Reconstruction from Point Cloud (GATE-540)

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Reconstruction from Point Cloud(GATE-540)
Dr.Çagatay ÜNDEGER Instructor Middle East
Technical University, GameTechnologies General
Manager SimBT Inc. e-mail cagatay_at_undeger.com
Reference Hugues et al, Surface Reconstruction
from Unorganized Points
Game Technologies Program Middle East Technical
University Spring 2010
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Outline

• Reconstruction from point clouds

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Goals

• Develop 3D Analysis Algorithms
• Reconstruction
• Segmentation
• Feature Detection
• Labeling
• Matching
• Classification
• Retrielval
• Recognition
• Clustering

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Goal of Surface Reconstruction

• Have a set of unorganized points
• Reconstruct a surface model that best
  approximates the real surface

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Data Sources

• Surfaces from range data
• Surfaces from contours (slices of images)
• Interactive surface sketching

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Terminology

• A surface compact, orientable two dimentional
  monifold
• A simplicial surface A piecewise linear surface
  with triangular faces
• X x1, ..., xn sampled data points on or
  near an unknown surface M
• M y1, ..., yn real points on unknown
  surface M that maps X

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Terminology

• p-dense ??
• ei or d maximum error of data source
• xi yi ei
• Features of M that are small compared to d will
  not be recoverable.
• It is not possible to recover features of M in
  regions where insufficient sampling has occured.

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Problem Statement Algorithm

• Goal
• To determine a surface N that approximates an
  unknown surface M
• An algorithm proposed by Hugues et al, 1992.
• Consists of two stages
• 1) Estimate signed geometric distance to the
  unknown surface M
• 2) Estimate unknown surface M using a contouring
  algorithm

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Define a Signed Distance Function

• Associate an oriented plane (tangent plane) with
  each of the data points.
• Tangent plane is a local linear approximation to
  the surface.
• Used to define signed distance function to
  surface.

sampled point
N
signed distance
tangent plane
estimated surface point
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Tangent Plane

• Nbhd(xi) k-neighborhood of xi
• Tangent plane center of xi (Oi) centroid of
  Nbhd(xi)
• Tangent plane normal of xi (Ni) determined
  using principle component analysis of Nbhd(xi)

sampled point xi
k-neighborhood of xi
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Principle Component Analysis

• Involves a mathematical procedure that transforms
  a number of possibly correlated variables into a
  smaller number of uncorrelated variables called
  principal components.

Normal of tangent plane might be found in
opposite direction
?
?
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Consistent Tangent Plane Orientation

• If two neigbors are consistently oriented,
• Their tangent planes should be facing almost the
  same direction.
• Otherwise one of them should be flipped.

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Consistent Tangent Plane Orientation

• Model the problem as a graph optimization
• Each Oi will have a corresponding Vi in graph
• Connect Vi and Vj is Oi and Oj are sufficiently
  close.
• Cost on edges encodes the degree to which Ni and
  Nj are consistently oriented.
• Maximize the total cost on the graph.
• NP-hard
• Use an approximation algorithm.

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Euclidian Minimum Spanning Three (EMST)

• Surface is assumed to be a single connected
  component,
• The graph should be connected.
• A simple connected graph for a set of points that
  tends to connect neighbors is EMST.
• EMST over tangent planes is not sufficiently
  dense!
• Enrich it by adding an edge (i,j) if oi is in the
  k-neighborhood of oj.
• Result is called Reimannian Graph.

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Reimannian Graph

• A connected graph

EMST over tangent planes
Reimannian Graph
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Simple Algorithm

• Arbitrarily choose an orientation for some plane
• Propogate the orientation to neigbors in
  Reimannian Graph.
• Order of propagation is important!

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A Good Propagation Order

• Favor propagation from oi to oj if unoriented
  planes are nearly parallel.
• Assign cost as 1 NiNj
• A cost is small if parallel
• A fovorable propagation order
• Travers mimimum spanning tree (MST)

dot product
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Assigning Orientations

• Assign z orientation to point in graph that has
  largest z coordinate.
• Travers the tree in depth first order.

Oriented tangent planes
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Computing Distance Function

• Signed distance f(p)
• f(p) of a point p to unknown surface M
• distance between p and closest point z ? M
• multiplied by 1 depending on the side of the
  surface p lies in
• z is unknown, thus use closest oi

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Computing Distance Function

• z p ((p-oi)Ni)Ni
• If d(z,X) lt (pd) then // graph is p-dense
• f(p) (p-oi)Ni
• Else
• f(p) undefined
• Defined ones create a zero set (estimate for M)

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Contour Tracing

• Contour tracing is to extract iso-surface from a
  scalar function.
• A variation of matching cubes is used
• Cube sizes should be less than pd
• But larger increases the speed and reduces the
  number of triangle facets created

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Contour Tracing

• Visit the cubes only intersect the zero set.
• No intersection if the signed distance is
  unefined in any vertex within a cube.

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Collapse Edges

• Contain triangles with arbitrary poor ascpect
  ratio.
• Collapse edges in post processing

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Collapse Edges
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Sample Results