Reconstructing and Analyzing Surfaces in 3Space

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Reconstructing and Analyzing Surfaces in 3-Space

• Jian Sun
• Adviser Prof. Tamal K. Dey

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Surface

• Surface topological space where every point has
  a 2-disk neighborhood.
• Analyze surfaces
• directly calculate area
• indirectly via induced structures
• Reconstruct surfaces
• exactly medial axis transformation
• approximately point cloud

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Induced structures
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Smooth Surface Reconstruction

• Input Points sampled from a smooth surface,
  possibly contaminated with noise.
• Task Reconstruct a piece-wise linear
  approximation of the sampled surface with
  theoretical guarantees.
• Topologically the same homeomorphic
• Geometrically close small Hausdorff distance

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Related Work

• Algorithms reconstructing surfaces with
  theoretical guarantees
• Delaunay/Voronoi based
• noise free CRUST AB99, COCONE ACDL00
• noise ROBUST COCONE DG04
• Implicit function based
• noise free Natural Neighborhood BC00
• Noise MLS Kol05 DGS05
• Distance function based
• Flow complex GJ02 DGRS05
• Wrap Edel03

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Our Work

• Define an adaptive MLS (AMLS) surface from a set
  of sample points
• Approx. the sampled surface with Theoretical
  guarantees
• What is new?
• smooth, instead of bumpyDG04
• Adaptive, instead of uniform Kol05 DGS05
• Give an algorithm to project all the sample
  points onto the AMLS surface

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Definition of AMLS surface

• Choice of weighting function
• Bounded influence of distant samples

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Sampling Conditions

• Sampling Conditions

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Theoretical Results

• Define map ? I-1(0)Å0.1? ! ?
• Map ? is a homeomorphism
• Surjective
• I(x) lt 0 in blue and
• I(x) gt 0 in red
• Injective
• nzI gt 0 in 0.3??
• I-1(0)Å0.1? and
• ? are isotopic CC04

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Approximation Algorithm for AMLS surfaces

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Normal Estimation Algorithm

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Justification
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Feature Estimation Algorithm

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Justification

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Experimental Result

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Analyzing Surfaces

• Analyze surfaces and the objects bounded by them
  via the induced structures
• not too complicated
• not too simple
• computable
• Curve-skeletons
• Handle and tunnel loops

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Curve-skeletons

• Curve-skeleton 1D representation of 3D shapes
  and useful in many applications
• Geometric modeling, computer vision, data
  analysis, etc
• Reduce dimensionality
• Build simpler algorithms
• Desirable properties Cor05
• Thin 1D curve or one voxel thick
• Centered w.r.t. distance
• Preserving topology connected components and
  tunnels
• Stable against noise

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Related Work

• Algorithms to extract curve-skeletons
• Topological thinning based PK99 LB04
• Remove voxel from boundary while keeping the
  topology
• Identify endpoints to keep the geometry
• Distance function based ZT99 ST03
• Thinning explore the derivative of the distance
  function
• General-field function based Grig98Cor05
• potential field, electrostatic field, visible
  repulsive force field.
• Issues
• No formal definition enjoying most of the
  desirable properties
• Existing algorithms often application specific

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Our Work

• Give a mathematical definition of curve-skeletons
  based on the medial geodesic function
• Enjoy most of the desirable properties
• Give an approximation algorithm to extract such
  curve-skeletons
• Practically plausible

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Roadmap
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Medial axis

• Medial axis set of centers of maximal inscribed
  balls
• The stratified structure Giblin-Kimia04
  generically, the medial axis of a surface
  consists of five types of points based on the
  number of tangential contacts.
• M2 inscribed ball with two contacts, form sheets
• M3 inscribed ball with three contacts, form
  curves
• Others

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Medial geodesic function (MGF)
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Properties of MGF

• Property 1 (proved) f is continuous everywhere
  and smooth almost everywhere. The singularity of
  f has measure zero in M2.
• Property 2 (observed) There is no local minimum
  of f in M2.
• Property 3 (observed) At each singular point x
  of f there are more than one shortest geodesic
  paths between ax and bx.

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Defining curve-skeletons

• Sk2SkÅM2 set of singular points of MGF or
  points with negative divergence w.r.t. rf
• Sk3SkÅM3 extending the view of divergence
• A point of other three types is on the
  curve-skeleton if it is the limit point of Sk2
  Sk3
• SkCl(Sk2 Sk3)

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Approximating curve-skeletons

• MA approximation Dey-Zhao03 subset of Voronoi
  facets
• MGF approximation f(F) and ?(F)
• Marking E is marked if ?(F)²n lt ? for all
  incident Voronoi facets
• Erosion proceed in collapsing manner and guided
  by MGF

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Examples
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Effect of ?
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Properties of curve-skeletons

• Thin (Algorithmically?)
• Preserve topology (Def?)
• Centered
• Stable
• Junction detective

Prop1 set of singular points of MGF is of
measure zero in M2
Medial axis homotopy equivalent to the original
shape Curve-skeleton homotopy equivalent to the
medial axis
Medial axis is in the middle of a shape Prop3
more than one shortest geodesic paths between its
contact points
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Two additional attributes

• Geodesic size g(E) indicate the size of the
  shape locally.
• Eccentricity e(E)g(E) / c(E) indicate whether
  the shape is tubular or flat locally.

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Shape decomposition

• Based on eccentricity

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Shape decomposition

• Based on bifurcation

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Timing
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Collapse 3D shapes to curves?

• Not always doable.
• Not doable anyway knotted surfaces
• Not doable via collapsing medial axis
• a thickening of a house with two room.

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Graph Retractable Surfaces

• A surface ? is graph retractable if both MÅI and
  MÅO deformation retract to a graph, denoted I and
  O, respectively.
• What are these loops and how to compute them?

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What are these loops?

• Generators for H1(?)
• Moreover
• Red Loops (tunnel) generators for H1(I), but
  trivial in H1(O)
• Green Loops (handle) generators for H1(O), but
  trivial in H1(I)
• Moreover
• Minimally linked ?j12glk(L, Kj) 1

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Existence

• Theorem 1 There exist 2g loops, denoted
  Jjj12g, such that lk(Ki, Jj)?ij. Half of them
  linked with KiIs, denoted JjIj1g, are handle
  loops, and the other half linked with KiOs,
  denoted JjOj1g, are tunnel loops. JjI j1g
  form a basis for H1(I) and JjOj1g form a
  basis for H1(O). Hence Jjj12g form a basis
  for H1(?).

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Criterion

• Theorem 2 A loop ? on M is a handle one iff
  lk(?, KiI) ? 0 for at least 1ig and lk(?,
  KiO)0 for all 1ig. A loop ? on M is a tunnel
  one iff lk(?, KiO) ? 0 for at least 1ig and
  lk(?, KiI)0 for all 1ig.

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Criterion
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How to compute?

• Topological algorithm to compute Jjj12g
• Step1 Compute Kii12g using the spanning tree
  of I and O
• Step2 Compute a system of 2g loops on ?
  VY90DS95, denoted ?jj12g.
• Step3 Compute lk(Ki, ?j) for all i and j. Let A
  be the matrix lk(Ki, ?j).
• A is the transform matrix from basis Jjj12g
  (defined in Theorem 1) to basis ?j j12g.
• A-1 aji exists and has integer entries.
• Jj ?i12g aji?i.
• Step4 Obtain Jj by concatenating ?is according
  to the above expression.

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How to compute incorporate geometry

• Compute Kjj12g
• Compute the maximal spanning tree for I and O
  using geodesic sizes as weight.
• Add the remaining edges, Ejs, to form Kjs.
• Compute Jjj12g
• Basic idea compute Jjs at different location
  indicated by Ejs.
• Let e be the skeleton edge with the smallest
  geodesic size in Ej. Let p be one of the vertices
  of the dual Delaunay triangle of e.
• Compute an optimal system of loops EW06 and
  apply topological algorithm.
• In all our experiments, one of the loop in the
  system of loops itself satisfies the condition to
  be Jjs.

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Results
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Results
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Existence on general surfaces

• Existence
• Example a thickened trefoil

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Conclusion

• Summary
• Define an adaptive MLS surface and give an
  implementation to approximate it.
• Give a mathematical definition for
  curve-skeletons and an algorithm to extract them.
  
• Define handle and tunnel loops for graph
  retractable surfaces and give a topologically
  guaranteed algorithm and an implementation that
  incorporates the geometry.
• Future work
• Reconstruct the surfaces with sharp features
• Consider high dimensional cases.

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Thank you!