Efficient segmentation of piecewise smooth images
A fast and robust segmentation model for piecewise smooth images is provided. Local statistics in an energy formulation are provided as a functional. The shape gradient of this new functional gives a contour evolution controlled by local averaging of image intensities inside and outside the contour. Fast computation is realized by expressing terms as the result of convolutions implemented via recursive filters. Results are similar to the general Mumford-Shah model but realized faster without having to solve a Poisson partial differential equation at each iteration. Examples are provided. A system to implement segmentation methods is also provided.
1. A method of segmentation of an image u 0 comprising:
utilizing a processor to create a plurality of smooth regions Ω i delimited by a boundary Γ;
approximating the image by a piecewise smooth function u σ inside each region Ω i ; and
evaluating a functional, wherein the piecewise smooth function is expressed as:
u
σ
(
x
,
Ω
i
)
=
∫
Ω
i
g
σ
(
x
-
y
)
u
0
(
y
)
ⅆ
y
∫
Ω
i
g
σ
(
x
-
y
)
ⅆ
y
,
wherein
g
σ
(
v
)
=
1
2
π
σ
exp
(
-
v
2
2
σ
2
)
is a Gaussian kernel.
2. The method as claimed in claim 1 , wherein the functional is evaluated in a level set implementation.
3. The method as claimed in claim 1 , wherein the functional is expressed as
E (Γ)=μ 2 ∫ S ( u 0 −u σ (Γ)) 2 dx+ν|Γ|.
4. The method as claimed in claim 1 , wherein the functional is expressed as
E
(
Γ
)
=
μ
2
∑
i
∫
(
u
0
-
u
σ
(
)
)
2
ⅆ
x
+
v
Γ
.
5. The method as claimed in claim 1 , wherein u σ represents an overall piecewise smooth approximation expressed as
u
σ
(
x
,
Γ
)
=
∑
i
χ
i
u
σ
(
x
,
Ω
i
)
.
6. The method as claimed in claim 1 , wherein a performance of segmentation can be tuned by selecting a value of σ.
7. The method as claimed in claim 1 , wherein the segmentation is a bi-partitioning, separating a region Ω from a complementary region Ω .
8. The method as claimed in claim 7 , further comprising:
evolving the boundary Γ between region Ω and region Ω according to an expression
∂
Γ
∂
t
(
x
)
=
[
μ
2
(
(
u
0
(
x
)
-
u
σ
(
x
,
Ω
_
)
)
2
-
(
u
0
(
x
)
-
u
σ
(
x
,
Ω
)
)
2
-
q
σ
(
x
,
Ω
_
)
+
q
σ
(
x
,
Ω
)
)
+
v
κ
]
N
(
x
)
.
9. The method as claimed in claim 8 , further comprising:
implementing the evolving of the boundary Γ with a level set representation.
10. The method as claimed in claim 9 , further comprising:
computing steps of the evolving of the boundary Γ with a recursive filter.
11. A system of segmentation of an image u 0 comprising:
a processor;
software operable on the processor to:
creating a plurality of smooth regions Ω i delimited by a boundary Γ;
approximating the image by a piecewise smooth function u σ inside each region Ω i ; and
evaluating a functional, wherein the piecewise smooth function is expressed as:
u
σ
(
x
,
Ω
i
)
=
∫
Ω
i
g
σ
(
x
-
y
)
u
0
(
y
)
ⅆ
y
∫
Ω
i
g
σ
(
x
-
y
)
ⅆ
y
,
wherein
g
σ
(
v
)
=
1
2
π
σ
exp
(
-
v
2
2
σ
2
)
is a Gaussian kernel.
12. The system as claimed in claim 11 , wherein the functional is evaluated in a level set implementation.
13. The system as claimed in claim 11 , wherein the functional is expressed as
E (Γ)=μ 2 ∫ S ( u 0 −u σ (Γ)) 2 dx+ν|Γ|.
14. The system as claimed in claim 11 , wherein the functional is expressed as
E
(
Γ
)
=
μ
2
∑
i
∫
(
u
0
-
u
0
(
)
)
2
ⅆ
x
+
v
Γ
.
15. The system as claimed in claim 11 , wherein u σ represents an overall piecewise smooth approximation expressed as
u
σ
(
x
,
Γ
)
=
∑
i
χ
i
u
σ
(
x
,
Ω
i
)
.
16. The system as claimed in claim 12 , wherein a performance of segmentation can be tuned by selecting a value of σ.
17. The system as claimed in claim 11 , wherein the segmentation is a bi-partitioning, separating a region Ω from a complementary region Ω .
18. The system as claimed in claim 17 , further comprising:
evolving the boundary Γ between region Ω and region Ω according to an expression
∂
Γ
∂
t
(
x
)
=
[
μ
2
(
(
u
0
(
x
)
-
u
σ
(
x
,
Ω
_
)
)
2
-
(
u
0
(
x
)
-
u
σ
(
x
,
Ω
)
)
2
-
q
σ
(
x
,
Ω
_
)
+
q
σ
(
x
,
Ω
)
)
+
v
κ
]
N
(
x
)
.
19. The system as claimed in claim 18 , further comprising:
implementing the evolving of the boundary Γ with a level set representation.
20. The system as claimed in claim 19 , further comprising:
computing steps of the evolving of the boundary Γ with a recursive filter.