Triangulations for computer graphics applications

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Triangulations for computer graphics applications

• I. Kolingerová (kolinger_at_kiv.zcu.cz)

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Contents

• 1. Introduction
• 2. On triangulated models
• 3. On triangulations
• 4. The best known planar triangulations
• 5. Generalizations incoming/outgoing points,
• refinement smoothing, moving points,
  pseudotriangulations
• 6. 3D triangulations
• 7. Applications to CG
• 8. Conclusion

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1. Introduction

• Goal of this paper - a brief survey of
  triangulations used for computer graphics image
  processing
• and experience with them in applications
• Triangulation - a set of triangles or tetrahedra
• on the given set of points,
• with possibility to add user-defined edges,
  sometimes with possibility of refinement

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2. On triangulated models

• A triangulated model planar/ triangulated
  boundary/ tetrahedral

a set of 2D of 2.5D points
planar triangulation
a triangulated model
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On triangulated models

• or

a set of 3D points
surface reconstruction in 3D (often via
tetrahedronization) or points projection and
triangulation in a parametric space
a triangulated surface mesh
AVG 7/33
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On triangulated models

• or

a set of 3D points
tetrahedronization
a volume model consisting of tetrahedra (i.e.
also of triangles)
AVG 7/33
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3. On triangulations
Triangulation T(P) of a set of n points in
E2 - a set of edges such that - no two
edges intersect at a point not in P - the
edges divide CH(P) into triangles
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On triangulations

• Many triangulations of a given points set
• many criteria of quality
• global
• local

• angular
• edge lengths (weight)

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On triangulations

• Local criteria an initial triangulation, checks
  of neighbouring simplices on local optimality if
  not optimal, flip to the other configuration if
  exists

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On triangulations

• Flipping converges in E2 for any criteria, in E3
  may not converge but works for the criteria used
  in practice
• Complexity of the triangulation O(n log n) in
  E2, O(n2) in E3, both can be done in O(n)
  expected case
• Flipping complexity O(n2) in the worst case and
  O(n) in the expected case

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On triangulations

• Global criteria polynomial algorithms for
  nearly all criteria unknown, locally optimal
  triangulations not globally optimal
• An exception Delaunay triangulation - maximizes
  minimal angles locally globally (in E2),
  minimizes the max. containment sphere radius (in
  E2 and E3)... etc.
• Solution for global extremes suboptimal
  algorithms, heuristics, probabilistic methods

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On triangulations

• Other triangulated models not defined by
  local/global criteria surface reconstruction
• "reconstruct a triangulated surface model to be
  as close as possible to the original geometric
  object given a set of surface points"
• gt weak definition

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4. The best known planar triangulations

• 4.1 Triangulations optimizing angles - most
  important group
• 4.2 Triangulations optimizing edge lengths
• 4.3 Multi-criteria optimized triangulations
• 4.4 Data-dependent triangulations

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4.1 Triangulations optimizing angles

• Most important group
• Maxmin angle (Delaunay triangulation), minmax
  angle, minmin angle, max sum of angles, minmax
  eccentricity (min.distance between CC centre and
  triangle vertices)

DT
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Delaunay triangulation DT(P)

• The circumcircle of any triangle in DT(P) does
  not contain any other point of P
• Maximizes the minimum angles
• The most equiangular triangles
• O(n log n) - worst case optimal

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Algorithms for DT(P)

• Local improvement by edge flipping
• Divide conquer
• Incremental insertion
• Incremental construction
• Sweeping
• High-dimensional embedding
• DC and sweeping the fastest but complicated and
  not flexible enough
• Our recommendation incremental insertion

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A real data set (20 000 points, 40 000 triangles)
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4.2 Triangulations optimizing edge lengths

• Less used in practice higher complexity
• Minimum weight triangulation (MWT),
• greedy triangulation (GT), minmax edge
  length

DT
GT
MWT
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Greedy triangulation GT(P)

• Consists of the shortest possible compatible
  edges
• Optimal algorithm yet not known
• expected O(n) case up to O(n3) worst case

DT
GT
GT
DT
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4.3 Multi-criteria optimized triangulations

• Optimization of several combined criteria
• Non-deterministic computation using local edge
  swaps
• Slow

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4.4 Data-dependent triangulations (DDT)

• For data with a steep slope
• Takes into consideration also heights of points,
  angles between triangle normals

DDT
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DDT is suitable for Triangulation
without considering heights does not respect the
break
Considering heights
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5. Generalizations

• 5.1 Constrained edges
• 5.2 Incoming/outgoing points
• 5.3 Refinement smoothing
• 5.4 Moving points
• 5.5 Pseudotriangulations

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5.1 Constrained edges

• The edges prescribed to a triangulation

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5.2 Incoming/outgoing points
Incoming points easy namely for
incremental algorithms
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Incoming/outgoing points
Outgoing points a bit more difficult (but
still OK) data structures must be able
to handle it
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5.3 Refinement smoothing
Triangulation on the given points may be bad
(even Delaunay)
Improvement possible by adding points and
smoothing the mesh
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5.4 Moving points

• All points move at the same time
• Constant velocities, linear trajectories
• Parameters are known in advance gt
• mesh changes can be planned
• An event queue
• DT most important events - 4 co-circular
  points,
• points depend on t
• gt a determinant of 4x4 matrix with
  variable items
• gt 4th order polynomial equation to be
  solved

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Moving points
A small demo one moving point Still not too much
stable
(c) T.Vomácka,2007
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5.5 Pseudotriangulations
Triangles not exciting enough? Too regular?
Boring?
(c) J.Trcka, 2007
What about pseudotriangulations?
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6. 3D triangulations

• Problems against 2D number of tetrahedra may
  differ for one set of points , maximally O(n2)
  tetrahedra, low readability
• In CG and image processing DT most often

n35
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Delaunay triangulation in 3D

• Min max radii of enclosing spheres nothing more
• Algorithms not all approaches converge
• Most often incremental insertion

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7. Applications to CG
7.1 Real time terrain modifications 7.2 Contour
lines computation and improvement 7.3
Reconstruction of planar domain boundaries 7.4
Image and video representation 7.5 Surface
reconstruction 7.6 Outliers removal 7.7 Tunnels
in protein molecules
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7.1 Real time terrain modifications
Circle tool demos
Drag demo
(c) J.Kadlec V. Purchart, 2007
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7.2 Contour lines computation and improvement
Contour lines computed automatically may have
problems
(c) I.Kolingerová, M.Dolák, V.Strych, 2003-7
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Contour lines computation and improvement

• Solution
• manual
• automatic
• Manual imposing constrained edges

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Contour lines computation and improvement

• Automatic to find artifacts automatically
• and use constrained edges
• ??
• At least flat areas

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Contour lines computation and improvement
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7.3 Reconstruction of planar domain boundaries
Remove triangles with at least one edge
longer than some limit
(c) I.Kolingerová, B.Žalik, 2006
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Reconstruction of planar domain boundaries
Sometimes difficult to decide OK for practical use
It can be used also for holes
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7.4 Image and video representation
A height array, volume data x an adaptive
mesh, tetrahedra
JPEG contra constrained-DT based coding
(c) J.Polec, M.Partyk, I.Kolingerová, 2003-5
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Image and video representation
Video as volume data
Original and tetrahedra-compressed images from
a video
(c) M.Varga, 2007
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7.5 Surface reconstruction
(c) M.Varnuška, 2005
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Surface reconstruction

• Modification of CRUST by Nina Amenta
• - DT 3D - a superset of surface triangles
• The surface approximation based on the shape of
  the dual Voronoi cells shape and on the surface
  normals estimates

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Surface reconstruction

• Post-processing to extract a manifold and to
  correct boundaries

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Surface reconstruction
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Surface reconstruction

• Noisy data - still a problem, post-processing
  necessary

Data VRVis TU Graz
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7.6 Outliers removal
Triangulation - a planar graph Delaunay
triangulation - a hypergraph of the Minimum
Spanning Tree
(c) I.Kolingerová, 2004
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Outliers removal
It can be used to find clusters and remove
outliers
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Outliers removal
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7.7 Tunnels in protein molecules
A protein
A protein a tunnel
(c) Implementation M. Zemek, 2007, Original
solution P.Medek,
P.Beneš, J.Sochor
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Tunnels in protein molecules
A regular triangulation, a power diagram and a
tunnel in protein molecules (2D version)
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Tunnels in protein molecules
A protein 3D RT
3D regular triangulation
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Tunnels in protein molecules
A protein a tunnel
3D regular triangulation a tunnel
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8. Conclusion

• Triangulations rich and interesting topic, very
  useful also in computer graphics and image
  processing

Much work done, still many challenges, namely in
3D An interesting and grateful topic for research
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Thanks for your attention