, e.g. moving system:
+( ) T T T0
Sc
x u Av A Ax A x
IMA Tutorial, 20. 4. 2001 www.geometrie.tuwien.ac.at
One parameter motions
1
2
3
with : there is
c
c
c
S xc c x
With we have T T TA A E A A A A 0
3 2
3 1
2 1
0
0
0
c c
c c
c c
S TS S 0 skew symmetricS
( ) cv cx x
velocity vect ( )or of :v x x
Linear vector field!
+ ( ) T T
Sc
Av uA Ax x
IMA Tutorial, 20. 4. 2001 www.geometrie.tuwien.ac.at
Discussion of the geometry of the velocity vector field
Special case 1: c o
( )v x c instantaneous translation
c oSpecial case 2:
( )v x c x
: there is ( ) .A x v xc o
For all points of the line
instantaneous rotation about axis A
A
o
c
x
v(x)
IMA Tutorial, 20. 4. 2001 www.geometrie.tuwien.ac.at
Discussion of the geometryof the velocity vector field
General case:Composition of an instantaneous rotationand an instantaneous translation… instantaneous screw motion
A
p
c
x
v(x)
c c p
c
Rotation axis A has direction vector c and passes throughpoints p with .
( … moment vector of the axis A)
Special case 3 (geometrically equivalent to case 2):
.c c o instantaneous rotation ( ) mitc cv x x
IMA Tutorial, 20. 4. 2001 www.geometrie.tuwien.ac.at
Discussion of the geometryof the velocity vector field
• translation with constant velocity
• uniform rotation about an axis
• uniform screw motion
One parameter motions with a constant
(time independent) velocity vector field:
IMA Tutorial, 20. 4. 2001 www.geometrie.tuwien.ac.at
Uniform screw motion
Composition of a rotation aboutan axis A and a proportionaltranslation parallel to A.
0 0
0 0
0
( ) cos( ) sin( )
( ) sin( ) cos( )
( )
x t x t y t
y t x t y t
z t z ptp … pitch
A curve generates a helical surface.
A
tpt
0 :p pure rotation, generates rotational surface.
' ' :p pure translation, generates cylindrical surface.
IMA Tutorial, 20. 4. 2001 www.geometrie.tuwien.ac.at
Reverse Engineering
Problem description:
Usually, in CAD/CAM systems are used todesign and produce technical objects.
Conversely, reverse engineering deals with thereconstruction of a computer model from anexisting object.
Different 3D scanning tools givea large number of data points(with measurement errors) on the surface of the technical object.
IMA Tutorial, 20. 4. 2001 www.geometrie.tuwien.ac.at
Reverse Engineering
Goal:
Automatic detection of simple surfaces in thereconstruction process the CAD model. This is important for a precise CAD representation!
Simple classes of surfaces include:
• planes,• cylinders & cones of revolution,• spheres,• tori,• general rotational surfaces,• helical surfaces.
IMA Tutorial, 20. 4. 2001 www.geometrie.tuwien.ac.at
Reconstruction ofhelical surfaces
Given: data points di
1. step: estimation of surface normals ni
di
ni
2. step: calculate an appropriate motion with the velocity vector field ( .) c cv x x
with ) ( .i i i i ii in ncv dnc n nd
Minimization of 2cos i is nonlinear in ( , ) : .c c C
v(di)i
instead: minimize
2
1
( )N
ii i
Tnc Cn Nc C
21 ,T DC C Dcwith side condition
1
1
1
0
0
0
IMA Tutorial, 20. 4. 2001 www.geometrie.tuwien.ac.at
Generalized eigenvalues
Lagrange: ( ) 0, 1.TD C CN DC
det( ) 0, cubic in .DN
For a gen. eigenvalue and gen. eigenvector C we have
( ) .F T TC DC C CNC
Therefore, the solution of F(C) = min is the generalized eigenvector C to the smallest generalized eigenvalue.
From the solution vector 6( , )C c c the axis ( , )A a aand the pitch p of the ‘best fitting’ screw motion is calculated:
,ac
c
,
pa
c c
c
2
.pc c
c
IMA Tutorial, 20. 4. 2001 www.geometrie.tuwien.ac.at
Reconstruction process
• Computation of the one parameter motion whose velocity vector field fits the estimated normals best.
• Estimation of surface normals in the data points.
• Reconstruction of the profile curve by moving the data into an appropriate plane.
• Reconstruction of the surface by applying the one parameter motion to the profile curve.
IMA Tutorial, 20. 4. 2001 www.geometrie.tuwien.ac.at
Curve reconstruction fromunorganized data points
• Spline curve through selected ordered data points.
• Compute minimal spanning tree of the complete graph by Delauny triangulation.
• Apply ‘moving least squares’ algorithm (see next slide).
IMA Tutorial, 20. 4. 2001 www.geometrie.tuwien.ac.at
Moving least squares
• Choose a point p and its neighborhood (distance <= H) in the ‘minimum spanning tree’.
• Fit parabola (weighted point data).
• Project p onto parabola.
p
x
y
IMA Tutorial, 20. 4. 2001 www.geometrie.tuwien.ac.at
Example: rotational surface
• Data points and estimated surface normal vectors
• Data points and computed axis of rotation
• Data points, rotated into a plane, and approximating curve
• Reconstructed surface of revolution
IMA Tutorial, 20. 4. 2001 www.geometrie.tuwien.ac.at
Example: tubular surface
• Data points and estimated surface normal vectors
• Reconstruction of spine curve with locally approximating tori
• Reconstructed tubular surface
IMA Tutorial, 20. 4. 2001 www.geometrie.tuwien.ac.at
Example: developable surface
• Developable surface
• Disturbed data points of the original surface
• Computation of locally best- fitting cones of revolution by region-growing
• Reconstruction of a G1-surface composed of segments of cones of revolution (right circular cones)
IMA Tutorial, 20. 4. 2001 www.geometrie.tuwien.ac.at
Example: developable surface
IMA Tutorial, 20. 4. 2001 www.geometrie.tuwien.ac.at
3D Inspection
Problem description:
Determine a one parameter motion that matches a point cloudto a surface (or another pointcloud) as good as possible.
Motivation:
• Inspection of technical objects / quality checks,
• 3D matching, e.g. joining different 3D laser scanner images to a 3D object.
IMA Tutorial, 20. 4. 2001 www.geometrie.tuwien.ac.at
3D inspection
xi
ni
di
v(xi)
( )d i in vi
xi ... data points
velocity vector fieldv(xi) of a one parameter motion:
( .) c cv x x
minimize 2( ( ))d iin c xci
determine distance di to surfaceand surface normal ni in thecorresponding surface point.
quadratic in , !c c
IMA Tutorial, 20. 4. 2001 www.geometrie.tuwien.ac.at
Industrial inspection
CAD-modelof a technicalobject
IMA Tutorial, 20. 4. 2001 www.geometrie.tuwien.ac.at
Industrial inspection
Image of theexisting object
IMA Tutorial, 20. 4. 2001 www.geometrie.tuwien.ac.at
Industrial inspection
Cloud of data pointsfrom the surface ofthe existing object
IMA Tutorial, 20. 4. 2001 www.geometrie.tuwien.ac.at
Industrial inspection
Point cloud withtransparentsurface
IMA Tutorial, 20. 4. 2001 www.geometrie.tuwien.ac.at
Industrial inspection
Input to thematchingalgorithm
IMA Tutorial, 20. 4. 2001 www.geometrie.tuwien.ac.at
Industrial inspection
Output of thematchingalgorithm
IMA Tutorial, 20. 4. 2001 www.geometrie.tuwien.ac.at
Industrial inspection
Color codingof deviations
IMA Tutorial, 20. 4. 2001 www.geometrie.tuwien.ac.at
Instantaneous Motions -
Applications to Problems in
Reverse Engineering
and 3D Inspection
H. PottmannInstitut für Geometrie, TU Wien