Parameterization-Free Projection For Geometry Reconstruction

Yaron Lipman            Daniel Cohen-Or           David Levin                   Hillel Tal-Ezer


ACM Transactions on Graphics (SIGGRAPH 2007)



 Example videos of the projection operator:     





We introduce a Locally Optimal Projection operator (LOP) for

surface approximation from point-set data. The operator is parameterization

free, in the sense that it does not rely on estimating a local

normal, fitting a local plane, or using any other local parametric

representation. Therefore, it can deal with noisy data which clutters

the orientation of the points. The method performs well in cases

of ambiguous orientation, e.g., if two folds of a surface lie near

each other, and other cases of complex geometry in which methods

based upon local plane fitting may fail. Although defined by a

global minimization problem, the method is effectively local, and it

provides a second order approximation to smooth surfaces. Hence

allowing good surface approximation without using any explicit or

implicit approximation space. Furthermore, we show that LOP is

highly robust to noise and outliers and demonstrate its effectiveness

by applying it to raw scanned data of complex shapes.


Technical Report:

Acrobat, ~5 MB      














Left: A prism point-cloud contaminated with ghost geometry

noise. Middle: MLS. Right: LOP. In both the point-set is

projected onto itself.


A noisy point-cloud of a surface with three holes (a). The

red points in (b) are projected onto the point-set in (a). The results

of the MLS and LOP projections are shown in (c),(d), respectively.



This example depicts the distribution of point by LOP operator.

(a): Starting from a crude initial guess (red points projected

onto the green point-set), the operator iteratively (b–d) distribute

the points regularly while respecting the geometry faithfully.


Hole-puncher scan which consists of a few registered

scans suffering from bad alignment, noise and outliers. (a) shows an

example of two scans which where registered using ICP. (b),(d-top)

are the whole input data seen from two angles. Note the high noise

and ghost geometry. The corresponding LOP reconstruction is depicted

in (c),(d-bottom). Note the zoomed-in views in (e) and (f).