Texturing a 3D modeled object
A computer-implemented method for designing a 3D modeled object representing a real object comprises providing a 3D mesh representing the real object, a texturing image and a mapping between the vertices of the 3D mesh and pixels of the texturing image; then maximizing a probability P(L(V)) of the form: P ( L ( V ) ) = 1 Z exp ( - ∑ i = 1 n φ i ′ ( L ( v i ) ) - ∑ f ∈ ℱ ψ f ′ ( { L ( v i ) } i ∈ f ) ) . Maximizing is performed with a predetermined discrete Markov Random Field optimization scheme viewing the 3D mesh and the pixel shifts associated to the texture coordinates of the vertices of the 3D mesh as a Markov Random Field of energy −log(P(L(V)))−log(Z). The method then comprises texturing the 3D mesh according to the texturing image, to the mapping, and to the result of the maximizing. This provides an improved solution for designing a 3D modeled object a real object.
1. A computer-implemented method for designing a 3D modeled object representing a real object, comprising:
providing, in a Product Lifecycle Management (PLM) system, a 3D mesh representing the real object and having vertices, a texturing image and a mapping between the vertices of the 3D mesh and pixels of the texturing image, wherein the PLM system is at least one of a Computer-Aided Design (CAD) system, Computer-Aided Engineering (CAE) system, and a Computer-Aided Manufacturing (CAM) system; then
maximizing a probability P(L(V)) of the form:
P
(
L
(
V
)
)
=
1
Z
exp
(
-
∑
i
=
1
n
φ
i
′
(
L
(
v
i
)
)
-
∑
f
∈
ℱ
ψ
f
′
(
{
L
(
v
i
)
}
i
∈
f
)
)
where:
V is a vector of vertices v i ,
n designates the number of vertices of the 3D mesh and v i designates the vertices of the 3D mesh,
L(v i ) designates a pixel shift to be applied after mapping vertex v i on the texturing image and selected in a predetermined finite set (L),
Z=Σ L(V) exp(−Σ i=1 n φ i ′( L ( v i ))−Σ fϵF ψ f ′({ L ( v i )} iϵf )),
F designates the set of sets of indices of mesh tiles of the 3D mesh,
φ i ′ designates a cost function associated to vertex v i and decreasingly depending on an extent to which the result of applying the pixel shift, selected for vertex v i , after mapping vertex v i on the texturing image respects a predetermined relation between vertices of the 3D mesh and pixels of the texturing image,
ψ f ′ designates a cost function associated to a tile f of the 3D mesh and depending on a global difference between pixel shifts selected for the vertices of tile f,
the maximizing being performed with a predetermined discrete Markov Random Field optimization scheme viewing the 3D mesh and the pixel shifts associated to the texture coordinates of the vertices of the 3D mesh as a Markov Random Field of energy:
−log( P ( L ( V )))−log( Z )=Σ i=1 n φ i ′( L ( v i ))+Σ fϵF ψ f ′({ L ( v i )} iϵf )); and
texturing the 3D mesh according to the texturing image, to the mapping, and to the result of the maximizing.
2. The method of claim 1 , wherein the cost function ψ f ′ is of the form:
ψ f ′({ L ( v i )} iϵf )=Σ {i,j}ϵp(f) ψ i,j ′( L ( v i ), L ( v j )),
where p(f) designates the set of pairs of indices of vertices of tile f, and
wherein the predetermined discrete Markov Random Field optimization scheme is a pairwise discrete Markov Random Field optimization scheme.
3. The method of claim 2 , wherein ψ i,j ′(L(v i ), L(v j )) is of the form
λ
3
(
L
(
v
i
)
-
L
(
v
j
)
1
,
where λ designates a positive scalar.
4. The method of claim 1 , wherein the predetermined relation between vertices of the 3D mesh and pixels of the texturing image amounts to a predetermined relation between 3D curvature values for a vertex of the 3D mesh and distance values to a nearest contour of the texturing image for a pixel of the texturing image.
5. The method of claim 4 , wherein:
3D curvature values (C i ) lower than a predetermined threshold (C) are in the predetermined relation with all distance values (T i (L(v i ))), and 3D curvature values higher than the predetermined threshold are in the predetermined relation with distance values according to an increasing one-to-one correspondence; and
T i (L(v i )) designates the value of a distance transform of the texturing image at the result of applying the pixel shift, selected for vertex v i , after mapping vertex v i on the texturing image, the distance transform being relative to a contour image of the texturing image.
6. The method of claim 5 , wherein φ i ′ is of the form:
φ
i
′
(
L
(
v
i
)
)
=
1
C
i
>
c
γ
C
i
-
T
i
(
L
(
v
i
)
)
,
where:
1 C i >c designates an indicator function, with C i designating the maximal 3D curvature of vertex v i and c designating a positive scalar, and
γ designates a positive scalar.
7. The method of claim 6 , wherein the contour image is determined with a Canny edge detector applied on the texturing image.
8. The method of claim 6 , wherein the distance transform of the texturing image is determined with a Chamfer mask applied on the texturing image relative to the contour image of the texturing image.
9. The method of claim 1 , wherein the 3D mesh, the texturing image and the mapping are all outputted by a predetermined structure-from-motion analysis scheme applied on the real object, the mapping corresponding to pose camera parameters determined for the texturing image in the structure-from-motion analysis.
10. A non-transitory data storage medium comprising:
a container holding recorded thereon a computer program for designing a 3D modeled object representing a real object,
the computer program including:
providing a 3D mesh representing the real object and having vertices, a texturing image and a mapping between the vertices of the 3D mesh and pixels of the texturing image; then
maximizing a probability P(L(V)) of the form:
P
(
L
(
V
)
)
=
1
Z
exp
(
-
∑
i
=
1
n
φ
i
′
(
L
(
v
i
)
)
-
∑
f
∈
ℱ
ψ
f
′
(
{
L
(
v
i
)
}
i
∈
f
)
)
where:
V is a vector of vertices v i ,
n designates the number of vertices of the 3D mesh and v i designates the vertices of the 3D mesh,
L(v i ) designates a pixel shift to be applied after mapping vertex v i on the texturing image and selected in a predetermined finite set (L),
Z=Σ L(V) exp(−Σ i=1 n φ i ′( L ( v i ))−Σ fϵF ψ f ′({ L ( v i )} iϵf )),
F designates the set of sets of indices of mesh tiles of the 3D mesh,
φ′ i designates a cost function associated to vertex v i and decreasingly depending on an extent to which the result of applying the pixel shift, selected for vertex v i , after mapping vertex v i on the texturing image respects a predetermined relation between vertices of the 3D mesh and pixels of the texturing image,
ψ f ′ designates a cost function associated to a tile f of the 3D mesh and depending on a global difference between pixel shifts selected for the vertices of tile f,
the maximizing being performed with a predetermined discrete Markov Random Field optimization scheme viewing the 3D mesh and the pixel shifts associated to the texture coordinates of the vertices of the 3D mesh as a Markov Random Field of energy:
−log( P ( L ( V )))−log( Z )=Σ i=1 n φ i ′( L ( v i ))+Σ fϵF ψ f ′({ L ( v i )} iϵf )); and
texturing the 3D mesh according to the texturing image, to the mapping, and to the result of the maximizing.
11. A computer system comprising:
a processor coupled to a memory,
the memory having recorded thereon a computer program comprising instructions for designing a 3D modeled object representing a real object, the instructions causing the processor to:
provide a 3D mesh representing the real object and having vertices, a texturing image and a mapping between the vertices of the 3D mesh and pixels of the texturing image; then
maximize a probability P(L(V)) of the form:
P
(
L
(
V
)
)
=
1
Z
exp
(
-
∑
i
=
1
n
φ
i
′
(
L
(
v
i
)
)
-
∑
f
∈
ℱ
ψ
f
′
(
{
L
(
v
i
)
}
i
∈
f
)
)
where:
V is a vector of vertices v i ,
n designates the number of vertices of the 3D mesh and v i designates the vertices of the 3D mesh,
L(v i ) designates a pixel shift to be applied after mapping vertex v i on the texturing image and selected in a predetermined finite set (L),
Z=Σ L(V) exp(−Σ i=1 n φ i ′( L ( v i ))−Σ fϵF ψ f ′({ L ( v i )} iϵf )),
F designates the set of sets of indices of mesh tiles of the 3D mesh,
φ′ i designates a cost function associated to vertex v i and decreasingly depending on an extent to which the result of applying the pixel shift, selected for vertex v i , after mapping vertex v i on the texturing image respects a predetermined relation between vertices of the 3D mesh and pixels of the texturing image,
ψ f ′ designates a cost function associated to a tile f of the 3D mesh and depending on a global difference between pixel shifts selected for the vertices of tile f,
the maximizing being performed with a predetermined discrete Markov Random Field optimization scheme viewing the 3D mesh and the pixel shifts associated to the texture coordinates of the vertices of the 3D mesh as a Markov Random Field of energy:
−log( P ( L ( V )))−log( Z )=Σ i=1 n φ i ′( L ( v i ))+Σ fϵF ψ f ′({ L ( v i )} iϵf )); and
texture the 3D mesh according to the texturing image, to the mapping, and to the result of the maximizing.