<s>
In	O
image	B-General_Concept
analysis	I-General_Concept
,	O
the	O
generalized	B-Algorithm
structure	I-Algorithm
tensor	I-Algorithm
(	O
GST	O
)	O
is	O
an	O
extension	O
of	O
the	O
Cartesian	O
structure	B-Algorithm
tensor	I-Algorithm
to	O
curvilinear	O
coordinates	O
.	O
</s>
<s>
It	O
is	O
mainly	O
used	O
to	O
detect	O
and	O
to	O
represent	O
the	O
"	O
direction	O
"	O
parameters	O
of	O
curves	O
,	O
just	O
as	O
the	O
Cartesian	O
structure	B-Algorithm
tensor	I-Algorithm
detects	O
and	O
represents	O
the	O
direction	O
in	O
Cartesian	O
coordinates	O
.	O
</s>
<s>
It	O
is	O
a	O
widely	O
known	O
method	O
in	O
applications	O
of	O
image	O
and	O
video	O
processing	O
including	O
computer	B-Application
vision	I-Application
,	O
such	O
as	O
biometric	O
identification	O
by	O
fingerprints	O
,	O
and	O
studies	O
of	O
human	O
tissue	O
sections	O
.	O
</s>
<s>
GST	O
represents	O
the	O
direction	O
along	O
which	O
the	O
image	O
can	O
undergo	O
an	O
infinitesimal	O
translation	B-Algorithm
with	O
minimal	O
(	O
total	B-Algorithm
least	I-Algorithm
squares	I-Algorithm
)	O
error	O
,	O
along	O
the	O
"	O
lines	O
"	O
fulfilling	O
the	O
following	O
conditions	O
:	O
</s>
<s>
where	O
are	O
errors	O
of	O
(	O
infinitesimal	O
)	O
translation	B-Algorithm
in	O
the	O
best	O
direction	O
(	O
designated	O
by	O
the	O
angle	O
)	O
and	O
the	O
worst	O
direction	O
(	O
designated	O
by	O
)	O
.	O
</s>
<s>
The	O
function	O
is	O
the	O
window	B-Language
function	I-Language
defining	O
the	O
"	O
outer	O
scale	O
"	O
wherein	O
the	O
detection	O
of	O
will	O
be	O
carried	O
out	O
,	O
which	O
can	O
be	O
omitted	O
if	O
it	O
is	O
already	O
included	O
in	O
or	O
if	O
is	O
the	O
full	O
image	O
(	O
rather	O
than	O
local	O
)	O
.	O
</s>
<s>
The	O
matrix	O
is	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Using	O
the	O
chain	O
rule	O
,	O
it	O
can	O
be	O
shown	O
that	O
the	O
integration	O
above	O
can	O
be	O
implemented	O
as	O
convolutions	B-Language
in	O
Cartesian	O
coordinates	O
applied	O
to	O
the	O
ordinary	O
structure	B-Algorithm
tensor	I-Algorithm
when	O
pair	O
the	O
real	O
and	O
imaginary	O
parts	O
of	O
an	O
analytic	B-Language
function	I-Language
,	O
</s>
<s>
Examples	O
of	O
analytic	B-Language
functions	I-Language
include	O
,	O
as	O
well	O
as	O
monomials	O
,	O
,	O
where	O
is	O
an	O
arbitrary	O
positive	O
or	O
negative	O
integer	O
.	O
</s>
<s>
The	O
monomials	O
are	O
also	O
referred	O
to	O
as	O
harmonic	O
functions	O
in	O
computer	B-Application
vision	I-Application
,	O
and	O
image	B-Algorithm
processing	I-Algorithm
.	O
</s>
<s>
Thereby	O
,	O
Cartesian	O
Structure	B-Algorithm
tensor	I-Algorithm
is	O
a	O
special	O
case	O
of	O
GST	O
where	O
,	O
and	O
,	O
i.e.	O
</s>
<s>
Thus	O
by	O
choosing	O
a	O
harmonic	O
function	O
,	O
one	O
can	O
detect	O
all	O
curves	O
that	O
are	O
linear	O
combinations	O
of	O
its	O
real	O
and	O
imaginary	O
parts	O
by	O
convolutions	B-Language
on	O
(	O
rectangular	O
)	O
image	O
grids	O
only	O
,	O
even	O
if	O
are	O
non-Cartesian	O
.	O
</s>
<s>
Furthermore	O
,	O
the	O
convolution	B-Language
computations	O
can	O
be	O
done	O
by	O
using	O
complex	O
filters	O
applied	O
to	O
the	O
complex	O
version	O
of	O
the	O
structure	B-Algorithm
tensor	I-Algorithm
.	O
</s>
<s>
Thus	O
,	O
GST	O
implementations	O
have	O
frequently	O
been	O
done	O
using	O
complex	O
version	O
of	O
the	O
structure	B-Algorithm
tensor	I-Algorithm
,	O
rather	O
than	O
using	O
the	O
(	O
1	O
,	O
1	O
)	O
tensor	B-Device
.	O
</s>
<s>
It	O
should	O
be	O
recalled	O
that	O
,	O
the	O
ordinary	O
structure	B-Algorithm
tensor	I-Algorithm
is	O
a	O
real	O
filter	O
,	O
usually	O
defined	O
by	O
a	O
sampled	O
and	O
scaled	O
Gaussian	B-Application
to	O
delineate	O
the	O
neighborhood	O
,	O
also	O
known	O
as	O
the	O
outer	O
scale	O
.	O
</s>
<s>
For	O
curve	O
families	O
defined	O
by	O
analytic	B-Language
functions	I-Language
,	O
it	O
can	O
be	O
shown	O
that	O
,	O
the	O
neighborhood	O
defining	O
function	O
is	O
complex	O
valued	O
,	O
</s>
<s>
a	O
so	O
called	O
symmetry	O
derivative	O
of	O
a	O
Gaussian	B-Application
.	O
</s>
<s>
Thus	O
,	O
the	O
orientation	O
wise	O
variation	O
of	O
the	O
pattern	O
to	O
be	O
looked	O
for	O
is	O
directly	O
incorporated	O
into	O
the	O
neighborhood	O
defining	O
function	O
,	O
and	O
the	O
detection	O
occurs	O
in	O
the	O
space	O
of	O
the	O
(	O
ordinary	O
)	O
structure	B-Algorithm
tensor	I-Algorithm
.	O
</s>
<s>
Efficient	O
detection	O
of	O
in	O
images	O
is	O
possible	O
by	O
image	B-Algorithm
processing	I-Algorithm
for	O
a	O
pair	O
,	O
.	O
</s>
<s>
Complex	O
convolutions	B-Language
(	O
or	O
the	O
corresponding	O
matrix	O
operations	O
)	O
and	O
point-wise	O
non-linear	O
mappings	O
are	O
the	O
basic	O
computational	O
elements	O
of	O
GST	O
implementations	O
.	O
</s>
<s>
In	O
analogy	O
with	O
the	O
Cartesian	O
structure	B-Algorithm
tensor	I-Algorithm
,	O
the	O
estimated	O
angle	O
is	O
in	O
double	O
angle	O
representation	O
,	O
i.e.	O
</s>
<s>
Logarithmic	O
spirals	O
,	O
including	O
circles	O
,	O
can	O
for	O
instance	O
be	O
detected	O
by	O
(	O
complex	O
)	O
convolutions	B-Language
and	O
non-linear	O
mappings	O
.	O
</s>
<s>
The	O
spirals	O
can	O
be	O
in	O
gray	O
(	O
valued	O
)	O
images	O
or	O
in	O
a	O
binary	B-Algorithm
image	I-Algorithm
,	O
i.e.	O
</s>
<s>
Generalized	B-Algorithm
structure	I-Algorithm
tensor	I-Algorithm
can	O
be	O
used	O
as	O
an	O
alternative	O
to	O
Hough	B-Algorithm
transform	I-Algorithm
in	O
image	B-Algorithm
processing	I-Algorithm
and	O
computer	B-Application
vision	I-Application
to	O
detect	O
patterns	O
whose	O
local	O
orientations	O
can	O
be	O
modelled	O
,	O
for	O
example	O
junction	O
points	O
.	O
</s>
<s>
Analogously	O
,	O
the	O
Cartesian	O
structure	B-Algorithm
tensor	I-Algorithm
is	O
a	O
representation	O
of	O
a	O
translation	B-Algorithm
too	O
.	O
</s>
<s>
Here	O
the	O
physical	O
process	O
consists	O
in	O
an	O
ordinary	O
translation	B-Algorithm
of	O
a	O
certain	O
amount	O
along	O
combined	O
with	O
translation	B-Algorithm
along	O
,	O
</s>
