Point Cloud Segmentation for 3D Reconstruction

1 Challenge 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 mutually intersecting

planes. Any two intersecting planes are selected, and all planes that intersect both 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


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Restricted 3d Hough Transform

Plane should contain (0,0,0) → z = ax + by

b = -ax/y + z/y (y != 0)


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Hough Filtering


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Final result


Questions?

Thank you!

[email protected]

mailto:[email protected]

Point Cloud Segmentation for 3D Reconstruction

Engineering

Transcript of Point Cloud Segmentation for 3D Reconstruction