Method for constructing surface parameterizations
A method, apparatus, and article of manufacture provide the ability to construct an optimal parameterization for a discrete surface patch. A three-dimensional image having a surface is selected. An initial parameterization of the surface onto a two-dimensional plane is constructed. A distortion of the initial parameterization is measured. A length parameter, an area parameter, and an angle parameter are specified. The initial parameterization is then optimized through a discrete deformation of a parameter domain that minimizes the distortion based on the length, area, and angle parameter.
1. A computer implemented method for constructing an optimal parameterization for a discrete surface patch, comprising:
selecting a three-dimensional image having a surface displayed by a computer device;
constructing an initial parameterization of the surface onto a two-dimensional plane displayed by the computer device;
measuring a distortion induced by the initial parameterization;
specifying a length parameter α l , an area parameter α a , and an angle parameter α c ; and
using the computer device for optimizing the initial parameterization through a discrete deformation of a parameter domain that minimizes the distortion of the parameterized surface displayed by the computer device on the two-dimensional plane based on the length, area, and angle parameters.
2. The method of claim 1 , wherein the initial parameterization has natural boundary conditions over a constrained boundary.
3. The method of claim 1 , wherein the initial parameterization comprises a valid parameterization wherein an inverse of the initial parameterization does not allow the surface to fold onto itself.
4. The method of claim 1 , wherein the distortion comprises
W
(
a
,
d
)
=
α
l
a
+
α
a
(
d
+
(
α
l
α
a
+
1
)
d
-
1
)
+
α
c
(
a
2
/
d
-
4
)
,
wherein α comprises a length control, d comprises an area control, and α 2 /d−4 comprises a conformality control.
5. The method of claim 1 , wherein a user specifies the α l , α a , α c parameters.
6. The method of claim 1 , wherein the optimizing comprises iteratively computing a search direction, calculating a step size, and updating the optimal deformation until a minimum distortion has been reached based on the α l , α a , α c parameters.
7. The method of claim 1 further comprising:
selecting matching features on two three dimensional surfaces;
evaluating a metric and mean curvature by scan converting a surface triangulation in the parameter domain;
applying a finite element discretization; and
optimizing the discretization using a multiscale approach to minimize an energy of the discretization Ε ε [φ] in accordance with
Ε ε [φ]=β bend Ε bend ε [φ]+β reg Ε reg [φ]+β F Ε F ε [φ], wherein β bend , β reg and β F are a bending energy, a feature energy, and a regularization energy that are each user controlled parameters.
8. An apparatus for constructing an optimal parameterization for a discrete surface patch in a computer system comprising:
(a) a computer having a memory;
(b) an application executing on the computer, wherein the application is configured to:
(i) select a three-dimensional image having a surface;
(ii) construct an initial parameterization of the surface onto a two-dimensional plane;
(iii) measure a distortion induced by the initial parameterization;
(iv) specify a length parameter α l , an area parameter α a , and an angle parameter α c ; and
(v) optimize the initial parameterization through a discrete deformation of a parameter domain that minimizes the distortion of the parameterized surface based on the length, area, and angle parameters.
9. The apparatus of claim 8 , wherein the initial parameterization has natural boundary conditions over a constrained boundary.
10. The apparatus of claim 8 , wherein the initial parameterization comprises a valid parameterization wherein an inverse of the initial parameterization does not allow the surface to fold onto itself.
11. The apparatus of claim 8 , wherein the distortion comprises
W
(
a
,
d
)
=
α
l
a
+
α
a
(
d
+
(
α
l
α
a
+
1
)
d
-
1
)
+
α
c
(
a
2
/
d
-
4
)
,
wherein a comprises α length control, d comprises an area control, and α 2 /d−4 comprises a conformality control.
12. The apparatus of claim 8 , wherein a user specifies the α l , α a , α c parameters.
13. The apparatus of claim 8 , wherein the application is configured to optimize by iteratively computing a search direction, calculating a step size, and updating the optimal deformation until a minimum distortion has been reached based on the α l , α a , α c parameters.
14. The apparatus of claim 8 , wherein the application is further configured to:
select matching features on two three dimensional surfaces;
evaluate a metric and mean curvature by scan converting a surface triangulation in the parameter domain;
apply a finite element discretization; and
optimize the discretization using a multiscale approach to minimize an energy of the discretization Ε ε [φ] in accordance with
Ε ε [φ]=β bend Ε bend ε [φ]+β reg Ε reg [φ]+β F Ε F ε [φ], wherein β bend , β reg , and β F are a bending energy, a feature energy, and a regularization energy that are each user controlled parameters.
15. An article of manufacture comprising a program storage medium readable by a computer and embodying one or more instructions executable by the computer to perform a method for constructing an optimal parameterization for a discrete surface patch in a computer system, the method comprising:
selecting a three-dimensional image having a surface;
constructing an initial parameterization of the surface onto a two-dimensional plane;
measure a distortion induced by the initial parameterization;
specifying a length parameter α l , an area parameter α a , and an angle parameter α c ; and
optimizing the initial parameterization through a discrete deformation of a parameter domain that minimizes the distortion of the parameterized surface based on the length, area, and angle parameters.
16. The article of manufacture of claim 15 , wherein the initial parameterization has natural boundary conditions over a constrained boundary.
17. The article of manufacture of claim 15 , wherein the initial parameterization comprises a valid parameterization wherein an inverse of the initial parameterization does not allow the surface to fold onto itself.
18. The article of manufacture of claim 15 , wherein the distortion comprises
W
(
a
,
d
)
=
α
l
a
+
α
a
(
d
+
(
α
l
α
a
+
1
)
d
-
1
)
+
α
c
(
a
2
/
d
-
4
)
,
wherein a comprises α length control, d comprises an area control, and α 2 /d−4 comprises a conformality control.
19. The article of manufacture of claim 15 , wherein a user specifies the α l , α a , α c parameters.
20. The article of manufacture of claim 15 , wherein the optimizing comprises iteratively computing a search direction, calculating a step size, and updating the optimal deformation until a minimum distortion has been reached based on the α l , α a , α c parameters.
21. The article of manufacture of claim 15 , the method further comprising:
selecting matching features on two three dimensional surfaces;
evaluating a metric and mean curvature by scan converting a surface triangulation in the parameter domain;
applying a finite element discretization; and
optimizing the discretization using a multiscale approach to minimize an energy of the discretization Ε ε [φ] in accordance with
Ε ε [φ]=β bend Ε bend ε [φ]+β reg Ε reg [φ]+β F Ε F ε [φ], wherein β bend , β reg , and β F are a bending energy, a feature energy, and a regularization energy that are each user controlled parameters.