Biomedical Engineering Reference
In-Depth Information
3D Structure Tensors
Structure tensors are a matrix representation of partial derivative information. As
they allow both orientation estimation and image structure analysis, they have
many applications in image processing and computer vision. 2D structure tensors
have been widely used in edge/corner detection and texture analysis; 3D structure
tensors have been used in low-level motion estimation and segmentation [6, 7].
Under the constant illumination model, the optic-flow (OF) equation of a spa-
tiotemporal image volume
I
(
x
)
centered at location
x
=[
x
,
y
,
t
]
is given by (3.1) [8]
where,
v
(
x
)=[
v
x
,
v
y
,
v
t
]
is the optic-flow vector at
x
,
d
I
(
x
)
¶
I
(
x
)
¶
I
(
x
)
¶
I
(
x
)
=
v
x
+
v
y
+
v
t
dt
¶
x
¶
y
¶
t
I
T
=
∇
(
x
)
v
(
x
)=
0
(3.1)
is estimated by minimizing (3.1) over a local 3D image patch W
and
v
,
centered at
x
. Note that
v
t
is not 1 since spatiotemporal orientation vectors will be
computed. Using Lagrange multipliers, a corresponding error functional
e
ls
(
(
x
)
(
x
,
y
)
)
to
minimize (3.1) using a least-squares error measure can be written as (3.2) where
W
x
is a
spatially invariant
weighting function (e.g., Gaussian) that emphasizes
the image gradients near the central pixel [7].
(
x
,
y
)
I
T
2
e
ls
(
)=
)
(
∇
(
)
(
))
W
(
,
)
x
y
v
x
x
y
d
y
W
(
x
,
y
l
1
T
v
−
+
v
(
x
)
(
x
)
(3.2)
Assuming a constant
v
)
to find the minimum, leads to the standard eigenvalue problem (3.3) for solving
v
(
x
)
within the neighborhood W
(
x
,
y
)
and differentiating
e
ls
(
x
(
x
)
the best estimate of
v
(
x
)
,
(
,
)
v
(
)=
l
v
(
)
J
x
W
x
x
(3.3)
The 3D structure tensor matrix
J
for the spatiotemporal volume centered at
x
can be written in expanded matrix form, without the spatial filter
W
(
(
x
,
W
)
x
,
y
)
and
the positional terms shown for clarity, as
⎡
⎣
⎤
⎦
¶
I
¶
x
¶
I
¶
x
d
y
¶
I
¶
x
¶
I
¶
y
d
y
¶
I
¶
x
¶
I
¶
t
d
y
W
W
W
¶
I
¶
y
¶
I
¶
x
d
y
¶
I
¶
y
¶
I
¶
y
d
y
¶
I
¶
y
¶
I
¶
t
d
y
J
=
(3.4)
W
W
W
¶
I
¶
t
¶
I
¶
x
d
y
¶
I
¶
t
¶
I
¶
y
d
y
¶
I
¶
t
¶
I
¶
t
d
y
W
W
W
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