Projecting points onto a point cloud
Speaker: Jun Chen
Mar 22, 2007
Data Acquisition
Point clouds
25893
Point clouds
56194
Unorganized, connectivity-free
topological
Surface Reconstruction
Applications
Reverse Engineering Virtual Engineering Rapid Prototyping Simulation Particle systems
Definition of “onto”
References
Parameterization of clouds of unorganized points using dynamic base surfaces
Phillip N. Azariadis (CAD,2004)
Drawing curves onto a cloud of points for point-based modeling
Phillip N. Azariadis, Nickolas S. Sapidis (CAD,2005)
References
Automatic least-squares projection of points onto point clouds with applications in reverse engineering
Yu-Shen Liu, Jean-Claude Paul et al. (CAD,2006)
Parameterization of clouds of unorganized points using dynamicbase surfaces
Phillip N. Azariadis
CAD, 2004, 36(7): p607-623
About the author
Instructor of the University of the Aegean, director of the Greek research institute “ELKEDE Technology & Design Centre SA”.
CAD , Design for Manufacture, Reverse Engineering, CG and Robotics.
Parameterization
each point
adequate parameter
well parameterized cloud
accurate surface fitting
2 D
Previous work
Mesh -- Starting from an underlying 3D triangulation of the cloud of points. Ref.[17]
Unorganized Projecting data points onto the base surface Hoppe’s method, ‘simplicial’ surfaces approxi
mating an unorganized set of points Piegl and Tiller’s method, base surfaceis fitted t
o the given boundary curves and to some of the data points.
no safe, universal
(0.3,1) (0,1)
Work of this paper
Algorithm Step 1
Initial base surface---- a Coons bilinear blended patch:
To get the n×m grid points, define: Ri(v)=S(ui,v), Rj(u)=S(u,vj),
pi,j= Ri(v)∩ Rj(u)=S(ui,vj),
so ni,j, Su(ui,vj, ), Sv(ui,vj, ) can be computed.
Error function: it is suitable for the point set with noise and irregular samples.
Step 2: Squared distances error
Step 2: Squared distances error
Step 2: Squared distances error
Let pi,j * be the result of the projection of the point pi,j onto the cloud of points following an
associated direction ni,j.
Proposition 1
Step 3: Minimizing the length of the projected grid sections
No crossovers or self-loops. Define: pi0,j(1<j<m-2) is a row.
closeness
length
identity
tridiagonal and symmetric
Combined projection :
O(m)
Bigger - >smoother
Step 3: Minimizing the length of the projected grid sections
Step 4: Fitting the DBS to the grid Given the set of n×m grid points, a (p,q)th-d
egree tensor product B-spline interpolating surface is Ref.[26,9.2.5]:
Step 5: Crossovers checking
If it fails 1. Terminate the algorithm. 2. Compute geodesic grid sections.The DBS is
re-fitted to the new grid. 3. Increase smoothing term. 4. Remove the grid sections which are stamped
as invalid.
Step 5:Terminating criterion
1. The DBS approximates the cloud of points with an accepted accuracy.
Step 5:Terminating criterion
1. The DBS approximates the cloud of points with an accepted accuracy.
2. The dimension of the grid has reached a predefined threshold.
3. The maximum number of iterations is surpassed.
A final refinement
Advantage
Only assumption:4 boundary curves
dense
thin
Contrarily to existing methods, there is
no restriction regarding the density
Conclusions
Error functions Smoothing Crossovers checking
Drawing curves onto a cloud of points for point-based modelling
Phillip N. Azariadis, Nickolas S. Sapidis
CAD, 2005, 37(1): p109-122
About the authors
Instructor of the University of the Aegean, the Advisory Editorial Board of CAD.
curve and surface modeling/fairing/visualization, discrete solid models, finite-element meshing, reverse engineering, solid modeling
Work of this paper
Projection vectors
pn
pf
Previous work
Dealing with 2D point set. Ref.[7,19,21,26] Appeared in Ref.[21,37]
(a) selection of the starting point is accomplished by trial and error,
(b) it involves four parameters that the user must specify,
(c) no proof of converge is presented, neither any measure for the required execution time.
Note ! Reconstructing an interpolating or fitting
surface is meaningless. Surface reconstruction is not make sense. They are not always work well. (smooth, closed,
density, complexity) Require the expenditure of large amounts of
time and space. Approximation causes some loss of information.
Error function
Error analysis
True location
Independent of the cloud of points
Weight function
distance between p
m and the axisstability
Weight function
distance between p
m and the axisstability
Weight function
Projection vectors
pn
pf
Algorithm
increase
Conclusions
Accuracy and robustness, directly without any reconstruction.
Method improved: Error analysis Weight function Iterative algorithm
Projection of polylines onto a cloud of points
Smoothing
Automatic least-squares projection of points onto point clouds with applications in reverse engineering
Yu-Shen Liua, Jean-Claude Paul, Jun-Hai Yong, Pi-Qiang Yu, Hui Zhang, Jia-Guang Sun, Karthik Ramanib
CAD, 2006, 37(12): p1251-1263
About the authors
Postdoctor of Purdue University
CAD
Senior researcher at CNRS
CAD, numerical analysis
Associate professor of Tsinghua University,
CAD, CG
Previous work
Ray tracing (need projection vector). Ref.[1,7,8,31] MLS (noise and irregular samples, resulting in large
r approximation errors). Ref.[2,3,8,20]
Review
Weight function
Projection vector is unknown before projecting.
Projection
Nonlinear optimization
Linear optimization
Make t(n) maximum or minimum
Proposition The weighted mean point p* that minimizes error function
is co-linear with the line defined by the test point p and the projection vector n computed.
Experimental results
Experimental results
Experimental results
Conclusions
Automatic projection of points.
Thank you!