Jean-Laurent Mallet. Geomodeling
Подождите немного. Документ загружается.
298
CHAPTER
6.
PIECE
WISE
LINEAR TRIANGULATED
SURFACES
Figure
6.25
Notion
of
isoparametric
constraint:
the
bold
curve
defines
an
isoparametric
constraint.
Preliminary
remarks
Let
p
fc
be a
point located
on an
isoparametric curve
J
I
(UQ)
of a
surface
S
and
let
t
l
be the
unit vector tangent
to
J
t
(u
l
Q
}
at
point
p
fe
.
By
definition
of
the
notion
of
gradient (see equation
(5.23)),
if s
represents
the
algebraic
curvilinear
abscissa
on
J
I
(UQ)
oriented
in the
direction
of
t\
then according
to the
definition (6.83)
This
clearly shows
that
gradw*
is
orthogonal
to
^(UQ):
Consequently,
as
suggested
in figure
(6.25)
and
according
to
equation
(6.69),
the
isometric constraints (6.80)
are
equivalent
to
Moreover,
if
N/,
is the
unit normal vector
to S at
point
p
fc
then, according
to
(5.11),
and
(6.69),
the
tangent
t*
to
J
I
(UQ)
at
point
p
fc
should
be
oriented
in
such
a way
that
Notion
of
isoparametric
constraints
As
shown
in figure
(6.25),
let
{p
l5
...,
p^,
p
fc+1
,...}
be an
ordered list
of
points
corresponding
to the
sampling
of a
curve
J~
l
drawn
on 5, and let
N&
be the
unit normal vector
to S at
point
p
fc
.
Let us
assume
that
the
points
{p^}
are
numbered
in
such
a way
that
This
ordering
is
compatible with condition (6.86) above, which suggests
the
possibility
of
making
one of the
isoparametric curves
of the
function
u
l
(x),
6.5.
GLOBAL PARAMETERIZATION
Figur e
6.26
Global
parameterization
of
a
triangulated
surface
with
isoparamet-
ric
DSI
constraints.
Isoparametric
constraints
allow
isoparametric
curves
to be fitted
to
geological
features,
such
as, for
example,
the
boundary
of
certain
fault
traces.
If
the
external
boundary
is
rectangular
in
shape,
then
isoparametric
constraints
can
also
be
used
to fit
this
boundary
to
isoparametric
curves.
(Data
courtesy
Elf)
for
example,
J
I
(UQ),
identical
to
J
l
for
some (unknown) value
UQ
of the
parameter
u
l
.
For
this purpose, according
to
equations
(6.84),
and
(6.85)-3,
we can say
that,
on the
curve
J7"
1
,
the
function
u
l
(x)
is
constant
and the
gradient
of
u
2
(x)
is
equal
to 1. In
other words,
for any
pair
of
points
(p
fc
,p
fc+1
),
this suggests
introducing
the
following
constraints
called
"^-isoparametric
constraints:"
The
"w
2
-isoparametric
constraints"
can be
denned
in a
similar
way and it is
straightforward
to
implement
all of
them
as
"fuzzy
Control-Delta" constraints,
already presented
on
page 256.
Comment
The
beauty
of the
notion
of the
^-isoparametric
constraint
is
that,
for
spec-
ifying
the
shape
of an
isoparametric curve
J
I
(UQ),
the
value
of
UQ
does
not
have
to be
known. This means
that
such
a
curve
can be
freely
specified
merely
by
picking
an
ordered list
of
points located
on
that
curve interactively
on S.
A
test
example
Figure
(6.26)
shows
a
global parameterization
of a
geological horizon
cut by
a
complex fault network.
The
isometric, Constant-Gradient,
and
C°
and
C
l
pseudo-continuity
DSI
constraints were used together with isoparametric
constraints
to
align some isoparametric curves with
the
boundary
of
certain
faults.
299
300
CHAPTER
6.
PIECEWISE LINEAR TRIANGULATED SURFACES
This
figure
also illustrates
the
importance
of
using higher certainty factors
for
the
isoparametric constraints
than
for
other
DSI
constraints
to
ensure
a
correct
fit for
isoparametric
curves.
Generalization
to 3D
The
parameterization
of
triangulated
surfaces
can be
extended
in a
straight-
forward
way to the
parameterization
of a 3D
region
71
modeled
as a
tetrahe-
dralized
volume
or a
regular 3-grid.
In
such
a
case,
the 3D
region
is
assumed
to be
represented
by a
discrete model
Ai
3
(O,
AT,
u,Cu)
where
the
vectorial
function
u has
three components representing
the
parameterization
of the
studied region:
For
any i
e
{1,2,3},
the
notion
of an
isoparametric surface
^(UQ)
is
defined
as
follows
in a
similar
way to the
notion
of
isoparametric curve:
Moreover,
it is
also possible
to
define
the
notion
of an
isoparametric constraint
to
specify
that
w
z
(x)
should have
one of its
isoparametric
surfaces,
say,
J
I
(UQ),
be
identical
to a
given surface
J
l
for
some unknown value
UQ
of the
ith
component
u
l
of the
parameter
u.
In
practice,
the
surface
J"
1
is
defined
by a
given (unordered)
set of
sampling
points
and the
associated isoparametric constraint
is
defined
as
follows
or,
equivalently:
These
linear equations
can
easily
be
transformed into
a set of K DSI
con-
straints
specifying
that
an
unknown isoparametric surface
J^
I
(UQ]
should
be
close
to the
given
surface
J"
1
for an
unknown value
UQ
of the
parameter
u
1
.
As
in the 2D
example represented
in figure
(6.26),
such
an
isoparametric
constraint allows
us to
align
the
isoparametric surfaces with horizons
and
faults
to
obtain
a 3D
parameterization compatible with
the
structure
of the
subsurface.
If
need
be,
these isoparametric surfaces
can in
turn
be
sampled
regularly
and
identified with
the
isoparametric surfaces
of a
regular curvilinear
3-grid.
6.5.5
Uncoupling properties from
the
geometry
Let
T(S)
be a
triangulated
surface holding
a
vectorial
p-dimensional
property
(p.
The
geometry
x and the
property
</?
of
T(S]
are
assumed
to be
modeled
6.5.
GLOBAL PARAMETERIZATION
Figur e
6.27 Uncoupling properties from
the
geometry:
a
property
y>
is
modeled
at the
nodes
of a fine
grid covering
the
parametric domain
D of the
triangulated
surface
T(«S).
Texture mapping
of
this grid
on
T(<5)
allows high-frequency variations
of
(p
to be
represented independently
of the
size
of the
triangles
of
T(«S).
Moreover,
the
continuity
of
(p
across faults
(if
any)
is
restored.
by
their values
{x(a)}
and
{(f>(o)}
sampled
at the
nodes
of
T(S}.
It can be
observed
that
• if the
surface
S is
very smooth, then very
few
triangles
are
needed
for
modeling
its
geometry and,
in the
extreme case where
S is a
perfectly
flat
rectangle,
T(S]
can be
composed
of
only
two
triangles;
• if the
property
</?
presents high-frequency variations
on
«S,
then
a lot of
tri-
angles
are
needed
for
modeling these variations accurately even
if S is
very
smooth.
This
clearly shows
the
need
for
uncoupling
the
properties
of a
surface
from
its
geometry.
In
addressing this problem,
the
next section
will
propose
a
solution
based
on a
parameterization
of
T(S):
Discrete
model
M
p
(ft
u
,
N
u
,
(p,C^)
attached
to
T(«S)
If
T(e>)
is a
globally parameterized open surface, according
to
section (3.4),
it is
always possible
to
cover
its
parametric domain
D
with
a set fi
u
of
nodes
{uy}
corresponding
to the
nodes
of a
regular 2-grid:
Such
a set of
nodes
Sl
u
can be
used
for
building
a 2D
graph
G(£l
u
,
N
u
]
where
the
neighborhood operator
N
u
is
defined
by
301
302
CHAPTER
6.
PIECEWISE LINEAR TRIANGULATED SURFACES
Let
(f>
be the
(m
xnxp)
vector collecting
all the
components
of
values
^(uy)
corresponding
to the
sampling
of
<^(x)
at
points
x(uy)
of
T(<S)
associated with
nodes
Ujj
e
fi
u
.
In the
parametric domain,
it is
possible
to
build
a
discrete
model
M
p
(£l
u
,
N
u
,(p,C
v
}
attached
to
T(5),
but
completely uncoupled
from
the
geometry
of
T(<5):
As
shown
in
figure
(6.27)
from
a
graphical point
of
view,
if p
=
1, it is
very
handy
to use a
well-known texture mapping technique [40,
82,
131]
for
painting
T(<5)
with
the
values stored
in the
matrix
(p.
Using
the
neighborhood operator
N
u
defined
by
equation (6.88) implies
a
"pseudo-continuity" across
the
images
of
fault traces
in the
parametric domain
D.
This means
that,
proceeding
in
this way,
(p
is
mapped continuously
on
T(<S)
as it was
prior
to
being
cut by
geological faults.
It
should
be
noted
that,
if the
discontinuities across faults have
to be
honored,
then
modifying
the
definition
of
N
u
to
remove
the
edges
of the
graph
G(£l
u
,N
u
]
cut by the
fault traces
in the
parametric domain
suffices.
Evaluating
</?(p)
at a
point
p on
T(«S)
Let
(itfc,
Vb]
be the
barycentric coordinates
of a
given point
p in a
given triangle
T
=
r(x(ao),x(ai),x(a
2
))
of
T(S):
The
image u(p)
in the
parametric domain
D of
T(«S)
is
denned
by
Thus
the
indexes
(i,j)
of the
left
bottom corner
u^-
of the
unit squared
2-
cell
Uij of
G(Slu,
NU)
containing
25
u(p)
=
(w
1
(p),w
2
(p))
can
easily
be
deter-
mined:
Let
(xij,yij]
be the
local normalized
coordinates
of
u(p)
in Uij
defined
by
and let
|A^j(p)
be the
four
coefficients,
as
defined
by
25
The
function
floor(x) is
defined
as the
greater integer either lower
or
equal
to the
real
value
x.
6.5.
GLOBAL PARAMETERIZATION
Fo r the
sake
of
conciseness,
the
following
notation
will
be
used
in the
rest
of
this section:
It is
verifiable
that
the
coefficients
{A^-(p)}
defined
by
equations (6.89)
are
such
that
In
other words,
the
coefficients
{A^j(p)},
as
denned
by
equation
(6.89),
appear
as a set of
barycentric coordinates
of
u(p) relative
to the
vertices
of the
unit
squared
2-cell
Uij of
Q(^t
u
,Nu}
containing
u(p).
Therefore,
it is
proposed
to
evaluate
<p(p)
on
T(<S)
as its
barycentric interpolation
</?(u(p))
relative
to the
vertices
of
Uij
in the
parametric
domain (see page 128):
Installing
a
fuzzy
Control-Property
constraint
As
suggested
in
figure
(4.7),
let IP be a
point holding
a
property
</>(IP)
pro-
jected
at
point
p on the
triangle
T(x(o:o),x(a!i),x(a2))
°f
^~(<S)-
Assume
that
we
want
to add a
fuzzy
Control-Property constraint
to the set
C
v
of
constraints related
to the
discrete model
A4
p
(Q
U
i
N
u
,
</?)£</?)
to
specify
that
In
such
a
case, according
to
equation (4.67),
the
following
terms have
to be
added
to the
local
form of the
DSI
equation
at
each
of the
four
nodes
{u^j}
of
the
cell
Uij
containing
the
image u(p)
of the
point
p in the
parametric
domain:
with:
303
304
CHAPTER
6.
PIECEWISE
LINEAR
TRIANGULATED
SURFACES
Figur e
6.28 Transmitter between
two
regular curvilinear 3-grids.
Notion
of
Transmitter
Geological
surfaces
such
as
faults,
unconformities,
and
horizons
introduce
dis-
continuities
in the
topology
and the
properties
of
geological
models.
These
surfaces
are
generally
open
and
orient
able
(e.g.,
see
figures 2.34, 6.22,
and
6.27)
so
that,
on
each
of
them,
it is
possible
to
•
compute
a
global parameterization
u;
•
build
a
2-grid
f2
u
defined
by
equation (6.87);
•
attach
a
pair
of
values
{ip~(uij),
<^
+
(uij)}
associated
to the
positive
and
neg-
ative
sides
of the
surface
at
each node
Uij
6
f2
u
;
and
•
build
a
discrete model
M
2
(£l
u
,
N
u
,
(f>,C<f,)
where
N
u
is
defined
by
equation
(6.88),
while
ip
is
such
that
The
discrete
model
A4
2
(£l
u
,-/V
u
,y?,
C^,)
so
denned
is
called
a
"transmitter"
and
can be
used
for
modeling
discontinuities
related
to the
topology
and the
properties
of the
model.
For
example:
• In
seismic processing,
the
refraction
of
seismic rays
is
controlled
by the
dis-
continuities
of the
seismic velocity
field
across geological surfaces.
This
kind
of
information
can
easily
be
modeled
as a
transmitter, where
ip~(uij)
and
tp
+
(uij)
are
defined
as
follows:
• In
reservoir
characterization,
the
regions
located
on the two
sides
of
geolog-
ical surfaces (horizons, faults, unconformities)
are
generally modeled
as
sets
of
polyhedral
3-cells.
It may be
interesting
to
identify
each
pair
of
adjacent
cells
{C~(\iij),
C
+
(uij)}
that
have
a
face
tangential
to
such
a
surface
at lo-
cation
u^.
This kind
of
topological information
can
easily
be
modeled
by a
transmitter, where
ip~(\iij)
and
(p
+
(uij)
are
defined
as
follows:
Figure (6.28) shows
an
example
of
such
a
transmitter.
6.6.
MODIFYING
THE
TOPOLOG
305
Figur e
6.29
Swapping
the
common
edge
(x(o:i),x(a2)}
of
two
adjacent
trian-
gles
of a
triangulated
surface
T(«S)
modifies
the
global
beauty
ofT(S).
Note
that
the
notion
of
transmitter
can be
extended
to
store properties
specific
to the
surface.
For
example,
in the
case
of a
fault, modeling
the
permeability
of
the
fault
at any
location
u^-
e
O
u
as an
additional property
may
prove
interesting.
It is
also possible
to use the
parametric domain
of the
transmitter
as a
"blackboard"
on
which
the
non-manifold junctions with
other
surfaces
can be
mapped
as
curves. This
is
particularly interesting
for
modeling Allan maps
(see
[5]).
6.6
Modifying
the
topology
By
definition
(see page 246),
a
triangulated
surface
T(S]
is no
more than
a
particular
cellular
partition
of an
unknown surface
S
sampled
at a finite
set of
points
{x(ct)
: a €
O}
corresponding
to the
vertices
of
T(S}.
A new
triangulated
surface
deduced
from
T(«S)
can be
built
in two
different
ways:
•
either
the new
triangulated
surface
is a new
cellular
partition
of the
same
surface
S,
• or the new
triangulated
surface
is a
cellular
partition
of a new
surface
whose
boundary
topology
differs
from
that
of S.
The first
approach involves
the
"beautifying"
and
"resampling" algorithms,
while
the
second approach corresponds
to the
"trimming"
and
"cut" algo-
rithms presented
in
this section.
6.6.1
Beautifying
algorithms
Beauty
of a
triangulated
surface
The
equilateral triangle
and the flat
triangle with
a
degenerated edge
are
generally
considered
as the
"nicest"
and the
"ugliest" triangles, respectively,
that
can be
built. This suggests measuring
the
"beauty"
b(T)
of a
triangle
T
306
CHAPTER
6.
PIECE
WISE
LINEAR
TRIANGULATED
SURFACES
Figur e
6.30
Examples
of
improvement
of
the
topology
of
a
triangulated
surface
(A).
The
resulting
"beautified"
surface
(B)
preserves
both
the
number
and the
location
of the
vertices
of the
original
surface.
The
resulting
"beautified"
surface
(C)
preserves only
the
number
of
vertices
of the
original
surface.
The
resulting
"resampled"
surface
(D)
preserves only
the
geometry
of the
original
surface.
as the
ratio
where
•
r(T)
is the
radius
of the
circle tangent
to the
three edges
of T; and
•
-R(T)
is the
radius
of the
circle passing
by the
three vertices
of T.
It can be
shown
(see
[46])
that
the
beauty
criterion
so
denned
can be
computed
as
follows,
where
di,
a2,
and
0:3
are the
internal
angles
of T:
Moreover,
for any
triangle
T, the
beauty
criterion
6(T)
has the
following
properties
(see
[46]):
•
b(T)
stays
in the
range
[0,
^];
•
the
maximum
|
of
b(T)
is
reached
if
T
is
equilateral;
and
• the
minimum
0 of
6(T)
is
reached
if T is flat.
This
means
that
the
global
beauty
of a
triangulated
surface
B(T(S)},
defined
as the sum of the
beauties
of all its
triangles,
is finite and
6.6.
MODIFYING
THE
TOPOLOGY
307
Algorithm
#1
As
suggested
in figure
(6.29),
considering
two
adjacent triangles
T and
T*
of
T(«S)
sharing
a
common edge,
let
cr(T\T*)
and
cr(T*|T)
be the two new
triangles deduced
from
T and
T*
by
"swapping"
the
common edge:
the
triangulated
surface
T(<S).
This
suggests
the
following "beautifying algo-
rithm," whose convergence
is
ensured because,
at
each
step
of the
iterative
value
(i -
|T(5)|):
//
beautifier#l
do
{
flag
<—
false
for_all(TeT*(<S)
) {
let
A(T)
be the set of
triangles
adjacent
to T
let T* be the
triangle
of
A(T)
such
that
6(cr(T*|T))
=
max
b(a(t\T}}
t€A(T)
if(6(a(T|T*))
+
b(a(T*|r))
>
6(T)
+
&(T*))
{
replace
T by
<r(T|T*)
replace
T* by
<r(T*|T)
flag
<—
£rue
} //
end:
if
} //
end:
for_all
}
while(
flag ) //
end:
do
figure
(6.30)-B
gives
an
example
of the
result obtained with this algorithm
when
applied
to the
triangulated surface shown
in figure
(6.30)-A.
Algorithm
#2
It
should
be
noted
that
the
beautifying algorithm presented above preserves
the
location
of the
vertices
of the
triangulated surface
to be
"beautified."
Obviously, relaxing
this
constraint would normally result
in a
nicer surface
T*(tS);
this suggests
the
following
algorithm derived
from
the
above algorithm:
//
beautifier#2
let
TO*
(S) be a
copy
of
T(«S)
let i = 0
do
{
i
<—
i +
1
In general, the swapping operation so defined modifies the global beanty of
process, the global beauty B(T(S)) increases and is upper bounded by the
let T*(S) be a copy of T(S)