<s>
In	O
the	O
fields	O
of	O
computer	B-Application
vision	I-Application
and	O
image	B-General_Concept
analysis	I-General_Concept
,	O
the	O
Harris	B-Algorithm
affine	I-Algorithm
region	I-Algorithm
detector	I-Algorithm
belongs	O
to	O
the	O
category	O
of	O
feature	O
detection	O
.	O
</s>
<s>
The	O
Harris	O
affine	O
detector	O
can	O
identify	O
similar	O
regions	O
between	O
images	O
that	O
are	O
related	O
through	O
affine	B-Algorithm
transformations	I-Algorithm
and	O
have	O
different	O
illuminations	O
.	O
</s>
<s>
Other	O
detectors	O
that	O
are	O
affine-invariant	O
include	O
Hessian	B-Algorithm
affine	I-Algorithm
region	I-Algorithm
detector	I-Algorithm
,	O
Maximally	B-Algorithm
stable	I-Algorithm
extremal	I-Algorithm
regions	I-Algorithm
,	O
Kadir	B-Algorithm
–	I-Algorithm
Brady	I-Algorithm
saliency	I-Algorithm
detector	I-Algorithm
,	O
edge-based	O
regions	O
(	O
EBR	O
)	O
and	O
intensity-extrema-based	O
regions	O
(	O
IBR	O
)	O
.	O
</s>
<s>
Earlier	O
works	O
in	O
this	O
direction	O
include	O
use	O
of	O
affine	B-Algorithm
shape	I-Algorithm
adaptation	I-Algorithm
by	O
Lindeberg	O
and	O
Garding	O
for	O
computing	O
affine	O
invariant	O
image	O
descriptors	O
and	O
in	O
this	O
way	O
reducing	O
the	O
influence	O
of	O
perspective	O
image	O
deformations	O
,	O
the	O
use	O
affine	O
adapted	O
feature	O
points	O
for	O
wide	O
baseline	O
matching	O
by	O
Baumberg	O
and	O
the	O
first	O
use	O
of	O
scale	O
invariant	O
feature	O
points	O
by	O
Lindeberg	O
;	O
for	O
an	O
overview	O
of	O
the	O
theoretical	O
background	O
.	O
</s>
<s>
The	O
Harris	O
affine	O
detector	O
relies	O
on	O
the	O
combination	O
of	O
corner	O
points	O
detected	O
through	O
Harris	O
corner	B-Algorithm
detection	I-Algorithm
,	O
multi-scale	O
analysis	O
through	O
Gaussian	B-Algorithm
scale	I-Algorithm
space	I-Algorithm
and	O
affine	O
normalization	O
using	O
an	O
iterative	O
affine	B-Algorithm
shape	I-Algorithm
adaptation	I-Algorithm
algorithm	O
.	O
</s>
<s>
For	O
each	O
initial	O
point	O
,	O
normalize	O
the	O
region	O
to	O
be	O
affine	O
invariant	O
using	O
affine	B-Algorithm
shape	I-Algorithm
adaptation	I-Algorithm
.	O
</s>
<s>
The	O
Harris	O
affine	O
detector	O
relies	O
heavily	O
on	O
both	O
the	O
Harris	O
measure	O
and	O
a	O
Gaussian	B-Algorithm
scale	I-Algorithm
space	I-Algorithm
representation	O
.	O
</s>
<s>
For	O
a	O
more	O
exhaustive	O
derivations	O
see	O
corner	B-Algorithm
detection	I-Algorithm
and	O
Gaussian	B-Algorithm
scale	I-Algorithm
space	I-Algorithm
or	O
their	O
associated	O
papers	O
.	O
</s>
<s>
The	O
Harris	O
corner	B-Algorithm
detector	I-Algorithm
algorithm	O
relies	O
on	O
a	O
central	O
principle	O
:	O
at	O
a	O
corner	O
,	O
the	O
image	O
intensity	O
will	O
change	O
largely	O
in	O
multiple	O
directions	O
.	O
</s>
<s>
Following	O
this	O
intuition	O
and	O
through	O
a	O
clever	O
decomposition	O
,	O
the	O
Harris	O
detector	O
uses	O
the	O
second	B-Algorithm
moment	I-Algorithm
matrix	I-Algorithm
as	O
the	O
basis	O
of	O
its	O
corner	O
decisions	O
.	O
</s>
<s>
(	O
See	O
corner	B-Algorithm
detection	I-Algorithm
for	O
more	O
complete	O
derivation	O
)	O
.	O
</s>
<s>
The	O
matrix	O
,	O
has	O
also	O
been	O
called	O
the	O
autocorrelation	O
matrix	O
and	O
has	O
values	O
closely	O
related	O
to	O
the	O
derivatives	B-Algorithm
of	I-Algorithm
image	I-Algorithm
intensity	I-Algorithm
.	O
</s>
<s>
where	O
and	O
are	O
the	O
respective	O
derivatives	B-Algorithm
(	O
of	O
pixel	O
intensity	O
)	O
in	O
the	O
and	O
direction	O
at	O
point	O
(	O
,	O
)	O
;	O
and	O
are	O
the	O
position	O
parameters	O
of	O
the	O
weighting	O
function	O
w	O
.	O
The	O
off-diagonal	O
entries	O
are	O
the	O
product	O
of	O
and	O
,	O
while	O
the	O
diagonal	O
entries	O
are	O
squares	O
of	O
the	O
respective	O
derivatives	B-Algorithm
.	O
</s>
<s>
A	O
Gaussian	B-Algorithm
scale	I-Algorithm
space	I-Algorithm
representation	O
of	O
an	O
image	O
is	O
the	O
set	O
of	O
images	O
that	O
result	O
from	O
convolving	B-Language
a	O
Gaussian	O
kernel	O
of	O
various	O
sizes	O
with	O
the	O
original	O
image	O
.	O
</s>
<s>
The	O
convolution	B-Language
with	O
a	O
Gaussian	O
kernel	O
smooths	O
the	O
image	O
using	O
a	O
window	O
the	O
size	O
of	O
the	O
kernel	O
.	O
</s>
<s>
Mikolajczyk	O
and	O
Schmid	O
(	O
2001	O
)	O
point	O
out	O
that	O
derivatives	B-Algorithm
and	O
other	O
measurements	O
must	O
be	O
normalized	O
across	O
scales	O
.	O
</s>
<s>
These	O
derivatives	B-Algorithm
,	O
or	O
any	O
arbitrary	O
measure	O
,	O
can	O
be	O
adapted	O
to	O
a	O
scale	B-Algorithm
space	I-Algorithm
representation	I-Algorithm
by	O
calculating	O
this	O
measure	O
using	O
a	O
set	O
of	O
scales	O
recursively	O
where	O
the	O
th	O
scale	O
is	O
.	O
</s>
<s>
See	O
scale	B-Algorithm
space	I-Algorithm
for	O
a	O
more	O
complete	O
description	O
.	O
</s>
<s>
The	O
Harris-Laplace	O
detector	O
combines	O
the	O
traditional	O
2D	O
Harris	O
corner	B-Algorithm
detector	I-Algorithm
with	O
the	O
idea	O
of	O
a	O
Gaussian	B-Algorithm
scale	I-Algorithm
space	I-Algorithm
representation	O
in	O
order	O
to	O
create	O
a	O
scale-invariant	O
detector	O
.	O
</s>
<s>
However	O
,	O
the	O
points	O
are	O
not	O
scale	O
invariant	O
and	O
thus	O
the	O
second-moment	B-Algorithm
matrix	I-Algorithm
must	O
be	O
modified	O
to	O
reflect	O
a	O
scale-invariant	O
property	O
.	O
</s>
<s>
Let	O
us	O
denote	O
,	O
as	O
the	O
scale	O
adapted	O
second-moment	B-Algorithm
matrix	I-Algorithm
used	O
in	O
the	O
Harris-Laplace	O
detector	O
.	O
</s>
<s>
The	O
operator	O
denotes	O
convolution	B-Language
.	O
</s>
<s>
and	O
are	O
the	O
derivatives	B-Algorithm
in	O
their	O
respective	O
direction	O
applied	O
to	O
the	O
smoothed	O
image	O
and	O
calculated	O
using	O
a	O
Gaussian	O
kernel	O
with	O
scale	O
.	O
</s>
<s>
In	O
terms	O
of	O
our	O
Gaussian	O
scale-space	B-Algorithm
framework	O
,	O
the	O
parameter	O
determines	O
the	O
current	O
scale	O
at	O
which	O
the	O
Harris	O
corner	O
points	O
are	O
detected	O
.	O
</s>
<s>
Building	O
upon	O
this	O
scale-adapted	O
second-moment	B-Algorithm
matrix	I-Algorithm
,	O
the	O
Harris-Laplace	O
detector	O
is	O
a	O
twofold	O
process	O
:	O
applying	O
the	O
Harris	O
corner	B-Algorithm
detector	I-Algorithm
at	O
multiple	O
scales	O
and	O
automatically	O
choosing	O
the	O
characteristic	O
scale	O
.	O
</s>
<s>
The	O
neighboring	O
scales	O
are	O
typically	O
chosen	O
from	O
a	O
range	O
that	O
is	O
within	O
a	O
two	O
scale-space	B-Algorithm
neighborhood	O
.	O
</s>
<s>
That	O
is	O
,	O
if	O
the	O
original	O
points	O
were	O
detected	O
using	O
a	O
scaling	O
factor	O
of	O
between	O
successive	O
scales	O
,	O
a	O
two	O
scale-space	B-Algorithm
neighborhood	O
is	O
the	O
range	O
.	O
</s>
<s>
where	O
and	O
are	O
the	O
second	O
derivatives	B-Algorithm
in	O
their	O
respective	O
directions	O
.	O
</s>
<s>
The	O
factor	O
(	O
as	O
discussed	O
above	O
in	O
Gaussian	O
scale-space	B-Algorithm
)	O
is	O
used	O
to	O
normalize	O
the	O
LoG	O
across	O
scales	O
and	O
make	O
these	O
measures	O
comparable	O
,	O
thus	O
making	O
a	O
maximum	O
relevant	O
.	O
</s>
<s>
The	O
scale	O
which	O
maximizes	O
this	O
LoG	O
measure	O
in	O
the	O
two	O
scale-space	B-Algorithm
neighborhood	O
is	O
deemed	O
the	O
characteristic	O
scale	O
,	O
,	O
and	O
used	O
in	O
subsequent	O
iterations	O
.	O
</s>
<s>
In	O
order	O
to	O
be	O
invariant	O
to	O
arbitrary	O
affine	B-Algorithm
transformations	I-Algorithm
(	O
and	O
viewpoints	O
)	O
,	O
the	O
mathematical	O
framework	O
must	O
be	O
revisited	O
.	O
</s>
<s>
The	O
second-moment	B-Algorithm
matrix	I-Algorithm
is	O
defined	O
more	O
generally	O
for	O
anisotropic	O
regions	O
:	O
</s>
<s>
Although	O
this	O
may	O
look	O
significantly	O
different	O
from	O
the	O
second-moment	B-Algorithm
matrix	I-Algorithm
in	O
the	O
Harris-Laplace	O
detector	O
;	O
it	O
is	O
in	O
fact	O
,	O
identical	O
.	O
</s>
<s>
The	O
goal	O
of	O
the	O
affine	O
invariant	O
detector	O
is	O
to	O
identify	O
regions	O
in	O
images	O
that	O
are	O
related	O
through	O
affine	B-Algorithm
transformations	I-Algorithm
.	O
</s>
<s>
We	O
thus	O
consider	O
a	O
point	O
and	O
the	O
transformed	O
point	O
,	O
where	O
A	O
is	O
an	O
affine	B-Algorithm
transformation	I-Algorithm
.	O
</s>
<s>
One	O
of	O
these	O
properties	O
is	O
that	O
the	O
square	O
root	O
of	O
the	O
second-moment	B-Algorithm
matrix	I-Algorithm
,	O
will	O
transform	O
the	O
original	O
anisotropic	O
region	O
into	O
isotropic	O
regions	O
that	O
are	O
related	O
simply	O
through	O
a	O
pure	O
rotation	O
matrix	O
.	O
</s>
<s>
The	O
rotation	O
matrix	O
can	O
be	O
recovered	O
using	O
gradient	O
methods	O
likes	O
those	O
in	O
the	O
SIFT	B-Algorithm
descriptor	O
.	O
</s>
<s>
As	O
discussed	O
with	O
the	O
Harris	O
detector	O
,	O
the	O
eigenvalues	O
and	O
eigenvectors	O
of	O
the	O
second-moment	B-Algorithm
matrix	I-Algorithm
,	O
characterize	O
the	O
curvature	O
and	O
shape	O
of	O
the	O
pixel	O
intensities	O
.	O
</s>
<s>
Using	O
this	O
mathematical	O
framework	O
,	O
the	O
Harris	O
affine	O
detector	O
algorithm	O
iteratively	O
discovers	O
the	O
second-moment	B-Algorithm
matrix	I-Algorithm
that	O
transforms	O
the	O
anisotropic	O
region	O
into	O
a	O
normalized	O
region	O
in	O
which	O
the	O
isotropic	O
measure	O
is	O
sufficiently	O
close	O
to	O
one	O
.	O
</s>
<s>
The	O
second-moment	B-Algorithm
matrix	I-Algorithm
is	O
computed	O
in	O
this	O
normalized	O
reference	O
frame	O
and	O
should	O
have	O
an	O
isotropic	O
measure	O
close	O
to	O
one	O
at	O
the	O
final	O
iteration	O
.	O
</s>
<s>
Because	O
the	O
detector	O
computes	O
the	O
second-moment	B-Algorithm
matrix	I-Algorithm
in	O
the	O
transformed	O
domain	O
,	O
it	O
's	O
convenient	O
to	O
denote	O
this	O
transformed	O
position	O
as	O
where	O
.	O
</s>
<s>
The	O
computational	O
complexity	O
of	O
the	O
Harris-Affine	B-Algorithm
detector	O
is	O
broken	O
into	O
two	O
parts	O
:	O
initial	O
point	O
detection	O
and	O
affine	O
region	O
normalization	O
.	O
</s>
<s>
For	O
a	O
pair	O
of	O
two	O
images	O
related	O
through	O
a	O
homography	B-Algorithm
matrix	O
,	O
two	O
points	O
,	O
and	O
are	O
said	O
to	O
correspond	O
if	O
:	O
</s>
<s>
(	O
2005	O
)	O
have	O
done	O
a	O
thorough	O
analysis	O
of	O
several	O
state-of-the-art	O
affine	O
region	O
detectors	O
:	O
Harris	O
affine	O
,	O
Hessian	B-Algorithm
affine	I-Algorithm
,	O
MSER	B-Algorithm
,	O
IBR	O
&	O
EBR	O
and	O
salient	B-Algorithm
detectors	O
.	O
</s>
<s>
This	O
should	O
be	O
expected	O
because	O
the	O
detectors	O
rely	O
heavily	O
on	O
relative	O
intensities	O
(	O
derivatives	B-Algorithm
)	O
and	O
not	O
absolute	O
intensities	O
.	O
</s>
<s>
Both	O
the	O
Harris-Affine	B-Algorithm
detector	O
and	O
Hessian-Affine	B-Algorithm
consistently	O
identify	O
double	O
the	O
number	O
repeatable	O
points	O
as	O
other	O
affine	O
detectors	O
:	O
~	O
1000	O
regions	O
for	O
an	O
800x640	O
image	O
.	O
</s>
<s>
However	O
,	O
for	O
some	O
structured	O
scenes	O
,	O
like	O
buildings	O
,	O
the	O
Harris-Affine	B-Algorithm
detector	O
performs	O
very	O
well	O
.	O
</s>
<s>
This	O
is	O
complementary	O
to	O
MSER	B-Algorithm
that	O
tends	O
to	O
do	O
better	O
with	O
well	O
structured	O
(	O
segmentable	O
)	O
scenes	O
.	O
</s>
<s>
Overall	O
the	O
Harris	O
affine	O
detector	O
performs	O
very	O
well	O
,	O
but	O
still	O
behind	O
MSER	B-Algorithm
and	O
Hessian-Affine	B-Algorithm
in	O
all	O
cases	O
but	O
blurred	O
images	O
.	O
</s>
<s>
Harris-Affine	B-Algorithm
and	O
Hessian-Affine	B-Algorithm
detectors	O
are	O
less	O
accurate	O
than	O
others	O
:	O
their	O
repeatability	O
score	O
increases	O
as	O
the	O
overlap	O
threshold	O
is	O
increased	O
.	O
</s>
<s>
:	O
K	O
.	O
Mikolajczyk	O
maintains	O
a	O
web	O
page	O
that	O
contains	O
Linux	O
binaries	O
of	O
the	O
Harris-Affine	B-Algorithm
detector	O
in	O
addition	O
to	O
other	O
detectors	O
and	O
descriptors	O
.	O
</s>
