My presentation of the article "Provably Good Sampling and Meshing of Surfaces" of J.-D. Boissonnat and S. Oudot. All Rights for text are Reserved by authors of this paper.
Date of presentation: May 2012
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2. Provably Good Sampling and Meshing of Surfaces
Not
a
smooth
surface
Jean-Daniel Boissonnat,
Steve Oudot
3. Provably Good Sampling and Meshing of Surfaces
Smooth
surface
Jean-Daniel Boissonnat,
Steve Oudot
4. Provably Good Sampling and Meshing of
Surfaces
Well
distributed
sample
points
Smooth
surface
Jean-Daniel Boissonnat,
Steve Oudot
5. Provably Good Sampling and
Meshing
of
Surfaces
Well
distributed
sample
points
Good
triangula9on
Smooth
surface
Jean-Daniel Boissonnat,
Steve Oudot
6. Provably Good Sampling and
Meshing
of
Surfaces
Well
distributed
sample
points
Good
triangula9on:
25
All
angles
are
greater
than
25
degrees
Smooth
surface
Jean-Daniel Boissonnat,
Steve Oudot
7. Provably Good Sampling and
Meshing
of
Surfaces
Well
distributed
sample
points
Good
triangula9on:
25
All
angles
are
greater
than
25
degrees
All
triangles
are
equilateral
Smooth
surface
Jean-Daniel Boissonnat,
Steve Oudot
8. Provably
Good
Sampling and
Meshing
of
Surfaces
Well
distributed
sample
points
Good
triangula9on:
25
All
angles
are
greater
than
25
degrees
All
triangles
are
equilateral
The
best
approximates
Smooth
surface
Jean-Daniel Boissonnat,
Steve Oudot
9. Provably
Good
Sampling and
Meshing
of
Surfaces
Well
distributed
sample
points
Good
triangula9on:
25
All
angles
are
greater
than
25
degrees
All
triangles
are
equilateral
The
best
approximates
Smooth
surface
Jean-‐Daniel
Boissonnat,
Steve
Oudot
10. Presented
by
Anisimov
Dmitry
1.
Take
a
smooth
surface
Compact,
orientable,
at
least
C2
–
con9nuous
closed
surface.
Not
completely
suitable
Completely
suitable
11. Presented
by
Anisimov
Dmitry
2.
Sample
this
surface
• Medial
axis
M
of
the
surface
S
2d
medial
axis
3d
medial
axis
12. Presented
by
Anisimov
Dmitry
2.
Sample
this
surface
• Distance
to
the
medial
axis
M
that
is
dM
dM
dM
2d
medial
axis
3d
medial
axis
13. Presented
by
Anisimov
Dmitry
2.
Sample
this
surface
• Minimum
distance
to
the
medial
axis
M
that
is
dM
inf
inf
dM = inf {dM (x), x ∈ S}
14. Presented
by
Anisimov
Dmitry
2.
Sample
this
surface
• Some
user-‐defined
func9on
σ : S → R
Ø Posi9ve
that
is
σ
> 0
Ø 1-‐Lipschitz
that
is
σ (x) − σ (y) ≤ x − y
15. Presented
by
Anisimov
Dmitry
2.
Sample
this
surface
• Ball
of
center
c
and
radius
r
that
is
B(c, r)
B
B(c, r)
16. Presented
by
Anisimov
Dmitry
2.
Sample
this
surface
• Ball
of
center
c
and
radius
r
that
is
B(c, r)
B
B(c, r)
17. Presented
by
Anisimov
Dmitry
2.
Sample
this
surface
• Construc9on
of
the
ini9al
point
sample
E
Pick
up
at
least
one
point
x
on
each
connected
component
of
S
and
insert
it
in
!
E
18. Presented
by
Anisimov
Dmitry
2.
Sample
this
surface
• Construc9on
of
the
ini9al
point
sample
E
Consider
a
ball
Bx
centered
at
x
of
radius
less
"1 1 %
min # dist(x, E ! {x}), dM (x), σ (x)&
$6 6 '
19. Presented
by
Anisimov
Dmitry
2.
Sample
this
surface
• Construc9on
of
the
ini9al
point
sample
E
Repeatedly
shoot
rays
inside
Bx
and
pick
up
three
points
(ux, vx, wx) of
S Bx
Insert
(ux, vx, wx) in
E
20. Presented
by
Anisimov
Dmitry
2.
Sample
this
surface
• Construc9on
of
the
ini9al
point
sample
E
Connec9ng
these
points
we
get
a
persistent facet
All
persistent facets are
Delaunay
facets
restricted
to
S
21. Presented
by
Anisimov
Dmitry
2.
Sample
this
surface
• Construc9on
of
the
ini9al
point
sample
E
All persistent facets remain
restricted
Delaunay
facets
throughout
the
course
of
algorithm
22. Presented
by
Anisimov
Dmitry
2.
Sample
this
surface
• Construc9on
of
the
ini9al
point
sample
E
All persistent facets remain
restricted
Delaunay
facets
throughout
the
course
of
algorithm
23. Presented
by
Anisimov
Dmitry
2.
Sample
this
surface
• Construc9on
of
the
ini9al
point
sample
E
All persistent facets remain
restricted
Delaunay
facets
throughout
the
course
of
algorithm
24. Presented
by
Anisimov
Dmitry
2.
Sample
this
surface
• Construc9on
of
the
ini9al
point
sample
E
All persistent facets remain
restricted
Delaunay
facets
throughout
the
course
of
algorithm
25. Presented
by
Anisimov
Dmitry
2.
Sample
this
surface
• Construc9on
of
the
ini9al
point
sample
E
All persistent facets remain
restricted
Delaunay
facets
throughout
the
course
of
algorithm
26. Presented
by
Anisimov
Dmitry
2.
Sample
this
surface
• Construc9on
of
the
ini9al
point
sample
E
All persistent facets remain
restricted
Delaunay
facets
throughout
the
course
of
algorithm
27. Presented
by
Anisimov
Dmitry
3.
Triangulate
this
surface
• Compute
the
3-‐dimensional
Delaunay
triangula9on
of
E
Del(E)
28. Presented
by
Anisimov
Dmitry
3.
Triangulate
this
surface
• Compute
the
set
of
all
edges
of
the
Voronoi
diagram
of
E
V(E)
29. Presented
by
Anisimov
Dmitry
3.
Triangulate
this
surface
• Compute
Delaunay
triangula9on
of
E
restricted
to
S
DelS(E)
30. Presented
by
Anisimov
Dmitry
3.
Triangulate
this
surface
• Compute
Delaunay
triangula9on
of
E
restricted
to
S
Not
constrained
DelS(E)
31. Presented
by
Anisimov
Dmitry
3.
Triangulate
this
surface
• Surface
Delaunay
ball
BD
of
restricted
Delaunay
facet
f
DelS(E)
32. Presented
by
Anisimov
Dmitry
3.
Triangulate
this
surface
• Surface
Delaunay
ball
BD
of
restricted
Delaunay
facet
f
Any
ball
centered
at
some
point
of
S f *
where
f* is
Voronoi
edge
dual
to
f
DelS(E)
33. Presented
by
Anisimov
Dmitry
3.
Triangulate
this
surface
• Bad
surface
Delaunay
ball
BD which
is
stored
in
L
It
is
ball
B(c, r) such
that
r > σ(c)
c
r
DelS(E)
34. Presented
by
Anisimov
Dmitry
3.
Triangulate
this
surface
• Surface
Delaunay
patch
The
intersec9on
of
a
surface
Delaunay
ball
with
S
DelS(E)
36. Presented
by
Anisimov
Dmitry
3.
Triangulate
this
surface
• E
is
a
loose
ε-‐sample
of
S
if:
dM
c
1.
∀c ∈ S V (E), E B(c, ε d M (c)) ≠ ∅
2.
DelS(E) has
ver9ces
on
all
the
connected
components
of
S
37. Presented
by
Anisimov
Dmitry
3.
Triangulate
this
surface
• DelS(E) has
ver9ces
on
all
the
connected
components
of
S
V(E)
38. Presented
by
Anisimov
Dmitry
3.
Triangulate
this
surface
• Algorithm
While
L
is
not
empty
• Take
an
element
B(c,r) from
L
• Insert
c
into
E
and
update
Del(E)
• Update
DelS(E) by
tes9ng
all
the
Voronoi
edges
that
have
changed
or
appeared:
Ø Delete
from
DelS(E) the
Delaunay
facets
whose
dual
Voronoi
edges
no
longer
intersect S
Ø Add
to
DelS(E) the
new
Delaunay
facets
whose
dual
Voronoi
edges
intersect
S
• Update
L
by
Ø Dele9ng
all
the
elements
of
L
which
are
no
longer
bad
surface
Delaunay
balls
Ø Adding
all
the
new
surface
Delaunay
balls
that
are
bad
39. Presented
by
Anisimov
Dmitry
3.
Triangulate
this
surface
• Termina9on
and
output
of
the
Algorithm
Ø The
Algorithm
terminates
40. Presented
by
Anisimov
Dmitry
3.
Triangulate
this
surface
• Termina9on
and
output
of
the
Algorithm
Ø The
Algorithm
outputs
E
and
DelS(E)
E
is
a
loose
ε-‐sample
of
S
DelS(E)
is
homeomorphic
to
the
input
surface
S
and
approximates
it
in
terms
of
its
Hausdorff
distance,
normals,
curvature,
and
area.
42. Presented
by
Anisimov
Dmitry
Magic
Epsilon
• To
find
ε
you
must
solve
this
simple
inequality:
2ε ε π
+ arcsin ≥
1− 8ε 1− ε 4
• Or
just
take
this:
ε
=
0.091
46. Presented
by
Anisimov
Dmitry
References
J.-‐D.
Boissonnat
and
S.
Oudot.
“Provably
Good
Sampling
and
Meshing
of
Surfaces.”
Graphical
Models
67
(2005),
405-‐51.
47. Presented
by
Anisimov
Dmitry
References
M.
Botsch,
L.
Kobbelt,
M.
Pauly,
P.
Alliez,
and
B.
Levy.
“Polygon
Mesh
Processing.”
Chapter
6,
Sec9on
6.5.1
(2010),
92-‐96.