<s>
An	O
eigenface	B-General_Concept
(	O
)	O
is	O
the	O
name	O
given	O
to	O
a	O
set	O
of	O
eigenvectors	O
when	O
used	O
in	O
the	O
computer	B-Application
vision	I-Application
problem	O
of	O
human	O
face	O
recognition	O
.	O
</s>
<s>
The	O
approach	O
of	O
using	O
eigenfaces	B-General_Concept
for	O
recognition	O
was	O
developed	O
by	O
Sirovich	O
and	O
Kirby	O
and	O
used	O
by	O
Matthew	O
Turk	O
and	O
Alex	O
Pentland	O
in	O
face	O
classification	O
.	O
</s>
<s>
The	O
eigenvectors	O
are	O
derived	O
from	O
the	O
covariance	O
matrix	B-Architecture
of	O
the	O
probability	O
distribution	O
over	O
the	O
high-dimensional	O
vector	O
space	O
of	O
face	O
images	O
.	O
</s>
<s>
The	O
eigenfaces	B-General_Concept
themselves	O
form	O
a	O
basis	O
set	O
of	O
all	O
images	O
used	O
to	O
construct	O
the	O
covariance	O
matrix	B-Architecture
.	O
</s>
<s>
The	O
eigenface	B-General_Concept
approach	O
began	O
with	O
a	O
search	O
for	O
a	O
low-dimensional	O
representation	O
of	O
face	O
images	O
.	O
</s>
<s>
Sirovich	O
and	O
Kirby	O
showed	O
that	O
principal	B-Application
component	I-Application
analysis	I-Application
could	O
be	O
used	O
on	O
a	O
collection	O
of	O
face	O
images	O
to	O
form	O
a	O
set	O
of	O
basis	O
features	O
.	O
</s>
<s>
If	O
the	O
training	O
set	O
consists	O
of	O
M	O
images	O
,	O
principal	B-Application
component	I-Application
analysis	I-Application
could	O
form	O
a	O
basis	O
set	O
of	O
N	O
images	O
,	O
where	O
N	O
<	O
M	O
.	O
The	O
reconstruction	O
error	O
is	O
reduced	O
by	O
increasing	O
the	O
number	O
of	O
eigenpictures	O
;	O
however	O
,	O
the	O
number	O
needed	O
is	O
always	O
chosen	O
less	O
than	O
M	O
.	O
For	O
example	O
,	O
if	O
you	O
need	O
to	O
generate	O
a	O
number	O
of	O
N	O
eigenfaces	B-General_Concept
for	O
a	O
training	O
set	O
of	O
M	O
face	O
images	O
,	O
you	O
can	O
say	O
that	O
each	O
face	O
image	O
can	O
be	O
made	O
up	O
of	O
"	O
proportions	O
"	O
of	O
all	O
the	O
K	O
"	O
features	O
"	O
or	O
eigenfaces	B-General_Concept
:	O
Face	O
image1	O
=	O
(	O
23%	O
of	O
E1	O
)	O
+	O
(	O
2%	O
of	O
E2	O
)	O
+	O
(	O
51%	O
of	O
E3	O
)	O
+	O
...	O
+	O
(	O
1%	O
En	O
)	O
.	O
</s>
<s>
In	O
1991	O
M	O
.	O
Turk	O
and	O
A	O
.	O
Pentland	O
expanded	O
these	O
results	O
and	O
presented	O
the	O
eigenface	B-General_Concept
method	O
of	O
face	O
recognition	O
.	O
</s>
<s>
In	O
addition	O
to	O
designing	O
a	O
system	O
for	O
automated	O
face	O
recognition	O
using	O
eigenfaces	B-General_Concept
,	O
they	O
showed	O
a	O
way	O
of	O
calculating	O
the	O
eigenvectors	O
of	O
a	O
covariance	O
matrix	B-Architecture
such	O
that	O
computers	O
of	O
the	O
time	O
could	O
perform	O
eigen-decomposition	O
on	O
a	O
large	O
number	O
of	O
face	O
images	O
.	O
</s>
<s>
Face	O
images	O
usually	O
occupy	O
a	O
high-dimensional	O
space	O
and	O
conventional	O
principal	B-Application
component	I-Application
analysis	I-Application
was	O
intractable	O
on	O
such	O
data	O
sets	O
.	O
</s>
<s>
Turk	O
and	O
Pentland	O
's	O
paper	O
demonstrated	O
ways	O
to	O
extract	O
the	O
eigenvectors	O
based	O
on	O
matrices	O
sized	O
by	O
the	O
number	O
of	O
images	O
rather	O
than	O
the	O
number	O
of	O
pixels	B-Algorithm
.	O
</s>
<s>
Once	O
established	O
,	O
the	O
eigenface	B-General_Concept
method	O
was	O
expanded	O
to	O
include	O
methods	O
of	O
preprocessing	O
to	O
improve	O
accuracy	O
.	O
</s>
<s>
Multiple	O
manifold	O
approaches	O
were	O
also	O
used	O
to	O
build	O
sets	O
of	O
eigenfaces	B-General_Concept
for	O
different	O
subjects	O
and	O
different	O
features	O
,	O
such	O
as	O
the	O
eyes	O
.	O
</s>
<s>
A	O
set	O
of	O
eigenfaces	B-General_Concept
can	O
be	O
generated	O
by	O
performing	O
a	O
mathematical	O
process	O
called	O
principal	B-Application
component	I-Application
analysis	I-Application
(	O
PCA	O
)	O
on	O
a	O
large	O
set	O
of	O
images	O
depicting	O
different	O
human	O
faces	O
.	O
</s>
<s>
Informally	O
,	O
eigenfaces	B-General_Concept
can	O
be	O
considered	O
a	O
set	O
of	O
"	O
standardized	O
face	O
ingredients	O
"	O
,	O
derived	O
from	O
statistical	O
analysis	O
of	O
many	O
pictures	O
of	O
faces	O
.	O
</s>
<s>
For	O
example	O
,	O
one	O
's	O
face	O
might	O
be	O
composed	O
of	O
the	O
average	O
face	O
plus	O
10%	O
from	O
eigenface	B-General_Concept
1	O
,	O
55%	O
from	O
eigenface	B-General_Concept
2	O
,	O
and	O
even	O
−3%	O
from	O
eigenface	B-General_Concept
3	O
.	O
</s>
<s>
Remarkably	O
,	O
it	O
does	O
not	O
take	O
many	O
eigenfaces	B-General_Concept
combined	O
together	O
to	O
achieve	O
a	O
fair	O
approximation	O
of	O
most	O
faces	O
.	O
</s>
<s>
Also	O
,	O
because	O
a	O
person	O
's	O
face	O
is	O
not	O
recorded	O
by	O
a	O
digital	B-Application
photograph	I-Application
,	O
but	O
instead	O
as	O
just	O
a	O
list	O
of	O
values	O
(	O
one	O
value	O
for	O
each	O
eigenface	B-General_Concept
in	O
the	O
database	O
used	O
)	O
,	O
much	O
less	O
space	O
is	O
taken	O
for	O
each	O
person	O
's	O
face	O
.	O
</s>
<s>
The	O
eigenfaces	B-General_Concept
that	O
are	O
created	O
will	O
appear	O
as	O
light	O
and	O
dark	O
areas	O
that	O
are	O
arranged	O
in	O
a	O
specific	O
pattern	O
.	O
</s>
<s>
Other	O
eigenfaces	B-General_Concept
have	O
patterns	O
that	O
are	O
less	O
simple	O
to	O
identify	O
,	O
and	O
the	O
image	O
of	O
the	O
eigenface	B-General_Concept
may	O
look	O
very	O
little	O
like	O
a	O
face	O
.	O
</s>
<s>
The	O
technique	O
used	O
in	O
creating	O
eigenfaces	B-General_Concept
and	O
using	O
them	O
for	O
recognition	O
is	O
also	O
used	O
outside	O
of	O
face	O
recognition	O
:	O
handwriting	B-Application
recognition	I-Application
,	O
lip	O
reading	O
,	O
voice	B-Application
recognition	I-Application
,	O
sign	O
language/hand	O
gestures	O
interpretation	O
and	O
medical	B-Application
imaging	I-Application
analysis	O
.	O
</s>
<s>
Therefore	O
,	O
some	O
do	O
not	O
use	O
the	O
term	O
eigenface	B-General_Concept
,	O
but	O
prefer	O
to	O
use	O
'	O
eigenimage	B-Application
 '	O
.	O
</s>
<s>
To	O
create	O
a	O
set	O
of	O
eigenfaces	B-General_Concept
,	O
one	O
must	O
:	O
</s>
<s>
They	O
must	O
also	O
be	O
all	O
resampled	O
to	O
a	O
common	O
pixel	B-Algorithm
resolution	O
(	O
r	O
×	O
c	O
)	O
.	O
</s>
<s>
Each	O
image	O
is	O
treated	O
as	O
one	O
vector	O
,	O
simply	O
by	O
concatenating	O
the	O
rows	O
of	O
pixels	B-Algorithm
in	O
the	O
original	O
image	O
,	O
resulting	O
in	O
a	O
single	O
column	O
with	O
r	O
×	O
c	O
elements	O
.	O
</s>
<s>
For	O
this	O
implementation	O
,	O
it	O
is	O
assumed	O
that	O
all	O
images	O
of	O
the	O
training	O
set	O
are	O
stored	O
in	O
a	O
single	O
matrix	B-Architecture
T	O
,	O
where	O
each	O
column	O
of	O
the	O
matrix	B-Architecture
is	O
an	O
image	O
.	O
</s>
<s>
Calculate	O
the	O
eigenvectors	O
and	O
eigenvalues	O
of	O
the	O
covariance	O
matrix	B-Architecture
S	O
.	O
Each	O
eigenvector	O
has	O
the	O
same	O
dimensionality	O
(	O
number	O
of	O
components	O
)	O
as	O
the	O
original	O
images	O
,	O
and	O
thus	O
can	O
itself	O
be	O
seen	O
as	O
an	O
image	O
.	O
</s>
<s>
The	O
eigenvectors	O
of	O
this	O
covariance	O
matrix	B-Architecture
are	O
therefore	O
called	O
eigenfaces	B-General_Concept
.	O
</s>
<s>
Usually	O
this	O
will	O
be	O
a	O
computationally	O
expensive	O
step	O
(	O
if	O
at	O
all	O
possible	O
)	O
,	O
but	O
the	O
practical	O
applicability	O
of	O
eigenfaces	B-General_Concept
stems	O
from	O
the	O
possibility	O
to	O
compute	O
the	O
eigenvectors	O
of	O
S	O
efficiently	O
,	O
without	O
ever	O
computing	O
S	O
explicitly	O
,	O
as	O
detailed	O
below	O
.	O
</s>
<s>
Choose	O
the	O
principal	B-Application
components	I-Application
.	O
</s>
<s>
The	O
number	O
of	O
principal	B-Application
components	I-Application
k	O
is	O
determined	O
arbitrarily	O
by	O
setting	O
a	O
threshold	O
ε	O
on	O
the	O
total	O
variance	O
.	O
</s>
<s>
These	O
eigenfaces	B-General_Concept
can	O
now	O
be	O
used	O
to	O
represent	O
both	O
existing	O
and	O
new	O
faces	O
:	O
we	O
can	O
project	O
a	O
new	O
(	O
mean-subtracted	O
)	O
image	O
on	O
the	O
eigenfaces	B-General_Concept
and	O
thereby	O
record	O
how	O
that	O
new	O
face	O
differs	O
from	O
the	O
mean	O
face	O
.	O
</s>
<s>
The	O
eigenvalues	O
associated	O
with	O
each	O
eigenface	B-General_Concept
represent	O
how	O
much	O
the	O
images	O
in	O
the	O
training	O
set	O
vary	O
from	O
the	O
mean	O
image	O
in	O
that	O
direction	O
.	O
</s>
<s>
Information	O
is	O
lost	O
by	O
projecting	O
the	O
image	O
on	O
a	O
subset	O
of	O
the	O
eigenvectors	O
,	O
but	O
losses	O
are	O
minimized	O
by	O
keeping	O
those	O
eigenfaces	B-General_Concept
with	O
the	O
largest	O
eigenvalues	O
.	O
</s>
<s>
In	O
practical	O
applications	O
,	O
most	O
faces	O
can	O
typically	O
be	O
identified	O
using	O
a	O
projection	O
on	O
between	O
100	O
and	O
150	O
eigenfaces	B-General_Concept
,	O
so	O
that	O
most	O
of	O
the	O
10,000	O
eigenvectors	O
can	O
be	O
discarded	O
.	O
</s>
<s>
Here	O
is	O
an	O
example	O
of	O
calculating	O
eigenfaces	B-General_Concept
with	O
Extended	O
Yale	O
Face	O
Database	O
B	O
.	O
</s>
<s>
Note	O
that	O
although	O
the	O
covariance	O
matrix	B-Architecture
S	O
generates	O
many	O
eigenfaces	B-General_Concept
,	O
only	O
a	O
fraction	O
of	O
those	O
are	O
needed	O
to	O
represent	O
the	O
majority	O
of	O
the	O
faces	O
.	O
</s>
<s>
For	O
example	O
,	O
to	O
represent	O
95%	O
of	O
the	O
total	O
variation	O
of	O
all	O
face	O
images	O
,	O
only	O
the	O
first	O
43	O
eigenfaces	B-General_Concept
are	O
needed	O
.	O
</s>
<s>
Performing	O
PCA	O
directly	O
on	O
the	O
covariance	O
matrix	B-Architecture
of	O
the	O
images	O
is	O
often	O
computationally	O
infeasible	O
.	O
</s>
<s>
If	O
small	O
images	O
are	O
used	O
,	O
say	O
100×100	O
pixels	B-Algorithm
,	O
each	O
image	O
is	O
a	O
point	O
in	O
a	O
10,000	O
-dimensional	O
space	O
and	O
the	O
covariance	O
matrix	B-Architecture
S	O
is	O
a	O
matrix	B-Architecture
of	O
10,000	O
×	O
10,000	O
=	O
108	O
elements	O
.	O
</s>
<s>
However	O
the	O
rank	O
of	O
the	O
covariance	O
matrix	B-Architecture
is	O
limited	O
by	O
the	O
number	O
of	O
training	O
examples	O
:	O
if	O
there	O
are	O
N	O
training	O
examples	O
,	O
there	O
will	O
be	O
at	O
most	O
N−1	O
eigenvectors	O
with	O
non-zero	O
eigenvalues	O
.	O
</s>
<s>
If	O
the	O
number	O
of	O
training	O
examples	O
is	O
smaller	O
than	O
the	O
dimensionality	O
of	O
the	O
images	O
,	O
the	O
principal	B-Application
components	I-Application
can	O
be	O
computed	O
more	O
easily	O
as	O
follows	O
.	O
</s>
<s>
Let	O
T	O
be	O
the	O
matrix	B-Architecture
of	O
preprocessed	O
training	O
examples	O
,	O
where	O
each	O
column	O
contains	O
one	O
mean-subtracted	O
image	O
.	O
</s>
<s>
Meaning	O
that	O
,	O
if	O
ui	O
is	O
an	O
eigenvector	O
of	O
TTT	O
,	O
then	O
vi	O
=	O
Tui	O
is	O
an	O
eigenvector	O
of	O
S	O
.	O
If	O
we	O
have	O
a	O
training	O
set	O
of	O
300	O
images	O
of	O
100×100	O
pixels	B-Algorithm
,	O
the	O
matrix	B-Architecture
TTT	O
is	O
a	O
300×300	O
matrix	B-Architecture
,	O
which	O
is	O
much	O
more	O
manageable	O
than	O
the	O
10,000	O
×	O
10,000	O
covariance	O
matrix	B-Architecture
.	O
</s>
<s>
Let	O
denote	O
the	O
data	O
matrix	B-Architecture
with	O
column	O
as	O
the	O
image	O
vector	O
with	O
mean	O
subtracted	O
.	O
</s>
<s>
The	O
eigenfaces	B-General_Concept
=	O
the	O
first	O
(	O
)	O
columns	O
of	O
associated	O
with	O
the	O
nonzero	O
singular	O
values	O
.	O
</s>
<s>
Using	O
SVD	O
on	O
data	O
matrix	B-Architecture
,	O
it	O
is	O
unnecessary	O
to	O
calculate	O
the	O
actual	O
covariance	O
matrix	B-Architecture
to	O
get	O
eigenfaces	B-General_Concept
.	O
</s>
<s>
Facial	O
recognition	O
was	O
the	O
motivation	O
for	O
the	O
creation	O
of	O
eigenfaces	B-General_Concept
.	O
</s>
<s>
For	O
this	O
use	O
,	O
eigenfaces	B-General_Concept
have	O
advantages	O
over	O
other	O
techniques	O
available	O
,	O
such	O
as	O
the	O
system	O
's	O
speed	O
and	O
efficiency	O
.	O
</s>
<s>
As	O
eigenface	B-General_Concept
is	O
primarily	O
a	O
dimension	O
reduction	O
method	O
,	O
a	O
system	O
can	O
represent	O
many	O
subjects	O
with	O
a	O
relatively	O
small	O
set	O
of	O
data	O
.	O
</s>
<s>
To	O
recognise	O
faces	O
,	O
gallery	O
images	O
–	O
those	O
seen	O
by	O
the	O
system	O
–	O
are	O
saved	O
as	O
collections	O
of	O
weights	O
describing	O
the	O
contribution	O
each	O
eigenface	B-General_Concept
has	O
to	O
that	O
image	O
.	O
</s>
<s>
When	O
a	O
new	O
face	O
is	O
presented	O
to	O
the	O
system	O
for	O
classification	O
,	O
its	O
own	O
weights	O
are	O
found	O
by	O
projecting	O
the	O
image	O
onto	O
the	O
collection	O
of	O
eigenfaces	B-General_Concept
.	O
</s>
<s>
Intuitively	O
,	O
the	O
recognition	O
process	O
with	O
the	O
eigenface	B-General_Concept
method	O
is	O
to	O
project	O
query	O
images	O
into	O
the	O
face-space	O
spanned	O
by	O
eigenfaces	B-General_Concept
calculated	O
,	O
and	O
to	O
find	O
the	O
closest	O
match	O
to	O
a	O
face	O
class	O
in	O
that	O
face-space	O
.	O
</s>
<s>
Given	O
input	O
image	O
vector	O
,	O
the	O
mean	O
image	O
vector	O
from	O
the	O
database	O
,	O
calculate	O
the	O
weight	O
of	O
the	O
kth	O
eigenface	B-General_Concept
as	O
:	O
</s>
<s>
Experiments	O
in	O
the	O
original	O
Eigenface	B-General_Concept
paper	O
presented	O
the	O
following	O
results	O
:	O
an	O
average	O
of	O
96%	O
with	O
light	O
variation	O
,	O
85%	O
with	O
orientation	O
variation	O
,	O
and	O
64%	O
with	O
size	O
variation	O
.	O
</s>
<s>
Various	O
extensions	O
have	O
been	O
made	O
to	O
the	O
eigenface	B-General_Concept
method	O
.	O
</s>
<s>
The	O
eigenfeatures	O
method	O
combines	O
facial	O
metrics	O
(	O
measuring	O
distance	O
between	O
facial	O
features	O
)	O
with	O
the	O
eigenface	B-General_Concept
representation	O
.	O
</s>
<s>
Fisherface	O
uses	O
linear	B-General_Concept
discriminant	I-General_Concept
analysis	I-General_Concept
and	O
is	O
less	O
sensitive	O
to	O
variation	O
in	O
lighting	O
and	O
pose	O
of	O
the	O
face	O
.	O
</s>
<s>
A	O
further	O
alternative	O
to	O
eigenfaces	B-General_Concept
and	O
Fisherfaces	O
is	O
the	O
active	B-General_Concept
appearance	I-General_Concept
model	I-General_Concept
.	O
</s>
<s>
This	O
approach	O
uses	O
an	O
active	B-General_Concept
shape	I-General_Concept
model	I-General_Concept
to	O
describe	O
the	O
outline	O
of	O
a	O
face	O
.	O
</s>
<s>
By	O
collecting	O
many	O
face	O
outlines	O
,	O
principal	B-Application
component	I-Application
analysis	I-Application
can	O
be	O
used	O
to	O
form	O
a	O
basis	O
set	O
of	O
models	O
that	O
encapsulate	O
the	O
variation	O
of	O
different	O
faces	O
.	O
</s>
<s>
Many	O
modern	O
approaches	O
still	O
use	O
principal	B-Application
component	I-Application
analysis	I-Application
as	O
a	O
means	O
of	O
dimension	O
reduction	O
or	O
to	O
form	O
basis	O
images	O
for	O
different	O
modes	O
of	O
variation	O
.	O
</s>
<s>
Eigenface	B-General_Concept
provides	O
an	O
easy	O
and	O
cheap	O
way	O
to	O
realize	O
face	O
recognition	O
in	O
that	O
:	O
</s>
<s>
Eigenface	B-General_Concept
adequately	O
reduces	O
statistical	O
complexity	O
in	O
face	O
image	O
representation	O
.	O
</s>
<s>
Once	O
eigenfaces	B-General_Concept
of	O
a	O
database	O
are	O
calculated	O
,	O
face	O
recognition	O
can	O
be	O
achieved	O
in	O
real	O
time	O
.	O
</s>
<s>
Eigenface	B-General_Concept
can	O
handle	O
large	O
databases	O
.	O
</s>
<s>
However	O
,	O
the	O
deficiencies	O
of	O
the	O
eigenface	B-General_Concept
method	O
are	O
also	O
obvious	O
:	O
</s>
<s>
Eigenface	B-General_Concept
has	O
difficulty	O
capturing	O
expression	O
changes	O
.	O
</s>
<s>
The	O
most	O
significant	O
eigenfaces	B-General_Concept
are	O
mainly	O
about	O
illumination	O
encoding	O
and	O
do	O
not	O
provide	O
useful	O
information	O
regarding	O
the	O
actual	O
face	O
.	O
</s>
<s>
To	O
cope	O
with	O
illumination	O
distraction	O
in	O
practice	O
,	O
the	O
eigenface	B-General_Concept
method	O
usually	O
discards	O
the	O
first	O
three	O
eigenfaces	B-General_Concept
from	O
the	O
dataset	O
.	O
</s>
<s>
Since	O
illumination	O
is	O
usually	O
the	O
cause	O
behind	O
the	O
largest	O
variations	O
in	O
face	O
images	O
,	O
the	O
first	O
three	O
eigenfaces	B-General_Concept
will	O
mainly	O
capture	O
the	O
information	O
of	O
3-dimensional	O
lighting	O
changes	O
,	O
which	O
has	O
little	O
contribution	O
to	O
face	O
recognition	O
.	O
</s>
<s>
By	O
discarding	O
those	O
three	O
eigenfaces	B-General_Concept
,	O
there	O
will	O
be	O
a	O
decent	O
amount	O
of	O
boost	O
in	O
accuracy	O
of	O
face	O
recognition	O
,	O
but	O
other	O
methods	O
such	O
as	O
fisherface	O
and	O
linear	O
space	O
still	O
have	O
the	O
advantage	O
.	O
</s>
