<s>
In	O
mathematics	O
,	O
the	O
structure	B-Algorithm
tensor	I-Algorithm
,	O
also	O
referred	O
to	O
as	O
the	O
second-moment	B-Algorithm
matrix	I-Algorithm
,	O
is	O
a	O
matrix	B-Architecture
derived	O
from	O
the	O
gradient	O
of	O
a	O
function	O
.	O
</s>
<s>
The	O
structure	B-Algorithm
tensor	I-Algorithm
is	O
often	O
used	O
in	O
image	B-Algorithm
processing	I-Algorithm
and	O
computer	B-Application
vision	I-Application
.	O
</s>
<s>
where	O
and	O
are	O
the	O
partial	O
derivatives	O
of	O
with	O
respect	O
to	O
x	O
and	O
y	O
;	O
the	O
integrals	O
range	O
over	O
the	O
plane	O
;	O
and	O
w	O
is	O
some	O
fixed	O
"	O
window	O
function	O
"	O
(	O
such	O
as	O
a	O
Gaussian	B-Error_Name
blur	I-Error_Name
)	O
,	O
a	O
distribution	O
on	O
two	O
variables	O
.	O
</s>
<s>
Note	O
that	O
the	O
matrix	B-Architecture
is	O
itself	O
a	O
function	O
of	O
.	O
</s>
<s>
If	O
the	O
gradient	O
of	O
is	O
viewed	O
as	O
a	O
2×1	O
(	O
single-column	O
)	O
matrix	B-Architecture
,	O
where	O
denotes	O
transpose	O
operation	O
,	O
turning	O
a	O
row	O
vector	O
to	O
a	O
column	O
vector	O
,	O
the	O
matrix	B-Architecture
can	O
be	O
written	O
as	O
the	O
matrix	B-Architecture
product	O
or	O
tensor	B-Device
or	O
outer	O
product	O
.	O
</s>
<s>
Note	O
however	O
that	O
the	O
structure	B-Algorithm
tensor	I-Algorithm
cannot	O
be	O
factored	O
in	O
this	O
way	O
in	O
general	O
except	O
if	O
is	O
a	O
Dirac	O
delta	O
function	O
.	O
</s>
<s>
In	O
image	B-Algorithm
processing	I-Algorithm
and	O
other	O
similar	O
applications	O
,	O
the	O
function	O
is	O
usually	O
given	O
as	O
a	O
discrete	O
array	B-Data_Structure
of	O
samples	O
,	O
where	O
p	O
is	O
a	O
pair	O
of	O
integer	O
indices	O
.	O
</s>
<s>
The	O
values	O
are	O
the	O
partial	O
derivatives	O
sampled	O
at	O
pixel	B-Algorithm
p	O
;	O
which	O
,	O
for	O
instance	O
,	O
may	O
be	O
estimated	O
from	O
by	O
by	O
finite	B-Algorithm
difference	I-Algorithm
formulas	O
.	O
</s>
<s>
The	O
importance	O
of	O
the	O
2D	O
structure	B-Algorithm
tensor	I-Algorithm
stems	O
from	O
the	O
fact	O
eigenvalues	O
(	O
which	O
can	O
be	O
ordered	O
so	O
that	O
)	O
and	O
the	O
corresponding	O
eigenvectors	O
summarize	O
the	O
distribution	O
of	O
the	O
gradient	O
of	O
within	O
the	O
window	O
defined	O
by	O
centered	O
at	O
.	O
</s>
<s>
The	O
formula	O
is	O
undefined	O
,	O
even	O
in	O
the	O
limit	B-Algorithm
,	O
when	O
the	O
image	O
is	O
constant	O
in	O
the	O
window	O
(	O
)	O
.	O
</s>
<s>
Aligned	O
but	O
oppositely	O
oriented	O
gradient	O
vectors	O
would	O
cancel	O
out	O
in	O
this	O
average	O
,	O
whereas	O
in	O
the	O
structure	B-Algorithm
tensor	I-Algorithm
they	O
are	O
properly	O
added	O
together	O
.	O
</s>
<s>
This	O
is	O
a	O
reason	O
for	O
why	O
is	O
used	O
in	O
the	O
averaging	O
of	O
the	O
structure	B-Algorithm
tensor	I-Algorithm
to	O
optimize	O
the	O
direction	O
instead	O
of	O
.	O
</s>
<s>
By	O
expanding	O
the	O
effective	O
radius	O
of	O
the	O
window	O
function	O
(	O
that	O
is	O
,	O
increasing	O
its	O
variance	O
)	O
,	O
one	O
can	O
make	O
the	O
structure	B-Algorithm
tensor	I-Algorithm
more	O
robust	O
in	O
the	O
face	O
of	O
noise	O
,	O
at	O
the	O
cost	O
of	O
diminished	O
spatial	O
resolution	O
.	O
</s>
<s>
The	O
formal	O
basis	O
for	O
this	O
property	O
is	O
described	O
in	O
more	O
detail	O
below	O
,	O
where	O
it	O
is	O
shown	O
that	O
a	O
multi-scale	O
formulation	O
of	O
the	O
structure	B-Algorithm
tensor	I-Algorithm
,	O
referred	O
to	O
as	O
the	O
multi-scale	O
structure	B-Algorithm
tensor	I-Algorithm
,	O
constitutes	O
a	O
true	O
multi-scale	O
representation	O
of	O
directional	O
data	O
under	O
variations	O
of	O
the	O
spatial	O
extent	O
of	O
the	O
window	O
function	O
.	O
</s>
<s>
The	O
interpretation	O
and	O
implementation	O
of	O
the	O
2D	O
structure	B-Algorithm
tensor	I-Algorithm
becomes	O
particularly	O
accessible	O
using	O
complex	O
numbers	O
.	O
</s>
<s>
where	O
and	O
is	O
the	O
direction	O
angle	O
of	O
the	O
most	O
significant	O
eigenvector	O
of	O
the	O
structure	B-Algorithm
tensor	I-Algorithm
whereas	O
and	O
are	O
the	O
most	O
and	O
the	O
least	O
significant	O
eigenvalues	O
.	O
</s>
<s>
where	O
is	O
the	O
identity	O
matrix	B-Architecture
in	O
2D	O
because	O
the	O
two	O
eigenvectors	O
are	O
always	O
orthogonal	O
(	O
and	O
sum	O
to	O
unity	O
)	O
.	O
</s>
<s>
The	O
first	O
term	O
in	O
the	O
last	O
expression	O
of	O
the	O
decomposition	O
,	O
,	O
represents	O
the	O
linear	O
symmetry	O
component	O
of	O
the	O
structure	B-Algorithm
tensor	I-Algorithm
containing	O
all	O
directional	O
information	O
(	O
as	O
a	O
rank-1	O
matrix	B-Architecture
)	O
,	O
whereas	O
the	O
second	O
term	O
represents	O
the	O
balanced	O
body	O
component	O
of	O
the	O
tensor	B-Device
,	O
which	O
lacks	O
any	O
directional	O
information	O
(	O
containing	O
an	O
identity	O
matrix	B-Architecture
)	O
.	O
</s>
<s>
Evidently	O
,	O
is	O
the	O
complex	O
equivalent	O
of	O
the	O
first	O
term	O
in	O
the	O
tensor	B-Device
decomposition	O
,	O
whereas	O
is	O
the	O
equivalent	O
of	O
the	O
second	O
term	O
.	O
</s>
<s>
where	O
is	O
the	O
(	O
complex	O
)	O
gradient	O
filter	O
,	O
and	O
is	O
convolution	O
,	O
constitute	O
a	O
complex	O
representation	O
of	O
the	O
2D	O
Structure	B-Algorithm
Tensor	I-Algorithm
.	O
</s>
<s>
As	O
discussed	O
here	O
and	O
elsewhere	O
defines	O
the	O
local	O
image	O
which	O
is	O
usually	O
a	O
Gaussian	B-Application
(	O
with	O
a	O
certain	O
variance	O
defining	O
the	O
outer	O
scale	O
)	O
,	O
and	O
is	O
the	O
(	O
inner	O
scale	O
)	O
parameter	O
determining	O
the	O
effective	O
frequency	O
range	O
in	O
which	O
the	O
orientation	O
is	O
to	O
be	O
estimated	O
.	O
</s>
<s>
The	O
elegance	O
of	O
the	O
complex	O
representation	O
stems	O
from	O
that	O
the	O
two	O
components	O
of	O
the	O
structure	B-Algorithm
tensor	I-Algorithm
can	O
be	O
obtained	O
as	O
averages	O
and	O
independently	O
.	O
</s>
<s>
In	O
turn	O
,	O
this	O
means	O
that	O
and	O
can	O
be	O
used	O
in	O
a	O
scale	B-Algorithm
space	I-Algorithm
representation	I-Algorithm
to	O
describe	O
the	O
evidence	O
for	O
presence	O
of	O
unique	O
orientation	O
and	O
the	O
evidence	O
for	O
the	O
alternative	O
hypothesis	O
,	O
the	O
presence	O
of	O
multiple	O
balanced	O
orientations	O
,	O
without	O
computing	O
the	O
eigenvectors	O
and	O
eigenvalues	O
.	O
</s>
<s>
A	O
functional	O
,	O
such	O
as	O
squaring	O
the	O
complex	O
numbers	O
have	O
to	O
this	O
date	O
not	O
been	O
shown	O
to	O
exist	O
for	O
structure	B-Algorithm
tensors	I-Algorithm
with	O
dimensions	O
higher	O
than	O
two	O
.	O
</s>
<s>
The	O
complex	O
representation	O
of	O
the	O
structure	B-Algorithm
tensor	I-Algorithm
is	O
frequently	O
used	O
in	O
fingerprint	O
analysis	O
to	O
obtain	O
direction	O
maps	O
containing	O
certainties	O
which	O
in	O
turn	O
are	O
used	O
to	O
enhance	O
them	O
,	O
to	O
find	O
the	O
locations	O
of	O
the	O
global	O
(	O
cores	O
and	O
deltas	O
)	O
and	O
local	O
(	O
minutia	O
)	O
singularities	O
,	O
as	O
well	O
as	O
automatically	O
evaluate	O
the	O
quality	O
of	O
the	O
fingerprints	O
.	O
</s>
<s>
The	O
structure	B-Algorithm
tensor	I-Algorithm
can	O
be	O
defined	O
also	O
for	O
a	O
function	O
of	O
three	O
variables	O
p	O
=(	O
x	O
,	O
y	O
,	O
z	O
)	O
in	O
an	O
entirely	O
analogous	O
way	O
.	O
</s>
<s>
thumb|180px|The	O
structure	B-Algorithm
tensor	I-Algorithm
ellipsoid	O
of	O
a	O
surface-like	O
neighborhood	O
(	O
"	O
surfel	B-General_Concept
"	O
)	O
,	O
where	O
.thumb	O
|180px|A	O
3D	O
window	O
straddling	O
a	O
smooth	O
boundary	O
surface	O
between	O
two	O
uniform	O
regions	O
of	O
a	O
3D	O
image.thumb	O
|180px|The	O
corresponding	O
structure	B-Algorithm
tensor	I-Algorithm
ellipsoid	O
.	O
</s>
<s>
thumb|180px|The	O
structure	B-Algorithm
tensor	I-Algorithm
of	O
a	O
line-like	O
neighborhood	O
(	O
"	O
curvel	O
"	O
)	O
,	O
where	O
.thumb	O
|180px|A	O
3D	O
window	O
straddling	O
a	O
line-like	O
feature	O
of	O
a	O
3D	O
image.thumb	O
|180px|The	O
corresponding	O
structure	B-Algorithm
tensor	I-Algorithm
ellipsoid	O
.	O
</s>
<s>
thumb|180px|The	O
structure	B-Algorithm
tensor	I-Algorithm
in	O
an	O
isotropic	O
neighborhood	O
,	O
where	O
.thumb	O
|180px|A	O
3D	O
window	O
containing	O
a	O
spherical	O
feature	O
of	O
a	O
3D	O
image.thumb	O
|180px|The	O
corresponding	O
structure	B-Algorithm
tensor	I-Algorithm
ellipsoid	O
.	O
</s>
<s>
The	O
structure	B-Algorithm
tensor	I-Algorithm
is	O
an	O
important	O
tool	O
in	O
scale	B-Algorithm
space	I-Algorithm
analysis	O
.	O
</s>
<s>
The	O
multi-scale	O
structure	B-Algorithm
tensor	I-Algorithm
(	O
or	O
multi-scale	O
second	O
moment	O
matrix	B-Architecture
)	O
of	O
a	O
function	O
is	O
in	O
contrast	O
to	O
other	O
one-parameter	O
scale-space	B-Algorithm
features	O
an	O
image	O
descriptor	O
that	O
is	O
defined	O
over	O
two	O
scale	O
parameters	O
.	O
</s>
<s>
Furthermore	O
,	O
let	O
denote	O
the	O
gradient	O
of	O
the	O
scale	B-Algorithm
space	I-Algorithm
representation	I-Algorithm
.	O
</s>
<s>
If	O
one	O
wants	O
the	O
multi-scale	O
structure	B-Algorithm
tensor	I-Algorithm
to	O
be	O
well-behaved	O
over	O
both	O
increasing	O
local	O
scales	O
and	O
increasing	O
integration	O
scales	O
,	O
then	O
it	O
can	O
be	O
shown	O
that	O
both	O
the	O
smoothing	O
function	O
and	O
the	O
window	O
function	O
have	O
to	O
be	O
Gaussian	B-Application
.	O
</s>
<s>
The	O
conditions	O
that	O
specify	O
this	O
uniqueness	O
are	O
similar	O
to	O
the	O
scale-space	B-Algorithm
axioms	I-Algorithm
that	O
are	O
used	O
for	O
deriving	O
the	O
uniqueness	O
of	O
the	O
Gaussian	B-Application
kernel	O
for	O
a	O
regular	O
Gaussian	B-Application
scale	B-Algorithm
space	I-Algorithm
of	O
image	O
intensities	O
.	O
</s>
<s>
If	O
we	O
keep	O
the	O
local	O
scale	O
parameter	O
fixed	O
and	O
apply	O
increasingly	O
broadened	O
versions	O
of	O
the	O
window	O
function	O
by	O
increasing	O
the	O
integration	O
scale	O
parameter	O
only	O
,	O
then	O
we	O
obtain	O
a	O
true	O
formal	O
scale	B-Algorithm
space	I-Algorithm
representation	I-Algorithm
of	O
the	O
directional	O
data	O
computed	O
at	O
the	O
given	O
local	O
scale	O
.	O
</s>
<s>
If	O
we	O
couple	O
the	O
local	O
scale	O
and	O
integration	O
scale	O
by	O
a	O
relative	O
integration	O
scale	O
,	O
such	O
that	O
then	O
for	O
any	O
fixed	O
value	O
of	O
,	O
we	O
obtain	O
a	O
reduced	O
self-similar	O
one-parameter	O
variation	O
,	O
which	O
is	O
frequently	O
used	O
to	O
simplify	O
computational	O
algorithms	O
,	O
for	O
example	O
in	O
corner	B-Algorithm
detection	I-Algorithm
,	O
interest	O
point	O
detection	O
,	O
texture	O
analysis	O
and	O
image	B-Algorithm
matching	I-Algorithm
.	O
</s>
<s>
A	O
conceptually	O
similar	O
construction	O
can	O
be	O
performed	O
for	O
discrete	O
signals	O
,	O
with	O
the	O
convolution	O
integral	O
replaced	O
by	O
a	O
convolution	O
sum	O
and	O
with	O
the	O
continuous	O
Gaussian	B-Application
kernel	O
replaced	O
by	O
the	O
discrete	O
Gaussian	B-Application
kernel	O
:	O
</s>
<s>
When	O
quantizing	O
the	O
scale	O
parameters	O
and	O
in	O
an	O
actual	O
implementation	O
,	O
a	O
finite	O
geometric	O
progression	O
is	O
usually	O
used	O
,	O
with	O
i	O
ranging	O
from	O
0	O
to	O
some	O
maximum	O
scale	O
index	O
m	O
.	O
Thus	O
,	O
the	O
discrete	O
scale	O
levels	O
will	O
bear	O
certain	O
similarities	O
to	O
image	B-Algorithm
pyramid	I-Algorithm
,	O
although	O
spatial	O
subsampling	O
may	O
not	O
necessarily	O
be	O
used	O
in	O
order	O
to	O
preserve	O
more	O
accurate	O
data	O
for	O
subsequent	O
processing	O
stages	O
.	O
</s>
<s>
The	O
eigenvalues	O
of	O
the	O
structure	B-Algorithm
tensor	I-Algorithm
play	O
a	O
significant	O
role	O
in	O
many	O
image	B-Algorithm
processing	I-Algorithm
algorithms	I-Algorithm
,	O
for	O
problems	O
like	O
corner	B-Algorithm
detection	I-Algorithm
,	O
interest	O
point	O
detection	O
,	O
and	O
feature	O
tracking	O
.	O
</s>
<s>
The	O
structure	B-Algorithm
tensor	I-Algorithm
also	O
plays	O
a	O
central	O
role	O
in	O
the	O
Lucas-Kanade	O
optical	O
flow	O
algorithm	O
,	O
and	O
in	O
its	O
extensions	O
to	O
estimate	O
affine	B-Algorithm
shape	I-Algorithm
adaptation	I-Algorithm
;	O
where	O
the	O
magnitude	O
of	O
is	O
an	O
indicator	O
of	O
the	O
reliability	O
of	O
the	O
computed	O
result	O
.	O
</s>
<s>
The	O
tensor	B-Device
has	O
been	O
used	O
for	O
scale	B-Algorithm
space	I-Algorithm
analysis	O
,	O
estimation	O
of	O
local	O
surface	O
orientation	O
from	O
monocular	O
or	O
binocular	O
cues	O
,	O
non-linear	O
fingerprint	O
enhancement	O
,	O
diffusion-based	B-Algorithm
image	I-Algorithm
processing	I-Algorithm
,	O
and	O
several	O
other	O
image	B-Algorithm
processing	I-Algorithm
problems	O
.	O
</s>
<s>
The	O
structure	B-Algorithm
tensor	I-Algorithm
can	O
be	O
also	O
applied	O
in	O
geology	O
to	O
filter	O
seismic	O
data	O
.	O
</s>
<s>
The	O
three-dimensional	O
structure	B-Algorithm
tensor	I-Algorithm
has	O
been	O
used	O
to	O
analyze	O
three-dimensional	O
video	O
data	O
(	O
viewed	O
as	O
a	O
function	O
of	O
x	O
,	O
y	O
,	O
and	O
time	O
t	O
)	O
.	O
</s>
<s>
To	O
obtain	O
true	O
Galilean	O
invariance	O
,	O
however	O
,	O
also	O
the	O
shape	O
of	O
the	O
spatio-temporal	O
window	O
function	O
needs	O
to	O
be	O
adapted	O
,	O
corresponding	O
to	O
the	O
transfer	O
of	O
affine	B-Algorithm
shape	I-Algorithm
adaptation	I-Algorithm
from	O
spatial	O
to	O
spatio-temporal	O
image	O
data	O
.	O
</s>
