Computer graphics using coarse level meshes
View Patent ↗Computer graphics systems, devices and methods adapted to enable display and/or storage of human-perceptible images on a display, device include an arrangement for generating a coarse level mesh representing a surface, from a finer level mesh surface representation. The arrangement includes an indicator value generator and a coarse level mesh generator. The indicator value generator, for respective ones of the points in the finer level mesh surface representation, evaluates an indicator function, the value indicating whether a subdivision-inverse filter methodology or a least-squares optimization methodology is to be used to determine a position for the corresponding point in the coarse level mesh representation. The coarse level mesh generator determines, for each of the points that is to be provided in the coarse level mesh representation, a position in response to the position of the corresponding point in the finer level mesh representation, in accordance with the one of the subdivision-inverse filter methodology and least-squares optimization methodology indicated by the indicator value generated by the indicator value generator.
1. A method, executable in a computer graphics system comprising a processor operable to generate mesh representations of surfaces the method comprising:
generating, using the processor, a coarse level mesh representation representing a surface, from a finer level mesh representation, the generating of a coarse level mesh representation comprising:
A. for respective ones of the points in the finer level surface representation, evaluating, using the processor, an indicator function to generate an indicator value, the indicator value indicating whether one of a subdivision-inverse filter methodology or a least-squares optimization methodology is to be used to determine a position for a corresponding point in the coarse level mesh representation;
B. determining, using the processor, for each of the points that are to be provided in the coarse level mesh representation, a position in the coarse level mesh representation in response to the position of the corresponding point in the finer level mesh representation, in accordance with the one of the subdivision-inverse filter methodology and least-squares optimization methodology as indicated by the generated indicator value; and
C. generating, using the processor, a Laplacian value for said respective ones of the points in the finer level mesh representation, thereby to enable the processor to generate a coarse level mesh representation;
wherein the generating of a Laplacian value comprises generating, using the processor, the Laplacian value L(k,j+1), for at least one of said points, said at least one of said points comprising a point on a boundary, crease line or the like in a triangular mesh representation, in accordance with
L
(
k
,
j
+
1
)
=
1
2
[
c
j
+
1
(
k
-
1
)
c
j
+
1
(
k
+
1
)
]
-
c
j
+
1
(
k
)
where c j+1 (k) represents the position of the point for which the Laplacian is being generated in the finer level mesh representation, and c j+1 (k−1) and c j+1 (k+1) represent the positions of neighboring points in the finer level mesh representation.
2. A method, executable in a computer graphics system comprising a processor operable to generate mesh representations of surfaces, the method comprising:
generating, using processor, a coarse level mesh representation representing a surface, from a finer level mesh representation, the generating of a coarse level mesh representation comprising:
A. for respective ones of the points in the finer level surface representation, evaluating, using the processor, an indicator function to generate an indicator value, the indicator value indicating whether one of a subdivision-inverse filter methodology or a least-squares optimization methodology is to be used to determine a position for a corresponding point in the coarse level mesh representation;
B. determining, using the processor, for each of the points that are to be provided in the coarse level mesh representation, a position in the coarse level mesh representation in response to the position of the corresponding point in the finer level mesh representation, in accordance with the one of the subdivision-inverse filter methodology and least-squares optimization methodology as indicated by the generated indicator value; and
C. generating, using the processor, a Laplacian value for said respective ones of the points in the finer level mesh representation, thereby to enable the processor to generate a coarse level mesh representation;
wherein the generating of a Laplacian value comprises:
generating, using the processor, a Laplacian value for said respective ones of the points in the finer level mesh representation; and
generating, using the processor, the Laplacian value L(k,j+1), for at least one of said points, said at least one of said points comprising a regular vertex, that is, for a vertex for which the valence “K” is equal to “six” in a triangular mesh representation, in accordance with
L
(
k
,
j
+
1
)
=
1
6
(
∑
l
∈
N
(
k
,
j
+
1
)
c
j
+
1
(
l
)
)
-
c
j
+
1
(
k
)
where c j+1 (k) represents the position of the point for which the Laplacian is being generated in the finer level mesh representation, and c j+1 ( 1 ) represents the positions of neighboring points in the finer level mesh representation.
3. A method, executable in a computer graphics system comprising a processor operable to generate mesh representations of surfaces, the method comprising:
generating, using the processor, a coarse level mesh representation representing a surface, from a finer level mesh representation, the generating of a coarse level mesh representation comprising:
A. for respective ones of the points in the finer level surface representation, evaluating, using the processor, an indicator function to generate an indicator value, the indicator value indicating whether one of a subdivision-inverse filter methodology or a least-squares optimization methodology is to be used to determine a position for a corresponding point in the coarse level mesh representation;
B. determining, using the processor, for each of the points that are to be provided in the coarse level mesh representation, a position in the coarse level mesh representation in response to the position of the corresponding point in the finer level mesh representation, in accordance with the one of the subdivision-inverse filter methodology and least-squares optimization methodology as indicated by the generated indicator value; and
C. generating, using the processor, Laplacian value for said respective ones of the points in the finer level mesh representation, thereby to enable the processor to generate a coarse level mesh representation;
wherein the generating of a Laplacian value comprises generating, using the processor, the Laplacian value L(k,j+1), for at least one of said points, said at least one of said points comprising a for a irregular vertex, that is, for a vertex for which the valence “K” is not equal to “six,” in a triangular mesh representation, in accordance with
L
(
k
,
j
+
1
)
ρ
[
1
K
∑
l
∈
N
(
k
,
j
+
1
)
c
j
+
1
(
l
)
-
c
j
+
1
(
k
)
]
where c j+1 (k) represents the position of the point for which the Laplacian is being generated in the finer level mesh representation, and c j+1 ( 1 ) represents the positions of neighboring points in the finer level mesh representation,
ρ
=
-
3
+
8
a
(
K
)
3
(
-
5
+
8
a
(
K
)
)
and
a
(
K
)
=
5
8
-
(
3
+
2
cos
(
2
π
K
)
8
)
2
.
4. A method, executable in a computer graphics system comprising a processor operable to generate mesh representations of surfaces, the method comprising:
generating, using the processor, a coarse level mesh representation representing a surface, from a finer level mesh representation, the generating of a coarse level mesh representation comprising:
A. for respective ones of the points in the finer level surface representation, evaluating, using the processor, an indicator function to generate an indicator value, the indicator value indicating whether one of a subdivision-inverse filter methodology or a least-squares optimization methodology is to be used to determine a position for a corresponding point in the coarse level mesh representation;
B. determining, using the processor, for each of the points that are to be provided in the coarse level mesh representation, a position in the coarse level mesh representation in response to the position of the corresponding point in the finer level mesh representation, in accordance with the one of the subdivision-inverse filter methodology and least-squares optimization methodology as indicated by the generated indicator value; and
C. generating, using the processor, a Laplacian value for said respective ones of the points in the finer level mesh representation, thereby to enable the processor to generate a coarse level mesh representation;
wherein the generating of a Laplacian value comprises generating, using the processor, the Laplacian value L(k,j+1), for at least one of said points, said at least one of said points comprising a point on a boundary, crease line or the like in a quadrilateral mesh representation, in accordance with
L
(
k
,
j
+
1
)
=
1
2
(
c
j
+
1
(
k
-
1
)
+
c
j
+
1
(
k
+
1
)
)
-
c
j
+
1
(
k
)
where c j+1 (k) is the position, of the vertex for which the Laplacian is being generated, and c j+1 (k−1) and c j+1 (k+1) are the positions of the neighboring points in the fine level mesh representation.
5. A method, executable in a computer graphics system comprising a processor operable to generate mesh representations of surfaces, the method comprising:
generating, using the processor, a coarse level mesh representation representing a surface, from a finer level mesh representation, the generating of a coarse level mesh representation comprising:
A. for respective ones of the points in the finer level surface representation, evaluating, using the processor an indicator function to generate an indicator value, the indicator value indicating whether one of a subdivision-inverse filter methodology or a least-squares optimization methodology is to be used to determine a position for a corresponding point in the coarse level mesh representation;
B. determining, using the processor, for each of the points that are to be provided in the coarse level mesh representation, a position in the coarse level mesh representation in response to the position of the corresponding point in the finer level mesh representation, in accordance with the one of the subdivision-inverse filter methodology and least-squares optimization methodology as indicated by the generated indicator value; and
C. generating, using the processor, a Laplacian value for said respective ones of the points in the finer level mesh representation, thereby to enable the processor to generate a coarse level mesh representation;
wherein the generating of a Laplacian value comprises generating, using the processor, Laplacian values L e (k,j+1) and L f (k,j+1), for at least one of said points, said at least one of said points comprising a point on a boundary, crease line or the like in a quadrilateral mesh representation, in accordance with
L
e
(
k
,
j
+
1
)
=
1
K
(
∑
l
∈
N
e
(
k
,
j
+
1
)
c
j
+
1
(
l
)
)
-
c
j
+
1
(
k
)
and
L
f
(
k
,
j
+
1
)
=
1
K
(
∑
l
∈
N
f
(
k
,
j
+
1
)
c
j
+
1
(
l
)
)
-
c
j
+
1
(
k
)
where N e (k,j+1) references a set of points comprising first order neighbors of the at least one of said points in the finer level mesh representation, and N f (k,j+1)references a set of points comprising second order neighbors of the at least one operating system said points in the finer level mesh representation.
6. The method of claim 5 , wherein the generating of a coarse level mesh representation further comprises determining, using the processor, for at least one of the points that are to be provided in the coarse level mesh representation, the position in the coarse level mesh representation as the position of the corresponding point in the finer level mesh representation if the magnitude of both generated Laplacian values L e (k,j+1) and L f (k,j+1) are below a predetermined threshold value.
7. A method, executable in a computer graphics system comprising a processor operable to generate mesh representations of surfaces, the method comprising:
generating, using the processor, a coarse level mesh representation representing a surface, from a finer level mesh representation, the generating of a coarse level mesh representation comprising:
A. for respective ones of the points in the finer level surface representation, evaluating, using the processor, an indicator function to generate a indicator value, the indicator value indicating whether one of a subdivision-inverse filter methodology or a least-squares optimization methodology is to be used to determine a position for a corresponding point in the coarse level mesh representation;
B. determining, using the processor, for each of the points that are to be provided in the coarse level mesh representation, a position in the coarse level mesh representation in response to the position of the corresponding point in the finer level mesh representation, in accordance with the one of the subdivision-inverse filter methodology and least-squares optimization methodology as indicated by the generated indicator value; and
C. generating, using the processor, a Laplacian value for said respective ones of the points in the finer level mesh representation, thereby to enable the processor to generate a coarse level mesh representation;
wherein the generating of a coarse level mesh representation further comprises:
determining, using the processor, for each of the points that are to be provided in the coarse level mesh representation, a position in the coarse level mesh representation in response to the position of the corresponding point in the finer level mesh representation, in accordance with the subdivision-inverse filter methodology if the magnitude of the indicator value is below a selected threshold value; and
determining, using the processor, for at least one of said points, comprising a point on a boundary crease line or the like in a triangular mesh representation, for which the magnitude of the indicator value is below the selected threshold value, a position c j (k) in the coarse level mesh representation in accordance with
c j ( k )= c j+1 ( k )+λ L ( k,j+ 1)
where c j+1 (k) represents the position of the corresponding point in the finer level mesh representation, L(k,j+1) represent the Laplacian value generated for the point in the finer level mesh representation, and λ represents parameter whose value is λ=−1.
8. A method, executable in a computer graphics system comprising a processor operable to generate mesh representations of surfaces, the method comprising:
generating, using the processor, a coarse level mesh representation representing a surface, from a finer level mesh representation, the generating of a coarse level mesh representation comprising:
A. for respective ones of the points in the finer level surface representation, evaluating, using the processor, an indicator function to generate an indicator value, the indicator value indicating whether one of a subdivision-inverse filter methodology or a least-squares optimization methodology is to be used to determine a position for a corresponding point in the coarse level mesh representation;
B. determining, using the processor, for each of the points that are to be provided in the coarse level mesh representation, a position in the coarse level mesh representation in response to the position of the corresponding point in the finer level mesh representation, in accordance with the one of the subdivision-inverse filter methodology and least-squares optimization methodology as indicated by the generated indicator value; and
C. generating, using the processor, a Laplacian value for said respective ones of the points in the finer level mesh representation, thereby to enable the processor to generate a coarse level mesh representation;
wherein the generating of a coarse level mesh representation further comprises:
determining, using the processor, for each of the points that are to be provided in the coarse level mesh representation, a position in the coarse level mesh representation in response to the position of the corresponding point in the finer level mesh representation, in accordance with the subdivision-inverse filter methodology if the magnitude of the indicator value is below a selected threshold value; and
determining, using the processor, for at least one of said points in a triangular mesh representation, comprising a regular point, that is, a point whose valence “K” is equal to “six,” and is not on a boundary, crease line or the like, for which the magnitude of the indicator value is below the selected threshold value, a position c j (k) in the coarse level mesh representation in accordance with
c j ( k )= c j+1 ( k )+λ L ( k,j +1)
where c j+1 (k) represents the position of the corresponding point in the finer level mesh representation, L(k,j+1) represents the Laplacian value generated for the point in the finer level mesh representation, and λ represents a parameter whose value is λ=−3/2.
9. A method, executable in a computer graphics system comprising a processor operable to generate mesh representations of surfaces, the method comprising:
generating, using the processor, a coarse level mesh representation representing a surface, from a finer level mesh representation, the generating of a coarse level mesh representation comprising:
A. for respective ones of the points in the finer level surface representation, evaluating, using the processor, an indicator function to generate an indicator value, the indicator value indicating whether one of a subdivision-inverse filter methodology or a least squares optimization methodology is to be used to determine a position for a corresponding point in the coarse level mesh representation;
B. determining, the using the processor, for each of the points that are to be provided in the coarse level mesh representation, a position in the coarse level mesh representation in response to the position of the corresponding point in the finer level mesh representation, in accordance with the one of the subdivision-inverse filter methodology and least-squares optimization methodology as indicated by the generated indicator value; and
C. generating, using the processor, a Laplacian value for said respective ones of the points in the finer level mesh representation, thereby to enable the processor to generate a coarse level mesh representation;
wherein the generating of a coarse level mesh representation further comprises:
determining, using the processor, for each of the points that are to be provided in the coarse level mesh representation, a position in the coarse level mesh representation in response to the position of the corresponding point in the finer level mesh representation, in accordance with the subdivision-inverse filter methodology if the magnitude of the indicator value is below a selected threshold value; and
determining, using the processor, for at least one of said points in a triangular mesh representation, comprising an irregular point, that is, a point whose valence “K” is not equal to “six,” and is not on a boundary, crease line, for which the magnitude of the indicator value is below the selected threshold value, a position c j (k) in the coarse level mesh representation in accordance with
c j ( k )= c j+1 ( k )+λ L ( k,j +1)
where c j+1 (k) represents the position of the corresponding point in the finer level mesh representation, L(k,j+1) represents the Laplacian value generated for the point in the finer level mesh representation, and λ represents a parameter whose value is generated in accordance with
λ
=
8
a
(
K
)
-
3
+
8
a
(
K
)
,
where
a
(
K
)
=
5
8
-
(
3
+
2
cos
(
2
π
K
)
8
)
2
.
10. A method, executable in a computer graphics system comprising a processor operable to generate mesh representations of surfaces, the method comprising:
generating, using the processor, a coarse level mesh representation representing a surface, from a finer level mesh representation, the generating of a coarse level mesh representation comprising:
A. for respective ones of the points in the finer level surface representation, evaluating, using the processor, an indicator function to generate an indicator value, the indicator value indicating whether one of a subdivision-inverse filter methodology or a least-squares optimization methodology is to be used to determine a position for a corresponding point in the coarse level mesh representation;
B. determining, using the processor, for each of the points that are to be provided in the coarse level mesh representation, a position in the coarse level mesh representation in response to the position of the corresponding point in the finer level mesh representation, in accordance with the one of the subdivision-inverse filter methodology and least-squares optimization methodology as indicated by the generated indicator value; and
C. generating, using the processor, Laplacian value for said respective ones of the points in the finer level mesh representation, thereby to enable the processor to generate a coarse level mesh representation;
wherein the generating of a coarse level mesh representation further comprises:
determining, using the processor, for each of the points that are to be provided in the coarse level mesh representation, a position in the coarse level mesh representation in response to the position of the corresponding point in the finer level mesh representation, in accordance with the subdivision-inverse filter methodology if the magnitude of the indicator value is below a selected threshold value; and
determining, using the processor, for at least one of said points, comprising a point on a boundary, crease line or the like in a quadrilateral mesh representation, for which the magnitude of the indicator value is below the selected threshold value, a position c j (k) in the coarse level mesh representation in accordance with
c j ( k )= c j+1 ( k )+λ L ( k,j +1)
where c j+1 (k) represents the position of the corresponding point in the finer level mesh representation, L(k,j+1) represents the Laplacian value generated for the point in the finer level mesh representation, and λ represents a parameter whose value is λ=−1.
11. A method, executable in a computer graphics system comprising a processor operable to generate mesh representations of surfaces, the method comprising:
generating, using the processor, a coarse level mesh representation representing a surface, from a finer level mesh representation the generating of a coarse level mesh representation comprising:
A. for respective ones of the points in the finer level surface representation, evaluating, using the processor, an indicator function to generate an indicator value, the indicator value indicating whether one of a subdivision-inverse filter methodology or a least-squares optimization methodology is to be used to determine a position for a corresponding point in the coarse level mesh representation;
B. determining, using the processor, for each of the points that are to be provided in the coarse level mesh representation, a position in the coarse level mesh representation in response to the position of the corresponding point in the finer level mesh representation, in accordance with the one of the subdivision-inverse filter methodology and least-squares optimization methodology as indicated by the generated indicator value; and
C. generating, using the processor, a Laplacian value for said respective ones of the points in the finer level mesh representation, thereby to enable the processor to generate a coarse level mesh representation;
wherein the generating of a coarse Level mesh representation further comprises:
determining, using the processor, for each of the points that are to be provided in the coarse level mesh representation, a position in the coarse level mesh representation in response to the position of the corresponding point in the finer level mesh representation, in accordance with the subdivision-inverse filter methodology if the magnitude of the indicator value is below a selected threshold value; and
determining, using the processor, for at least one of said points, in a quadrilateral mesh representation, which is not on a boundary, crease line or the like, for which the magnitude of the indicator value is below the selected threshold value, a position c j (k) in the coarse level mesh representation in accordance with
c j ( k )= c j+1 ( k )+λ 1 L e ( k,j +1)+λ 2 L f ( k,j +1)
where c j+1 (k) represents the position of the corresponding point in the finer level mesh representation, L e (k,j+1) and L f (k,j+1) represent the Laplacian value generated for the point in the finer level mesh representation, and λ 1 and λ 2 represent parameters whose values are generated in accordance with, if the valence “K” of the point not equal to “three,”
λ
1
=
-
4
K
-
3
λ
2
=
1
K
-
3
and, if the valence “K” for the vertex is equal to “three,” λ 1 =−8, λ 2 =−2.
12. A method, executable in a computer graphics system comprising a processor operable to generate mesh representations of surfaces, the method comprising:
generating, using the processor, a coarse level mesh representation representing a surface, from a finer level mesh representation, the generating of a coarse level mesh representation comprising:
A. for respective ones of the points in the finer level surface representation, evaluating, using the processor, an indicator function to generate an indicator value, the indicator value indicating whether one of a subdivision-inverse filter methodology or a least-squares optimization methodology is to be used to determine a position for a corresponding point in the coarse level mesh representation;
B. determining, using the processor, for each of the points that are to be provided in the coarse level mesh representation, a position in the coarse level mesh presentation in response to the position of the corresponding point in the finer level mesh representation, in accordance with the one of the subdivision-inverse filter methodology and least-squares optimization methodology as indicated by the generated indicator value; and
C. generating, using the processor, a Laplacian value for said respective ones of the points in the finer level mesh representation, thereby to enable the processor to generate a coarse level mesh representation;
wherein the generating of a coarse level mesh representation further comprises:
determining, using the processor, for each of the points that are to be provided in the coarse level mesh representation, a position in the coarse level mesh representation in response to the position of the corresponding point in the finer level mesh representation, in accordance with the least-squares optimization methodology if the magnitude of the indicator value is above a selected threshold value; and
determining, using the processor, for at least one of said points, comprising a point on a boundary, crease line or the like in a triangular mesh representation, for which the magnitude of the indicator value is not below the selected threshold value, a position c j (k) in the coarse level mesh representation in accordance with
c j ( k )= c j+1 ( k )=λ L ( k,j+ 1)
where c j+1 (k) represents the position of the corresponding point in the finer level mesh representation, L(k,j+1) represents the Laplacian values generated for the point in the finer level mesh representation, and λ represents a parameter whose value is generated in accordance with
λ
=
1
L
(
k
)
[
b
0
1
D
L
(
k
)
+
1
2
b
1
1
D
(
L
(
k
-
1
)
+
L
(
k
+
1
)
)
]
where
b
0
1
D
=
-
12
35
and
b
1
1
D
=
-
23
49
,
and L(k−1) and L(k+1) represent Laplacian value generated for neighboring points in the finer level mesh representation.
13. A method, executable in a computer graphics system comprising a processor operable to generate mesh representations of surfaces, the method comprising:
generating, using the processor, a coarse level mesh representation representing a surface, from a finer level mesh representation, the generating of a coarse level mesh representation comprising:
A. for respective ones of the points in the finer level surface representation, evaluation, using the processor, an indicator function to generate an indicator value, the indicator value indicating methodology is to be used to determine a position for a least-squares optimization methodology is to be used to determine a position for a corresponding point in the coarse level mesh representation;
B. determining, using the processor, for each of the points that are to be provided in the coarse level mesh representation, a position in the coarse level mesh representation in response to the position of the corresponding point in the finer level mesh representation, in accordance with the one of the subdivision-inverse filter methodology and least-squares optimization methodology as indicated by the generated indicator value; and
C. generating, using the processor, a Laplacian value for said respective ones of the points in the finer level mesh representation, thereby to enable the processor to generate a coarse level mesh representation;
wherein the generating of a coarse level mesh representation further comprises:
determining, using the processor, for each of the points that are to be provided in the coarse level mesh representation, a position in the coarse level mesh representation in response ti the position of the corresponding point in the finer level mesh representation, in accordance with the least-squares optimization methodology if the magnitude of the indicator value is above a selected threshold value; and
determining, using the processor, for at least one of said points in a triangular mesh representation, comprising a regular point, that is, a point whose valence “K” is equal to “six,” and is not on a boundary, crease line or the like, for which the magnitude of the indicator value is not below the selected threshold value, a position c j (k) in the coarse level mesh representation in accordance with
c j ( k )= c j+1 ( k )+λ L ( k,j +1)
where c j+1 (k) represents the position of the corresponding point in the finer level mesh representation, L(k,j+1) represents the Laplacian value generated for the point in the finer level mesh representation, and λ represents a parameter whose value is
λ
=
1
L
(
k
)
[
b
0
reg
L
(
k
)
+
1
6
b
1
reg
∑
l
∈
N
(
k
,
j
+
1
)
L
(
l
)
]
where
b
0
reg
=
-
61
5720
and
b
1
reg
=
-
14403
5720
and L( 1 ) represent Laplacian Value Generated for points, identified by indices N(k,j+1), that neighbor the at least one of said points.
14. A method, executable in a computer graphics system comprising a processor operable to generate mesh representations of surfaces, the method comprising:
generating, using the processor, a coarse level mesh representation representing a surface, from a finer level mesh representation, the generating of a coarse level mesh representation comprising:
A. for respective ones of the points in the finer level surface representation, evaluating, using the processor, an indicator function to generate an indicator value, the indicator value indicating whether one of a subdivision-inverse filter methodology or a least-squares optimization methodology is to be used to determine a position for a corresponding point in the coarse level mesh representation;
B. determining, using the processor, for each of the points that are to be provided in the coarse level mesh representation, a position in the coarse level mesh representation in response to the position of corresponding point in the finer level mesh representation, in accordance with the one of the subdivision-inverse filter methodology and least-squares optimization methodology as indicated by the generated indicator value; and
C. generating using the processor, a Laplacian value for said respective ones of the points in the finer level mesh representation, thereby to enable the processor to generate a coarse level mesh representation;
wherein the generating of a coarse level mesh representation further comprises:
determining, using the processor, for each of the points that are to be provided in the coarse level mesh representation, a position in the coarse level mesh representation in response to the position of the corresponding point in the finer level mesh representation, in accordance with the least-squares optimization methodology if the magnitude of the indicator value is above a selected threshold value; and
determining, using the processor, for at least one of said points in a triangular mesh representation, comprising an irregular point, that is, a point whose valence “K” is not equal to “six,” and is not on a boundary, crease line or the like, for which the magnitude of the indicator value is not below the selected threshold value, a position c j (k) in the coarse level mesh representation in accordance with
c j ( k )= c j+1 ( k )+λ L ( k,j+ 1)
where c j+1 (k) represents the position of the corresponding point in the finer level mesh representation, L(k,j+1) represents the Laplacian value generated for the point in the finer level mesh representation, and λ represents a parameter whose value is generated in accordance with
λ
=
1
L
(
k
)
[
b
0
reg
L
(
k
)
+
1
K
b
1
reg
∑
l
∈
N
(
k
,
j
+
1
)
L
(
l
)
]
,
where
b
0
reg
=
2
(
5
-
8
a
(
K
)
)
(
14647
K
-
391848
a
(
K
)
+
391848
a
(
k
)
2
)
715
(
3
+
8
a
(
k
)
)
(
256
+
41
K
-
512
a
(
k
)
+
256
a
(
k
)
2
)
and
b
1
reg
=
16
(
-
5531
K
-
24521
a
(
K
)
+
24521
a
(
K
)
2
)
715
(
256
+
41
K
-
512
a
(
K
)
+
25
a
(
K
)
2
)
and
where
a
(
K
)
=
5
8
-
(
3
+
2
cos
(
2
π
K
)
8
)
2
.
15. A method, executable in a computer graphics system comprising a processor operable to generate mesh representations of surfaces, the method comprising:
generating, using the processor, a coarse level representation representing a surface, from a finer level mesh representation, the generating of a coarse level mesh representation comprising:
A. for respective ones of the points in the finer level surface representation, evaluating, using the processor, an indicator function to generate an indicator value, the indicator value indicating whether one of a subdivision-inverse filter methodology or a least-squares optimization methodology is to be used to determine a position for a corresponding point in the coarse level mesh representation;
B. determining, using the processor, for each of the points that are to be provided in the coarse level mesh representation, a position in the coarse level mesh representation in response to the position of the corresponding point in the finer level mesh representation, in accordance with the one of the subdivision-inverse filter methodology and least-squares optimization methodology as indicated by the generated indicator value; and
C. generating, using the processor, a Laplacian value for said respective ones of the points in the finer level mesh representation, thereby to enable the processor to generate a coarse level mesh representation;
wherein the generating of a coarse level mesh representation further comprises:
determining, using the processor, for each of the points that are to be provided in the coarse level mesh representation, a position in the coarse level mesh representation in response to the position of the corresponding point in the finer level mesh representation, in accordance with the least-squares optimization methodology if the magnitude of the indicator value is above a selected threshold value; and
determining, using the processor, for at least one of said points, comprising a point on a boundary, crease line or the like in a quadrilateral mesh representation, for which the magnitude of the indicator value is not below the selected threshold value, a position c j (k) in the coarse level mesh representation in accordance with
c j ( k )= c j+1 ( k )+λ L ( k,j+ 1)
where c j+1 (k) represents the position of the corresponding point in the finer level mesh representation, L(k,j+1) represents the Laplacian value generated for the point in the finer level mesh representation, and λ represents a parameter whose value is generated in accordance with
λ
=
1
L
(
k
)
[
b
0
1
D
L
(
k
)
+
1
2
b
1
1
D
(
L
(
k
-
1
)
+
L
(
k
+
1
)
)
]
where
b
0
1
D
=
-
12
35
and
b
1
1
D
=
-
23
49
,
and L(k−1) and L(k+1) represent Laplacian values generated for neighboring points in the finer level mesh representation.
16. A method, executable in a computer graphics system comprising a processor operable to generate mesh representation of surfaces, the method comprising:
generating, using the processor, a coarse level mesh representation representing a surface, from a finer level mesh representation, the generating of a coarse level mesh representation comprising:
A. for respective ones of the points in the finer level surface representation, evaluating, using the processor, an indicator function to generate an indicator value, the indicator value indicating whether one of a subdivision-inverse filter methodology or a least-square optimization methodology is to be used to determine a position for a corresponding point in the coarse level mesh representation;
B. determining, using the processor, for each of the points that are to be provided in the coarse level mesh representation, a position in the coarse level mesh representation in response to the corresponding point in the finer level mesh representation, in accordance with one of the subdivision-inverse filter methodology and least-squares optimization methodology as indicated by the generated indicator value; and
C. generating, using the processor, a Laplacian value for said respective ones of the points in the finer level mesh representation, thereby to enable the processor to generate a coarse level mesh representation;
wherein the generating of a coarse level mesh representation further comprises:
determining, for each of the points that are to be provided in the coarse level mesh representation, a position in the coarse level mesh representation in response to the position of the corresponding point in the finer level mesh representation, in accordance value is above a selected threshold value; and
determining, using the processor, for at least one of said points in a quadrilateral mesh representation, which is not on a boundary, crease line or the like, for which the magnitude of the indicator value is below the selected threshold value, a position c j (k) in the coarse level mesh representation in accordance with
c j ( k )= c j+1 ( k )+λ 1 L e ( k,j+ 1)+λ 2 L f ( k,j+ 1)
where c j+1 (k) represents the position of the corresponding point in the finer level mesh representation, L e (k,j+1) and L f (k,j+1) represent Laplacian values generated for the point in the finer level mesh representation, and λ 1 and λ 2 represent parameters whose values are generated in accordance with
λ
1
=
1
L
e
(
k
,
j
+
1
)
[
b
10
cc
L
e
(
k
,
j
+
1
)
+
1
K
b
11
cc
∑
l
∈
N
(
k
,
j
+
1
)
L
e
(
l
,
j
+
1
)
]
and
λ
2
=
1
L
f
(
k
,
j
+
1
)
[
b
20
cc
L
f
(
k
,
j
+
1
)
+
1
K
b
21
cc
∑
l
∈
N
(
k
,
j
+
1
)
L
f
(
l
,
j
+
1
)
]
where, if the at least one of said points is regular, that is, if its valence “K” is “four,”
b
10
cc
=
-
9946871
4862025
b
11
cc
=
-
1024
405
b
20
cc
=
1644032
972405
b
21
cc
=
-
1338874
972405
and, if the at least one point is irregular, that is, its valence “K” is other than “four,”
b
10
cc
=
162307143936
-
92746939392
K
-
8924282387
K
3
4862025
(
12544
-
14336
K
+
4096
K
2
+
901
K
3
)
b
11
cc
=
1024
(
2793728
-
1596416
K
-
244001
K
3
)
99225
(
12544
-
14336
K
+
4096
K
2
901
K
3
)
b
20
cc
=
512
(
-
113305472
+
64745984
K
+
17391149
K
3
)
4862025
(
12544
-
14336
K
+
4096
K
2
+
901
K
3
)
b
21
cc
=
4
(
8660934688
-
4949105536
K
-
1876158821
K
3
)
4862025
(
12544
-
14336
K
+
4096
K
2
+
901
K
3
)
.