<s>
In	O
image	B-Algorithm
processing	I-Algorithm
,	O
computer	B-Application
vision	I-Application
and	O
related	O
fields	O
,	O
an	O
image	B-Algorithm
moment	I-Algorithm
is	O
a	O
certain	O
particular	O
weighted	O
average	O
(	O
moment	O
)	O
of	O
the	O
image	O
pixels	O
 '	O
intensities	O
,	O
or	O
a	O
function	O
of	O
such	O
moments	O
,	O
usually	O
chosen	O
to	O
have	O
some	O
attractive	O
property	O
or	O
interpretation	O
.	O
</s>
<s>
Image	B-Algorithm
moments	I-Algorithm
are	O
useful	O
to	O
describe	O
objects	O
after	O
segmentation	B-Algorithm
.	O
</s>
<s>
Simple	O
properties	O
of	O
the	O
image	O
which	O
are	O
found	O
via	O
image	B-Algorithm
moments	I-Algorithm
include	O
area	O
(	O
or	O
total	O
intensity	O
)	O
,	O
its	O
centroid	O
,	O
and	O
information	O
about	O
its	O
orientation	O
.	O
</s>
<s>
The	O
central	B-General_Concept
moments	I-General_Concept
of	O
order	O
up	O
to	O
3	O
are	O
:	O
</s>
<s>
Central	B-General_Concept
moments	I-General_Concept
are	O
translational	O
invariant	O
.	O
</s>
<s>
Information	O
about	O
image	O
orientation	O
can	O
be	O
derived	O
by	O
first	O
using	O
the	O
second	O
order	O
central	B-General_Concept
moments	I-General_Concept
to	O
construct	O
a	O
covariance	O
matrix	O
.	O
</s>
<s>
The	O
term	O
invariant	B-Algorithm
moments	I-Algorithm
is	O
often	O
abused	O
in	O
this	O
context	O
.	O
</s>
<s>
However	O
,	O
while	O
moment	B-Algorithm
invariants	I-Algorithm
are	O
invariants	O
that	O
are	O
formed	O
from	O
moments	O
,	O
the	O
only	O
moments	O
that	O
are	O
invariants	O
themselves	O
are	O
the	O
central	B-General_Concept
moments	I-General_Concept
.	O
</s>
<s>
The	O
central	B-General_Concept
moments	I-General_Concept
μi	O
j	O
of	O
any	O
order	O
are	O
,	O
by	O
construction	O
,	O
invariant	O
with	O
respect	O
to	O
translations	B-Algorithm
.	O
</s>
<s>
Invariants	O
ηi	O
j	O
with	O
respect	O
to	O
both	O
translation	B-Algorithm
and	O
scale	O
can	O
be	O
constructed	O
from	O
central	B-General_Concept
moments	I-General_Concept
by	O
dividing	O
through	O
a	O
properly	O
scaled	O
zero-th	O
central	B-General_Concept
moment	I-General_Concept
:	O
</s>
<s>
Note	O
that	O
translational	O
invariance	O
directly	O
follows	O
by	O
only	O
using	O
central	B-General_Concept
moments	I-General_Concept
.	O
</s>
<s>
invariants	O
with	O
respect	O
to	O
translation	B-Algorithm
,	O
scale	O
,	O
and	O
rotation	O
can	O
be	O
constructed	O
:	O
</s>
<s>
These	O
are	O
well-known	O
as	O
Hu	O
moment	B-Algorithm
invariants	I-Algorithm
.	O
</s>
<s>
A	O
general	O
theory	O
on	O
deriving	O
complete	O
and	O
independent	O
sets	O
of	O
rotation	O
moment	B-Algorithm
invariants	I-Algorithm
was	O
proposed	O
by	O
J	O
.	O
Flusser	O
.	O
</s>
<s>
He	O
showed	O
that	O
the	O
traditional	O
set	O
of	O
Hu	O
moment	B-Algorithm
invariants	I-Algorithm
is	O
neither	O
independent	O
nor	O
complete	O
.	O
</s>
<s>
In	O
the	O
original	O
Hu	O
's	O
set	O
there	O
is	O
a	O
missing	O
third	O
order	O
independent	O
moment	B-Algorithm
invariant	I-Algorithm
:	O
</s>
<s>
applied	O
Hu	O
moment	B-Algorithm
invariants	I-Algorithm
to	O
solve	O
the	O
Pathological	O
Brain	O
Detection	O
(	O
PBD	O
)	O
problem	O
.	O
</s>
<s>
Doerr	O
and	O
Florence	O
used	O
information	O
of	O
the	O
object	O
orientation	O
related	O
to	O
the	O
second	O
order	O
central	B-General_Concept
moments	I-General_Concept
to	O
effectively	O
extract	O
translation	B-Algorithm
-	O
and	O
rotation-invariant	O
object	O
cross-sections	O
from	O
micro-X-ray	O
tomography	O
image	O
data	O
.	O
</s>
