Image Processing Reference
In-Depth Information
−
k
T
Sk
e
(
k
)=Trace(
S
)
(10.62)
Accordingly, minimizing Eq. (10.62) yields the direction of
minimum translation
error
. Evidently maximizing Eq. (10.62) yields the direction of
maximum translation
error
. Both minimum and maximum error directions are given by the eigenvectors of
S
. Paraphrased, the structure tensor encodes the
minimum resistance direction
in the
spatial domain, which is identical to the direction of the line fit to the power spectrum
in the TLS sense.
10.11 Discrete Structure Tensor by Direct Tensor Sampling
Until this section, the theory for detection of the orientation of a scalar function in
2D space has been based on continuous signals. One such technique was summarized
by theorem 10.1 which we will attempt to approximate by use of discrete functions.
We call this approach
direct tensor sampling
since the suggested method examines
whether or not the spectrum of an image consists of a line, by directly estimating the
matrix
S
, with the elements given by Eq. (10.28) without first estimating the power
spectrum by a discrete local spectrum.
We need to approximate the continuous integrand of Eq. (10.28) from a discrete
image. To that end, we need the approximation of
∂f
(
r
)
∂x
i
∂f
(
r
)
∂x
j
,
with
i, j
=1
,
2
,
1
=
x,
and
x
2
=
y,
(10.63)
on a Cartesian grid i.e.,
r
=
r
l
, where
r
l
is the coordinates of the grid nodes. In
analogy with the theory presented in Sects. 8.2, 8.3, and 9.2, we can do this by
filtering the original image linearly:
=
k
∂f
(
r
l
)
∂x
i
f
(
r
l
+
r
k
)
∂μ
(
r
l
)
∂x
i
with
i
=1
,
2
(10.64)
∂μ
(
r
l
)
∂x
i
=
∂μ
(
r
)
∂x
i
∂f
(
r
l
)
∂x
i
=
∂f
(
r
)
where
∂x
i
|
r
=
r
l
, and then applying pointwise multi-
plication between the two thus-obtained discrete partial derivative images:
|
r
=
r
l
,
∂f
(
r
l
)
∂x
i
∂f
(
r
l
)
∂x
j
,
with
i, j
=1
,
2
,
1
=
x,
and
x
2
=
y.
(10.65)
The latter is an estimate of Eq. (10.63) on a Cartesian grid. Note that the continuous
form of Eq. (10.63) is not known, but we estimated nevertheless its discrete version
by applying a linear discrete filtering to the discrete
f
(
r
l
), followed by a pointwise
multiplication on the grid
r
l
.
To estimate the structure tensor elements, Eq. (10.28), we first reconstruct (10.63)
from its samples (10.65):
=
l
∂f
(
r
)
∂x
i
∂f
(
r
)
∂x
j
∂f
(
r
l
)
∂x
i
∂f
(
r
l
)
∂x
j
μ
(
r
−
r
l
)
i, j
:1
,
2
(10.66)
