Avoiding Slivers in Practice and with a Guarantee

1
Avoiding Slivers in Practice and with a
Guarantee

• Daniel Dumitriu, Stefan Funke, Martin Kutz,
  Nikola Milosavljevic
• Presented by Kuiyu Li for 888
• April 5, 2007

2
Overview

• A new method for reconstructing a 2-manifold from
  a point sample in R3
• Throws away geometry info and operates
  combinatorially on a graph
• Conservative in creating adjacencies between
  samples in the vicinity of slivers

3
References (3 papers)

• To understand the paper, we need at least
• 1 Smooth-surface reconstruction in near-linear
  time, SODA 2002
• 2 Network Sketching or How Much Geometry
  Hides in Connectivity? Part II, SODA 2007
• 3 Surface reconstruction by voronoi filtering
  (CRUST), SOCG 1998

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Background and Motivation

• CRUST is a early method for reconstructing smooth
  closed surface T based on a point sample V
• It defines local feature size
• V should bee-sample of T
• The reconstructed surface is the set of delaunay
  triangles that are dual to the Voronoi edges in
  the Voronoi diagram of V that are intersected by
  the surface T

3 Surface reconstruction by voronoi filtering
(CRUST), SOCG 1998
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Background and Motivation

• Slivers
• Def 4 almost cocircular points that are nearby
  on the surface and have an empty diametral ball
• Because of this, its impossible to decide the
  correct reconstruction without knowing T
• Local decisions might disagree
• For each sample p, we need to know which of the
  candidate triangles to keep for the final
  reconstruction
• Sliver induce a Voronoi vertex for the involved
  sample points

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Background and Motivation

• One possibility is to decide adjacencies
  conservatively by only creating adjacencies that
  are sure
• But its unclear how much connectivity is lost,
  i.e.
• the resulting graph is connected at all ?
• how big potential holes/faces are ?

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Contribution of the Paper

• Show that it is actually possible to make local
  decisions
• Guarantee that the resulting graph exhibits
  topological equivalence to the original surface

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Contribution of the Paper

• A Graph-based algorithm
• Operate combinatorially on a graph
• Created adjacencies are conservative
• 2 samples are connected if there is a safe
  sliver-free region around them
• It also guarantees constant-size faces in the
  reconstruction
• No manifold extraction step

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Conservative Adjacencies in R2

• Input a uniform e-sampling V
• Create a locally neighborhood graph G(V,E)
• G(V,E) is also a a-quasi-unit-disk-graph
• Call S ? V a tight k-subsample of V, if

2 Network Sketching or How Much Geometry
Hides in Connectivity? Part II, SODA 2007
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GVD and CDM

• Determining adjacencies among nodes in S based on
  a Graph Voronoi Diagram (GVD)
• GVD is created on the locally neighborhood graph
  G(V,E)
• GVD partitions the plane into simply connected
  disjoint regions, with each of them containing
  one node of S
• The constructed graph G(S,E'), called
  Combinatorial Delaunay Map, denoted as CDM(S), of
  S is the dual graph of GVD

2 Network Sketching or How Much Geometry
Hides in Connectivity? Part II, SODA 2007
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How to determine adjacencies ?

• How to determine adjacencies in of S ?
• labeling of G(V,E), given S ? V
• Determining Adjacencies of nodes of S to get
  CDM(S)

2 Network Sketching or How Much Geometry
Hides in Connectivity? Part II, SODA 2007
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Theorem 2.1
2 Network Sketching or How Much Geometry
Hides in Connectivity? Part II, SODA 2007
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CDM(S) is dense !(ß-Skeleton of S ? CDM(S)
ß-Skeleton is dense)

• Consider theß-Skeleton of S
• Def ß-skeleton of S has an edge between p q ?
  S iff any ball of radius ßpq/2 touching p and
  q is empty of other points
• Whenß1, its Gabriel graph
• ß-Skeleton is dense
• ß-skeleton of S is a s-power spanner
• s-power distance ds(p,q) between p q ? S is
  determined by a sequence of points (power minimal
  / efficient path)
  such that
  is minimal

2 Network Sketching or How Much Geometry
Hides in Connectivity? Part II, SODA 2007
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CDM(S) is dense !(ß-Skeleton of S ? CDM(S)
ß-Skeleton is dense)

• Lemma 2.3
• For a good choice of s and ß, all power efficient
  paths use only edges of the ß-skeleton and that
  the ß-skeleton-graph is connected
• Finally, to show thatß-Skeleton is dense, only
  need to show that the induced faces of ß-skeleton
  are of constant size !

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CDM(S) is dense !(ß-Skeleton of S ? CDM(S)
ß-Skeleton is dense)

• At this point, we have shown that the ß-skeleton
  of S induces a connected planar graph which has
  constant-size faces
• For a suitable choice of k,eandß
• ß-skeleton ? CDM(S)
• Corollary 2.1

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How about in R3 ? a 3D Algorithm

• Input
• e-sample V of a manifold T
• Output
• reconstruction of V for T as a subcomplex of the
  Delaunay tetrahedralization of V

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3D Algorithm Steps

• (1) Determine a Lipschitz function f(v) for
  every v ? V, which lower-bounds elfs(v)
• (2) Construct a local neighborhood graph G(V) by
  creating an edge from every point v to all other
  points v'with vv' 6f(v)
• (3) Compute a tight k-subsample S of V
• (4) Create certified adjacencies among samples
  of S using the Graph Voronoi Didgram of G(V)
• (5) Use geometric positions of the points in S
  to identify faces (which are of constant size) of
  the graph induced by certified adjacencies
• (6) Triangulate all non-triangular faces
• (7) re-insert points in V - S for final
  reconstruction

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Smooth-surface reconstruction in near-linear
time, SODA 2002

• Lipschitz function
• Foragt0, fS-gtR has the Lipschitz property if for
  any x, y?S, f(x) f(y)axy
• Locally uniforme-sampling
• A sample P?S is a locally uniforme-sampling, if
  there is a control functionfS-gtR and constant
  0lta,ß,dlt1, such that

1 Smooth-surface reconstruction in near-linear
time, SODA 2002
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Smooth-surface reconstruction in near-linear
time, SODA 2002

• Input e-sampling P of S
• Output triangulationSof P to approximate S
• Algorithm steps
• (1) Estimation of decimation radius for each pt
  of P
• (2) Do the decimation to get a locally uniform
  sub-sampling Q?P
• (3) Use Crust to construct surfaceS from Q
• (4) Re-insert samples P-Q" for final
  reconstruction
• Time complexity
• Worst time O(nlgn)

1 Smooth-surface reconstruction in near-linear
time, SODA 2002
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Core Correctness Proof

• local neighborhood graph corresponds locally to
  aa-quasi-unit-disk graph for a set of points in
  the plane
• Corollary 2.1 also apply in here
• The certified adjacencies locally form a planar
  graph
• The faces of this graph have constant size
• The constructed graph on S is a mesh that is
  locally planar and covers the whole 2-manifold T

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Core Correctness Proof

• Theorem 2.2

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Implementation

• The paper doesnt implement step (6) and (7)
• It assumes that the input points V is a locally
  uniform sampling
• so it doesnt compute Lipschitz function f(v)
• Local neighborhood graph is constructed by
  connecting a sample to its 15 nearest neighbors
• A Simplified determination of adjacencies of
  CDM(S)
• two samples s1 s2 ? S are adjacent in CDM(S) if
  the number of edges linking a sample v1 (? s1s
  graph Voronoi cell) to a sample v2 (? s2s graph
  Voronoi cell) is more than 7

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Results

• Point cloud Graph Voronoi Diagram

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Results

• CDM(S) Identifies faces

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Results

• Time spend in different stages
• (step 2) Ngb local neighborhood graph
• (step 3) MIS get tight k-subsample S from V
• (step 4) Vor Graph Voronoi Diagram (GVD)
• (step 4) Del Combinatorial Delaunay Map (CDM)
• (step 5) Cye identify faces

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Results

• Parameter choice
• According to the theory, k must be chosen very
  large to guarantee sufficient density of CDM(S)

k2
k5
k14
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Results
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Results
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Thanks