Method for robust scale-space analysis of 3D local structures in medical images
View Patent ↗A method for determining the location, shape and orientation of a tumor in a medical image includes finding a plurality of spatial extrema μ of a D-dimensional spatial signal f for a set of bandwidths H by performing mean shift-based gradient-ascent iterations for a set of bandwidths H and then determining a D-dimensional spread and orientation of the signal about each extrema μ by estimating a covariance Σ of the signal f for each extrema μ. The optimal estimate of μ and Σ is determined by performing a Jensen-Shannon divergence on the full set of μ and Σ.
1. A method for analyzing three-dimensional structures in medical images, in order to determine the location, shape and orientation of a tumor, said method comprising the steps of:
finding at least one spatial extrema μ of a D-dimensional spatial signal f for a set of bandwidths H;
estimating a D-dimensional spread and orientation of the signal about each extrema μ by estimating a covariance Σ of the signal f for each extrema μ; and
finding a most stable estimate of μ and Σ from the at least one extrema μ and covariances Σ to find from the signal f an optimal estimate of a target structure
wherein the spatial extrema μ of signal f are found by solving an equation defined by
m
(
x
;
H
)
≡
∫
μ
Φ
(
x
-
μ
;
H
)
f
(
μ
)
ⅆ
μ
∫
Φ
(
x
-
μ
;
H
)
f
(
μ
)
ⅆ
μ
-
x
where m(x; H) is an extended fixed-bandwidth mean shift vector, x is a spatial location corresponding to a signal measurement, and Φ(x−u;H) is a Gaussian kernel for a bandwidth H.
2. The method of claim 1 , wherein Φ(x−μ; H) can be defined as
exp
(
-
1
2
D
2
(
x
,
μ
;
H
)
)
wherein D 2 (x,μ; H)=(x−μ) T H −1 (x−μ) and H −1 is a weighted harmonic mean of the bandwidth matrices,
H
-
1
(
x
)
=
∑
i
=
1
n
w
i
(
x
)
H
i
,
and wherein the weights can be defined as w i
w
i
(
x
)
=
1
H
i
1
2
exp
(
-
1
2
D
2
(
x
,
μ
;
H
i
)
)
∑
i
=
1
n
1
H
i
1
2
exp
(
-
1
2
D
2
(
x
,
μ
;
H
i
)
)
and can be normalized to unity.
3. The method of claim 1 , wherein finding the spatial extrema μ of signal f comprises the steps of:
making an estimate of an extrema and evaluating the extended fixed-bandwidth mean shift vector for this extrema, wherein y 1 is used to denote the first term of m(x; H) for the initial estimate of μ 1 ;
evaluating the extended fixed-bandwidth mean shift vector m(x; H), replacing the second term with y 1 and replacing the initial estimate of μ 1 with the previous evaluation of m(x; H);
repeating said evaluations of the extended fixed-bandwidth mean shift vector m(x; H), each time each time replacing the second term with the first term from the previous iteration, and evaluating the first term on the previous evaluation of m (x; H), until a value of μ k is found for which the extended mean shift vector m (x; H) is sufficiently close to zero; and
partitioning data points of the signal by grouping said data points that converge into the same extrema.
4. The method of claim 1 , wherein the step of estimating the covariance comprises the steps of defining the covariance by
AΣ=B
where A=(m 1 ; . . . ; m t u ) T H −T is a D×D dimensional positive definite matrix and B=(b i ; . . . ; b t u ) T , with ΣH −1 m i ≈b i ≡μ−x i −m i ; and
evaluating the equation
Σ*= U p Σ P −1 U {tilde over (Q)} Σ {tilde over (Q)} U {tilde over (Q)} T Σ P −1 U P T ;
where A T A=U P Σ P 2 U P T and B T B≡Q with {tilde over (Q)}=Σ P U P T QU P Σ P =U {tilde over (Q)} Σ {tilde over (Q)} 2 U {tilde over (Q)} T are symmetric Schur decompositions.
5. The method of claim 1 , wherein the most stable estimate of μ and Σ is found by calculating, for a neighborhood a about each bandwidth value h, a Jensen-Shannon divergence defined by
JS
(
h
)
=
1
2
log
1
2
a
+
1
∑
j
=
h
-
a
h
+
a
Σ
j
∏
j
=
h
-
a
h
+
a
2
a
+
1
Σ
j
+
1
2
∑
j
=
h
-
a
h
+
a
(
μ
j
-
μ
)
T
(
∑
j
=
h
-
a
h
+
a
Σ
j
)
-
1
(
μ
j
-
μ
)
where
μ
=
1
2
a
+
1
∑
h
-
a
h
+
a
μ
j
,
and finding an extrema of said divergences.
6. A method for analyzing three-dimensional structures in medical images, in order to determine the shape and orientation of a tumor whose location has been marked, said method comprising the steps of:
sampling a set of starting points from a neighborhood of a marker;
estimating an extrema μ by performing mean shift-based gradient-ascent iterations for a set of bandwidths H;
performing a regular sampling of points in a neighborhood of the extrema estimate μ;
estimating a spread and orientation about the extrema μ by estimating a covariance Σ of the sampling of points of the extrema μ; and
estimating the stability of μ and Σ by calculating a Jensen-Shannon divergence.
7. A method for determining the location, shape and orientation of a tumor in a medical image, said method comprising the steps of:
defining an extended fixed-bandwidth mean shift vector m(x; H) by an equation defined by
m
(
x
;
H
)
≡
∫
u
Φ
(
x
-
u
;
H
)
f
(
u
)
ⅆ
u
∫
Φ
(
x
-
u
;
H
)
f
(
u
)
ⅆ
u
-
x
wherein x is a spatial location corresponding to a signal measurement, and wherein Φ(x−μ; H) can be defined as
exp
(
-
1
2
D
2
(
x
,
μ
;
H
)
)
where D 2 (x,μ; H)=(x−μ) T H −1 (x−μ) and H −1 is a weighted harmonic mean of the bandwidth matrices,
H
-
1
(
x
)
=
∑
i
=
1
n
w
i
(
x
)
H
i
,
and where the weights can be defined as
w
i
(
x
)
=
1
H
i
1
2
exp
(
-
1
2
D
2
(
x
,
μ
;
H
i
)
)
∑
i
=
1
n
1
H
i
1
2
exp
(
-
1
2
D
2
(
x
,
μ
;
H
i
)
)
and can be normalized to unity Φ(x−μ; H);
making an estimate of an extrema and evaluating the extended fixed-bandwidth mean shift vector for this extrema, wherein y 1 is used to denote the first term of m(x; H) for the initial estimate of μ 1 ;
evaluating the extended fixed-bandwidth mean shift vector m(x; H), replacing the second term with y 1 and replacing the initial estimate of μ 1 with the previous evaluation of m(x; H);
repeating said evaluations of the extended fixed-bandwidth mean shift vector m(x; H), each time each time replacing the second term with the first term from the previous iteration, and evaluating the first term on the previous evaluation of m (x; H), until a value of μ k is found for which the extended mean shift vector m (x; H) is sufficiently close to zero;
partitioning data points of the signal by grouping said data points that converge into the same extrema;
estimating the D-dimensional spread and orientation of the signal by defining the covariance as
AΣ=B
where A=(m 1 ; . . . ; m t u ) T H −T is a D×D dimensional positive definite matrix and B=(b i ; . . . ; b t u ) T , with ΣH −1 m i ≈b i ≡μ−x i −m i and evaluating the equation
Σ
*
=
U
P
Σ
P
-
1
U
Q
~
Σ
Q
~
U
Q
~
T
Σ
P
-
1
U
P
T
;
where
A
T
A
=
U
P
Σ
P
2
U
P
T
and
B
T
B
≡
Q
with
Q
~
=
Σ
P
U
P
T
QU
P
Σ
P
=
U
Q
~
Σ
Q
~
2
U
Q
~
T
are
symmetric Schur decompositions; and
finding a most stable estimate of μ and Σ from the at least one extrema μ and covariances Σ by calculating, for a neighborhood a about each bandwidth value h, a Jensen-Shannon divergence defined by
JS
(
h
)
=
1
2
log
1
2
a
+
1
∑
j
=
h
-
a
h
+
a
Σ
j
∏
j
=
h
-
a
h
+
a
2
a
+
1
Σ
j
+
1
2
∑
j
=
h
-
a
h
+
a
(
μ
j
-
μ
)
T
(
∑
j
=
h
-
a
h
+
a
Σ
j
)
-
1
(
μ
j
-
μ
)
where
μ
=
1
2
a
+
1
∑
h
-
a
h
+
a
μ
j
,
and finding an extrema of said divergences, to find from the signal f an optimal estimate of a target structure.
8. A program storage device readable by a computer, tangibly embodying a program of instructions executable by the computer to perform the method steps for analyzing three-dimensional structures in medical images, in order to determine the location, shape and orientation of a tumor, said method steps comprising:
finding at least one spatial extrema μ of a D-dimensional spatial signal f for a set of bandwidths H;
estimating a D-dimensional spread and orientation of the signal about each extrema μ by estimating a covariance Σ of the signal f for each extrema μ; and
finding a most stable estimate of μ and Σ from the at least one extrema μ and covariances Σ to find from the signal f an optimal estimate of a target structure wherein the spatial extrema μ of signal f are found by solving an equation defined by
m
(
x
;
H
)
≡
∫
u
Φ
(
x
-
u
;
H
)
f
(
u
)
ⅆ
u
∫
Φ
(
x
-
u
;
H
)
f
(
u
)
ⅆ
u
-
x
where m(x; H) is an extended fixed-bandwidth mean shift vector, x is a spatial location corresponding to a signal measurement, and Φ(x−u; H) is a Gaussian kernel for a bandwidth H.
9. The computer readable program storage device of claim 8 , wherein Φ(x−μ; H) can be defined as
exp
(
-
1
2
D
2
(
x
,
μ
;
H
)
)
wherein D 2 (x,μ; H)=(x−μ) T H −1 is a weighted harmonic mean of the bandwidth matrices,
w
i
(
x
)
=
1
H
i
1
2
exp
(
-
1
2
D
2
(
x
,
μ
;
H
i
)
)
∑
i
=
1
n
1
H
i
1
2
exp
(
-
1
2
D
2
(
x
,
μ
;
H
i
)
)
and wherein the weights can be defined as
w
i
(
x
)
=
1
H
i
1
2
exp
(
-
1
2
D
2
(
x
,
μ
;
H
i
)
)
∑
i
=
1
n
1
H
i
1
2
exp
(
-
1
2
D
2
(
x
,
μ
;
H
i
)
)
and can be normalized to unity.
10. The computer readable program storage device of claim 8 , the method steps for finding the spatial extrema μ of signal f comprises:
making an estimate of an extrema and evaluating the extended fixed-bandwidth mean shift vector for this extrema, wherein y 1 is used to denote the first term of m(x; H) for the initial estimate of μ 1 ;
evaluating the extended fixed-bandwidth mean shift vector m(x; H), replacing the second term with y 1 and replacing the initial estimate of μ 1 with the previous evaluation of m(x; H);
repeating said evaluations of the extended fixed-bandwidth mean shift vector m(x; H), each time each time replacing the second term with the first term from the previous iteration, and evaluating the first term on the previous evaluation of m (x; H), until a value of μ k is found for which the extended mean shift vector m (x; H) is sufficiently close to zero; and
partitioning data points of the signal by grouping said data points that converge into the same extrema.
11. The computer readable program storage device of claim 8 , wherein the method steps for estimating the covariance comprises defining the covariance by
AΣ=B
where A=(m 1 ; . . . ; m t u ) T H −T is a D×D dimensional positive definite matrix and B=(b i ; . . . ; b t u ) T , with ΣH −1 m i ≈b i ≡μ−x i −m i ; and
evaluating the equation
Σ
*
=
U
p
Σ
P
-
1
U
Q
~
Σ
Q
~
U
Q
~
T
Σ
P
-
1
U
P
T
;
where
A
T
A
=
U
P
Σ
P
2
U
P
T
and
B
T
B
≡
Q
with
Q
~
=
Σ
P
U
P
T
Q
U
P
Σ
P
=
U
Q
~
Σ
Q
~
2
U
Q
~
T
are symmetric Schur decompositions.
12. The computer readable program storage device of claim 8 , wherein the method steps for finding the most stable estimate of μ and Σ comprise calculating, for a neighborhood a about each bandwidth value h, a Jensen-Shannon divergence defined by
JS
(
h
)
=
1
2
log
1
2
a
+
1
∑
j
=
h
-
a
h
+
a
Σ
j
∏
j
=
h
-
a
h
+
a
2
a
+
1
Σ
j
+
1
2
∑
j
=
h
-
a
h
+
a
(
μ
j
-
μ
)
T
(
∑
j
=
h
-
a
h
+
a
Σ
j
)
-
1
(
μ
j
-
μ
)
where
μ
=
1
2
a
+
1
∑
h
-
a
h
+
a
μ
j
,
and finding an extrema of said divergences.