Point Cloud Segmentation for 3D Reconstruction

1Challenge the future
Point Cloud Segmentation &
Surface Reconstruction
An overview of methods
Ir. Pirouz Nourian
PhD candidate & Instructor, chair of Design Informatics, since 2010
MSc in Architecture 2009
BSc in Control Engineering 2005
MSc Geomatics, GEO1004, Directed by Dr. Sisi Zlatanova

Topics
A very short introduction to segmentation and surface reconstruction
• Big Picture, Problem Statement
• Goal is to make a [simple] Brep, with few faces
• General Approaches to this problem (surface
reconstruction):
• Volumetric, Implicit, using voxels and iso-surfaces
• Alpha Shapes (Ball-Pivoting)
• Delaunay or Voronoi Based?
• Poisson?
• General Approaches to this problem (surface
reconstruction):

Topics
A very short introduction to segmentation and surface reconstruction
• Big Picture, Problem Statement
• Goal is to make a [simple] Brep, with few faces
Images courtesy of Ajith Sampath
?

A few approaches to surface reconstruction
• Voxels  Field of Signed-Distance Iso-Surface
• Ball-Pivoting (alpha shapes) triangulation
• Planar Segments Neighbourhood Matrix Edge List Simple Mesh
• Poisson Surface Reconstruction
• Implicit Function
• [ Hoppe et al. 92 ]
• Volumetric Reconstruction
• [ Curless and Levoy 96 ]
• Alpha Shapes
• [ Edelsbrunner and Mucke 94 ]
• 3D Voronoi-Based Reconstruction
• [ Amenta , Bern & Kamvysselis 98 ]

Segmentation for Detecting Features
Points belonging to a set of surfaces considered as Brep faces
Images courtesy of Dr. Shi Pu, Faculty of Feo Information Science and Earth Observations, University of Twente (ITC)

3D reconstruction from AHN pointcloud: Carl Chen, Rusne Sileryte, Kaixuan Zhou; Ed. Pirouz Nourian, Sisi Zlatanova

Building Reconstruction
 The distance between two planar segments (P & Q) in a roof is defined as:
 A neighborhood Matrix is then generated. The matrix shows all mutuallyintersecting
planes.
 Any two intersecting planes are selected,and all planes thatintersectboth of them are
enumerated,and solved.
 For instance Planes {1,4,10} and {1,4,13} are solved to get the breakline (A-B)as shown.

Problem
• Estimated Normal Vectors For Each Point
• Estimated Curvature For Each Point
Preliminaries…
• Estimate Normal Vectors For Each Point Using the Covariance Matrix
• (Fit a Plane to a Bunch of Points)
• Cross Product of the Two Eigen Vectors Corresponding to the Two Largest Eigen Values

Covariance Matrix Computation
Check the attached code for the latest version!
Dim Neighbors = Pts.FindAll(Function(V) V.distanceto(point) < D)
Dim Centroid As New Point3d
For Each neighbor As point3d In Neighbors
Centroid = Centroid + neighbor
Next
For k As Integer=0 To Neighbors.count - 1
Dim CiCBar As New Matrix(3, Neighbors.count)
Dim Diff As point3d = Neighbors(k) - Centroid
CiCBar(0, k) = Diff.X
CiCBar(1, k) = Diff.y
CiCBar(2, k) = Diff.z
Dim CiCBarOld As New Matrix(3, Neighbors.count)
CiCBarOld = CiCBar.Duplicate()
CiCBar.Transpose()
CovM = CiCBarOld * CiCBar ‘Why is this a matrix of correct size?
Next
CovM.Scale(1 / Neighbors.count)
CovMatrices.Add(CovM)

Efficient Eigen Value Computation
• Find the roots of this characteristic function (why? & how?)
• Using a Trigonometric Method
• (The following code is my version of it, test it first on a cubic function)
A special case in which a 3x3 Matrix is dealt with
Function CubicRoots(a, b, c, d)
Dim t As Double
Dim p,q As Double
p = (3 * a * c - b ^ 2) / (3 * a ^ 2)
If p < 0 Then
q = (2 * b ^ 3 - 9 * a * b * c + 27 * a ^ 2 * d) / (27 * a ^ 3)
Dim roots As New list(Of Double)
For k As Integer = 0 To 2
t = 2 * Math.Sqrt(-p / 3) * Math.Cos((1 / 3) * Math.Acos(((3 * q) / (2 * p)) * Math.Sqrt(-3 / p)) - k * ((2 * Math.PI) / 3))
Dim x As Double = t - b / (3 * a)
roots.add(x)
Next
Return roots
Else
Return Nothing
End If
End Function

Ready for Segmentation!
We do an iterative segmentation called Region Growing
Mesh Segmentation in Action
low angle threshold
high angle threshold

Mesh Segmentation
• Connected Faces
• Start Making Regions out of Neighbours of Similar Aspects
(normal vectors/orientations)
• Recursively Grow Regions with an Angle Tolerance (how?)
How to detect faces of a Brep given a Mesh?
Define Faces=M.Faces
Define Regions As New list(Of List(Of Integer))
For i in range M.Faces.Count
If clusters.count = 0 Or There is no region containing(i) Then
Define region As New list(Of Integer)
region.Append(i)
For j in range adjacentfaces(of i)
If isSimilar(M.FaceNormals(i), M.FaceNormals(j), t) Then region.Append(j)
End For
region = RecursivelyGrowRegion(M, region, angle threshold)
Regions.Append(region)
End If
End For

Point Cloud Segmentation
Region Growing Segmentation Pseudocode

A few approaches to surface reconstruction
• 3d Hough Transform [Vosselman et al.]
• Implicit Function [ Hoppe et al. 92 ]
• Alpha Shapes [ Edelsbrunner and Mucke 94 ]
• Many more:
Berger, Matthew, et al. "State of the art in surface reconstruction from point clouds." EUROGRAPHICS star reports. Vol. 1. No. 1. 2014.
(e.g. Ball-Pivoting algorithm) The space generated by point pairs that can
be touched by an empty disc of radius alpha.
Using Voxels  Field of Signed-Distance Iso-Surface

2D Hough Transform

2d Hough
Transform
b = -ax + y

2d Hough
Transform
R=x cos θ+ y sin θ

3d Hough transform
c = -ax -by + z
R = x cos θ cos Φ + y sin θ cos Φ+ z sin Φ

20
Restricted
3d Hough Transform
Plane should contain
(0,0,0) → z = ax + by
b = -ax/y + z/y (y != 0)

24
Hough
Filtering

25
Final
result

Questions?
Thank you!
p.nourian@tudelft.nl

Point Cloud Segmentation for 3D Reconstruction

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