Point Set Processing and Surface Reconstruction (www.cgal.org) Nader Nader Salman, INRIA In collaboration with Pierre Alliez, INRIA Laurent Saboret, INRIA Gaël Guennebaud, INRIA Bordeaux 9th ACCV Xi’an China, Sep 25-27 2009

Outline Common pipeline Structure Documentation Demo Roadmap

Common Pipeline Physical Acquisition Registration Pre-processing Reconstruction Digital Geometry

Example Pipeline (1) laser range data Acquisition Registration Pre-processing (simplification, denoising, smoothing, …) Reconstruction

Example Pipeline (2) multi-view passive stereo Acquisition & calibration Point cloud generation Pre-processing (simplification, denoising, smoothing, …) Reconstruction

Structure (pipeline-oriented) Point set Analysis Processing Normals Reconstruction Contouring Bounding box Bounding sphere Centroid Average spacing Simplification Outlier removal Smoothing Estimation Orientation Poisson Algebraic point set surfaces Surface mesh generator (grayed out = in other CGAL packages)

Entering the Pipeline… Point set Clean point set Clean points with oriented normals Analysis Processing Normals Reconstruction Contouring We can enter the pipeline later if we have a clean point set, or points with oriented normals, or points with unoriented normals, etc. Bounding box Bounding sphere Centroid Average spacing Simplification Outlier removal Smoothing Estimation Orientation Poisson Algebraic point set surfaces Surface mesh generator Points with unoriented normals (grayed out = in other CGAL packages)

Clean points with oriented normals Output… Point set Analysis Processing Normals Reconstruction Contouring Bounding box Bounding sphere Centroid Average spacing Simplification Outlier removal Smoothing Estimation Orientation Poisson Algebraic point set surfaces Surface mesh generator Clean points with oriented normals Implicit function Surface triangle mesh (grayed out = in other CGAL packages)

Point_set_processing_3 Introduction Left: 275k points sampled on an elephant (Minolta laser scanner) Right: point set cleaned and simplified to 17K points

Analysis Average spacing Bounding box Bounding sphere Centroid reuse orthogonal search for K nearest neighbors used by reconstruction / contouring algorithms takes an iterator range of 3D points and parameter K

Processing Simplification Outlier removal Smoothing by random selection by clustering (regular grid) Outlier removal sort w.r.t. sum of squared distances to KNN and cut at specified percentile. Smoothing jet fitting + projection

Example Simplification from 100K to 50k points by random simplification

Example Simplification from 100K to 50k points by clustering

Example Outlier Removal (a lion and a bunch of outliers)

Example Smoothing For each point find KNN fit jet (smooth parametric surface) project onto jet (noisy point set) (smoothed point set)

Normals Estimation (no orientation) Orientation KNN + PCA (fit a plane) KNN + jet fitting (better for noisy data) Orientation KNN + BGL MST (Minimum Spanning Tree) [Hoppe 92]

Example Normal Estimation Normals estimation through PCA (7 KNN) Orientation through MST (7 KNN)

Example Normal Orientation Normal Orientation through MST

Surface_reconstruction_points_3 Introduction reconstructed surface using Poisson 17K points sampled on an elephant (Minolta laser scanner) reconstructed surface using APSS (in progress)

Reconstruction (1) Poisson surface reconstruction [Kazhdan-Bolitho-Hoppe, SGP 2006] Solves for an implicit function (~indicator function) Isosurface extracted by CGAL surface mesh generator

Poisson Surface Reconstruction Reconstruct the surface of the model by solving for the indicator function of the shape. M Indicator function 1 1 1

Poisson Surface Reconstruction There is a relationship between the normal field and gradient of indicator function M Indicator gradient points + oriented normals

Poisson Surface Reconstruction Represent the points by a vector field Find the function whose gradient best approximates : Applying the divergence operator, we can transform this into a Poisson problem: 

Poisson Surface Reconstruction We solve for the Poisson equation onto the vertices of a (refined) 3D Delaunay triangulation [TAUCS linear solver]

Example Poisson

Example Poisson Left: 120K points sampled on child statue (Minolta laser scanner) Right: reconstructed surface

Example Poisson Left: 120K points sampled on a statue (Minolta laser scanner) Right: reconstructed surface

Example Poisson

Example Poisson Left: 70K points with (emphasized) outliers Right: reconstructed surface

Example Poisson Left: Bimba 120K reconstructed with distance = 0.25*average spacing Right: Bimba 120K reconstructed with distance = 0.15*average spacing

Right: reconstructed surface Example Poisson Left: 65K points sampled on a hand with no data at the wrist base (Kreon laser scanner) Right: reconstructed surface

Example Poisson

Example Poisson Left: 50K points sampled on Neptune trident Right: point set simplified to 1K then reconstructed

Example Poisson Left: points sampled on a sphere with flipped normals Right: reconstructed surface

Example Poisson Left: 4K points sampled on a mechanical piece with sharp edges Right: reconstructed surface

Poisson duration wrt #input points

Surface meshing duration and error wrt approximation distance

Memory wrt #input points

Reconstruction error wrt #input points

Reconstruction (2) Algebraic point set surfaces (APSS) [Guennebaud, SIGGRAPH 2007] Evaluates an implicit function on the fly (~signed distance function) Isosurface extracted by CGAL surface mesh generator

APSS Based on Moving Least Squares fitting on algebraic spheres

APSS Point projection

APSS

APSS can evaluate an implicit function at any point

APSS

Poisson vs APSS? INPUT = 275K points sampled on an elephant (minolta laser scanner)

Poisson APSS Approximation error in surface mesh generator: 0.004

Poisson APSS Approximation error in surface mesh generator: 0.002

Poisson APSS Approximation error in surface mesh generator: 0.001 always a bit smoother one extra component

Documentation Demo Current: 98 pages Reference manual available online User manual available online Demo Current: 3D point set demo (QT4 + QGLViewer)

The End