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 dynamic base surfaces Phillip N. Azariadis CAD, 2004, 36(7): p

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 approximating an unorganized set of points Piegl and Tiller’s method, base surfaceis fitted to 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: R i (v)=S(u i,v), R j (u)=S(u,v j ), p i,j = R i (v) ∩ R j (u)=S(u i,v j ), so n i,j, S u (u i,v j, ), S v (u i,v j, ) can be computed.

Error function: it is suitable for the point set with noise and irregular samples. Step 2: Squared distances error

Let p i,j * be the result of the projection of the point p i,j onto the cloud of points following an associated direction n i,j.

Proposition 1

Step 3: Minimizing the length of the projected grid sections No crossovers or self-loops. Define: p i0,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- degree 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 thi n 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): p

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 pnpn pfpf

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 axis stability

Weight function distance between p m and the axis stability

Weight function

Projection vectors pnpn pfpf

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): p

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 larger 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

Conclusions Automatic projection of points.

Thank you!