Image Processing Reference
In-Depth Information
Fig. 14.4. The gradient directions (
left
) in the local images of FMTEST3 and the correspond-
ing group directions (
right
). To avoid clutter, the six gradient direction types that can occur in
local images, e.g., the
colored region
, are drawn at the same point
in the image is higher than one. We will call such images
n
-folded symmetric
,asthe
following definition describes:
Definition 14.1.
Let
F
be the Fourier transform of an image
f
.If
F
is zero except
on a set of lines, and any of these lines can be obtained from another by an integer
multiple of
2
n
rotations, then
f
is
n
-folded symmetric.
Because the FT is linear, the
n
-folded symmetric images can be viewed as the
superposition of linearly symmetric images that differ directionwise maximally. The
isocurves of linearly symmetric images are parallel lines. Accordingly, the definition
suggests an extension of the linear symmetry concept to high-order symmetries by
isocurve directions of ordinary lines that are jointly present in the image. Note that
this extension is different than the generalized structure tensor discussed in Chap. 11,
which measures the amount of linear symmetry in non-Cartesian coordinates, i.e. it
still represents the presence of a single direction, albeit in a different coordinate
system.
Because
is invariant to translations of
f
, a quantity that represents fitting of
multiple lines to it will also be invariant to translations. Consequently, fitting multiple
lines to
|
F
|
|
2
, which is easier to do than to
, has useful consequences to image
analysis. If the fitting error is reasonably low, the obtained directions of individual
lines are the gradient directions of the individual isocurves of
f
. Using the mappings
discussed in Sect. 14.1, it will then be straightforward to assign a meaningful and
unique group direction to
f
.
We will first discuss fitting a cross to
F
in the TLS sense. The error function
associated with this problem is not as straightforward to formulate as in Chapter 10.
The approach taken here is first to map
F
to another function through a two-to-one
transformation, and then to view it as a line-fitting problem, which we know how to
|
F
|
F
|
