Image Processing Reference
In-Depth Information
The order number and the symmetry number of a complex moment refer to
p
+
q
and
p
q
, respectively. The integrals above should be interpreted as summations when the
complex moments of discrete functions are to be computed. In the following sections
we will make use of (real and complex) moments as a spectral regression tool, i.e.,
to fit a line to the FT of a function.
−
10.3 The Structure Tensor in 2D
The
structure tensor
1
or the
direction tensor
models linearly symmetric structures
that are frequently found in images. To represent certain geometric properties of
images, it associates 2
2 symmetric matrices that are tensors to them. This is not
different from the fact that in a color image there are several color components per
image point, e.g., HSB, to every point. Typically, however, the structure tensor is used
to quantify shape properties of local images. As such, structure tensors are assigned
to every image point to represent properties of neighborhoods.
Let the scalar function
f
(
r
), taking the two-dimensional vector
r
=(
x, y
)
T
as ar-
gument, represent an image, which is usually a neighborhood around an image point.
As before, the (capitalized) letter
F
is the Fourier transform of
f
. We denote with
|
×
=(
ω
x
,ω
y
)
T
F
(
ω
)
|
the magnitude spectrum of
f
, where
ω
is the Fourier transform
|
2
we denote the power spectrum
coordinates in angular frequencies, and with
|
F
(
ω
)
of
f
. We will use the power spectrum rather than
to measure the significance of a
given frequency in the signal because it will turn out that the average values of
|
F
|
|
2
|
F
are easier to measure in practice than
.
The direction of a linearly symmetric function
f
(
r
)=
g
(
k
T
r
) is well-defined by
the vector
k
, but only up to a sign factor. According to lemma 10.1, if and only if
f
is
linearly symmetric is its power spectrum,
|
F
|
|
2
concentrated to a central line with the
direction
k
. The direction of this line represents the direction of the linear symmetry.
We will approach estimating
k
by fitting the image power spectrum,
|
F
|F |
2
, a line in
the total least square TLS sense. Consequently, it will be possible to “measure” if
f
is linearly symmetric by studying the error of the fit. If the error is “small” in the
sense that has been defined, then our method will take this as a provision that the fit
was successful and that the image approximates a linearly symmetric image
g
(
k
T
r
)
well. It turns out that, in this procedure,
g
need not be known beforehand. It will be
automatically determined when the error of the fit is near zero, because we will then
obtain a reliable direction along which to “cut” the image. In turn the 1D function
obtained by cutting the image is
g
only if
f
is linearly symmetric. Owing to the
continuity of the TLS error function, the decrease or the increase of the error will be
graceful when
f
approaches to a linearly symmetric function or departs from one.
We discuss the details of the line-fitting next.
We wish to fit an axis through the origin of the Fourier transform of an image,
f
,
which
may or may not
be linearly symmetric. Fitting an axis to a finite set of points
is classically performed by minimizing the error function:
1
Other names of this tensor include the “second order moment tensor”, “inertia tensor”,
“outer product tensor”, and “covariance matrix”.
