Image Processing Reference
In-Depth Information
11
Direction in Curvilinear Coordinates
This chapter provides a general technique for detection of patterns possessing lin-
ear symmetry, with respect to curvilinear coordinates in 2D images. Curves given
by a
harmonic function pair
(HFP) will be discussed in detail. The idea is to “bend
and twist” the image by means of an HFP so that the patterns can be detected by
the same formalism that we developed for the structure tensor. This will lead us to
the concepts of
coordinate transformations
(CT) and
generalized structure tensor
(GST), which will be discussed further from the viewpoint of pattern recognition.
Since very intricate patterns can be described by such CTs, the technique is a general
toolbox for pattern detection. We will also develop a unifying concept for geometric
shape quantitation and detection, to the effect that the generalized structure tensor
becomes an extension of the generalized Hough transform, with the additional prop-
erty that it is also capable of handling negative votes as well as complex-valued votes.
In the generalized structure tensor theory, the detection of intricate target objects is
equivalent to a problem of symmetry detection in the HFP coordinate system. The
generalized structure tensor does not necessitate explicit coordinate transformations.
Instead, via Lie operators, the “bending and twisting” occurs in the complex-valued
filters implicitly once for all, rather than being performed explicitly on every image
to be recognized.
11.1 Curvilinear Coordinates by Harmonic Functions
Let
ξ
(
x, y
) be a
harmonic function,
that is, its partial derivatives of the first two
orders are continuous and it satisfies the
Laplace equation:
∂x
2
+
∂
2
ξ
∂
2
ξ
Δξ
=
=0
.
(11.1)
∂y
2
Due to the linearity of Laplace's equation, linear combinations of harmonic functions
are also harmonic. If two harmonic functions
ξ
and
η
satisfy the
Cauchy-Riemann
equations:
