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!