<s>
Corner	B-Algorithm
detection	I-Algorithm
is	O
an	O
approach	O
used	O
within	O
computer	B-Application
vision	I-Application
systems	O
to	O
extract	O
certain	O
kinds	O
of	O
features	O
and	O
infer	O
the	O
contents	O
of	O
an	O
image	O
.	O
</s>
<s>
Corner	B-Algorithm
detection	I-Algorithm
is	O
frequently	O
used	O
in	O
motion	O
detection	O
,	O
image	B-Algorithm
registration	I-Algorithm
,	O
video	B-Operating_System
tracking	I-Operating_System
,	O
image	O
mosaicing	O
,	O
panorama	B-Algorithm
stitching	I-Algorithm
,	O
3D	B-Algorithm
reconstruction	I-Algorithm
and	O
object	O
recognition	O
.	O
</s>
<s>
Corner	B-Algorithm
detection	I-Algorithm
overlaps	O
with	O
the	O
topic	O
of	O
interest	O
point	O
detection	O
.	O
</s>
<s>
In	O
practice	O
,	O
most	O
so-called	O
corner	B-Algorithm
detection	I-Algorithm
methods	O
detect	O
interest	O
points	O
in	O
general	O
,	O
and	O
in	O
fact	O
,	O
the	O
term	O
"	O
corner	O
"	O
and	O
"	O
interest	O
point	O
"	O
are	O
used	O
more	O
or	O
less	O
interchangeably	O
through	O
the	O
literature	O
.	O
</s>
<s>
Examples	O
of	O
edge	B-Algorithm
detection	I-Algorithm
that	O
can	O
be	O
used	O
with	O
post-processing	O
to	O
detect	O
corners	O
are	O
the	O
Kirsch	B-Algorithm
operator	I-Algorithm
and	O
the	O
Frei-Chen	O
masking	O
set	O
.	O
</s>
<s>
Specifically	O
,	O
there	O
are	O
several	O
blob	B-Algorithm
detectors	I-Algorithm
that	O
can	O
be	O
referred	O
to	O
as	O
"	O
interest	O
point	O
operators	O
"	O
,	O
but	O
which	O
are	O
sometimes	O
erroneously	O
referred	O
to	O
as	O
"	O
corner	B-Algorithm
detectors	I-Algorithm
"	O
.	O
</s>
<s>
Corner	B-Algorithm
detectors	I-Algorithm
are	O
not	O
usually	O
very	O
robust	O
and	O
often	O
require	O
large	O
redundancies	O
introduced	O
to	O
prevent	O
the	O
effect	O
of	O
individual	O
errors	O
from	O
dominating	O
the	O
recognition	O
task	O
.	O
</s>
<s>
One	O
determination	O
of	O
the	O
quality	O
of	O
a	O
corner	B-Algorithm
detector	I-Algorithm
is	O
its	O
ability	O
to	O
detect	O
the	O
same	O
corner	O
in	O
multiple	O
similar	O
images	O
,	O
under	O
conditions	O
of	O
different	O
lighting	O
,	O
translation	O
,	O
rotation	O
and	O
other	O
transforms	O
.	O
</s>
<s>
A	O
simple	O
approach	O
to	O
corner	B-Algorithm
detection	I-Algorithm
in	O
images	O
is	O
using	O
correlation	O
,	O
but	O
this	O
gets	O
very	O
computationally	O
expensive	O
and	O
suboptimal	O
.	O
</s>
<s>
This	O
is	O
one	O
of	O
the	O
earliest	O
corner	B-Algorithm
detection	I-Algorithm
algorithms	O
and	O
defines	O
a	O
corner	O
to	O
be	O
a	O
point	O
with	O
low	O
self-similarity	O
.	O
</s>
<s>
Harris	O
and	O
Stephens	O
improved	O
upon	O
Moravec	O
's	O
corner	B-Algorithm
detector	I-Algorithm
by	O
considering	O
the	O
differential	O
of	O
the	O
corner	O
score	O
with	O
respect	O
to	O
direction	O
directly	O
,	O
instead	O
of	O
using	O
shifted	O
patches	O
.	O
</s>
<s>
where	O
A	O
is	O
the	O
structure	B-Algorithm
tensor	I-Algorithm
,	O
</s>
<s>
If	O
a	O
Box	B-Algorithm
filter	I-Algorithm
is	O
used	O
the	O
response	O
will	O
be	O
anisotropic	O
,	O
but	O
if	O
a	O
Gaussian	O
is	O
used	O
,	O
then	O
the	O
response	O
will	O
be	O
isotropic	O
.	O
</s>
<s>
The	O
Shi	O
–	O
Tomasi	O
corner	B-Algorithm
detector	I-Algorithm
directly	O
computes	O
because	O
under	O
certain	O
assumptions	O
,	O
the	O
corners	O
are	O
more	O
stable	O
for	O
tracking	O
.	O
</s>
<s>
Note	O
that	O
this	O
method	O
is	O
also	O
sometimes	O
referred	O
to	O
as	O
the	O
Kanade	O
–	O
Tomasi	O
corner	B-Algorithm
detector	I-Algorithm
.	O
</s>
<s>
If	O
can	O
be	O
interpreted	O
as	O
the	O
precision	B-General_Concept
matrix	I-General_Concept
for	O
the	O
corner	O
position	O
,	O
the	O
covariance	O
matrix	O
for	O
the	O
corner	O
position	O
is	O
,	O
i.e.	O
</s>
<s>
Note	O
that	O
is	O
the	O
structure	B-Algorithm
tensor	I-Algorithm
.	O
</s>
<s>
The	O
computation	O
of	O
the	O
second	O
moment	O
matrix	O
(	O
sometimes	O
also	O
referred	O
to	O
as	O
the	O
structure	B-Algorithm
tensor	I-Algorithm
)	O
in	O
the	O
Harris	O
operator	O
,	O
requires	O
the	O
computation	O
of	O
image	B-Algorithm
derivatives	I-Algorithm
in	O
the	O
image	O
domain	O
as	O
well	O
as	O
the	O
summation	O
of	O
non-linear	O
combinations	O
of	O
these	O
derivatives	B-Algorithm
over	O
local	O
neighbourhoods	O
.	O
</s>
<s>
Since	O
the	O
computation	O
of	O
derivatives	B-Algorithm
usually	O
involves	O
a	O
stage	O
of	O
scale-space	B-Algorithm
smoothing	O
,	O
an	O
operational	O
definition	O
of	O
the	O
Harris	O
operator	O
requires	O
two	O
scale	O
parameters	O
:	O
(	O
i	O
)	O
a	O
local	O
scale	O
for	O
smoothing	O
prior	O
to	O
the	O
computation	O
of	O
image	B-Algorithm
derivatives	I-Algorithm
,	O
and	O
(	O
ii	O
)	O
an	O
integration	O
scale	O
for	O
accumulating	O
the	O
non-linear	O
operations	O
on	O
derivative	O
operators	O
into	O
an	O
integrated	O
image	O
descriptor	O
.	O
</s>
<s>
and	O
let	O
and	O
denote	O
the	O
partial	O
derivatives	B-Algorithm
of	O
.	O
</s>
<s>
Thus	O
,	O
we	O
can	O
compute	O
the	O
multi-scale	O
Harris	O
corner	O
measure	O
at	O
any	O
scale	O
in	O
scale-space	B-Algorithm
to	O
obtain	O
a	O
multi-scale	O
corner	B-Algorithm
detector	I-Algorithm
,	O
which	O
responds	O
to	O
corner	O
structures	O
of	O
varying	O
sizes	O
in	O
the	O
image	O
domain	O
.	O
</s>
<s>
is	O
computed	O
at	O
every	O
scale	O
in	O
scale-space	B-Algorithm
and	O
scale	O
adapted	O
corner	O
points	O
with	O
automatic	O
scale	O
selection	O
(	O
the	O
"	O
Harris-Laplace	O
operator	O
"	O
)	O
are	O
computed	O
from	O
the	O
points	O
that	O
are	O
simultaneously	O
:	O
</s>
<s>
An	O
earlier	O
approach	O
to	O
corner	B-Algorithm
detection	I-Algorithm
is	O
to	O
detect	O
points	O
where	O
the	O
curvature	O
of	O
level	O
curves	O
and	O
the	O
gradient	O
magnitude	O
are	O
simultaneously	O
high	O
.	O
</s>
<s>
and	O
to	O
detect	O
positive	O
maxima	O
and	O
negative	O
minima	O
of	O
this	O
differential	O
expression	O
at	O
some	O
scale	O
in	O
the	O
scale	B-Algorithm
space	I-Algorithm
representation	I-Algorithm
of	O
the	O
original	O
image	O
.	O
</s>
<s>
This	O
approach	O
is	O
the	O
first	O
corner	B-Algorithm
detector	I-Algorithm
with	O
automatic	O
scale	O
selection	O
(	O
prior	O
to	O
the	O
"	O
Harris-Laplace	O
operator	O
"	O
above	O
)	O
and	O
has	O
been	O
used	O
for	O
tracking	O
corners	O
under	O
large	O
scale	O
variations	O
in	O
the	O
image	O
domain	O
and	O
for	O
matching	O
corner	O
responses	O
to	O
edges	O
to	O
compute	O
structural	O
image	O
features	O
for	O
geon-based	O
object	O
recognition	O
.	O
</s>
<s>
These	O
detectors	O
are	O
more	O
completely	O
described	O
in	O
blob	B-Algorithm
detection	I-Algorithm
.	O
</s>
<s>
To	O
improve	O
the	O
corner	B-Algorithm
detection	I-Algorithm
ability	O
of	O
the	O
differences	O
of	O
Gaussians	O
detector	O
,	O
the	O
feature	O
detector	O
used	O
in	O
the	O
SIFT	B-Algorithm
system	O
therefore	O
uses	O
an	O
additional	O
post-processing	O
stage	O
,	O
where	O
the	O
eigenvalues	O
of	O
the	O
Hessian	O
of	O
the	O
image	O
at	O
the	O
detection	O
scale	O
are	O
examined	O
in	O
a	O
similar	O
way	O
as	O
in	O
the	O
Harris	O
operator	O
.	O
</s>
<s>
The	O
scale	O
selection	O
properties	O
,	O
affine	O
transformation	O
properties	O
and	O
experimental	O
properties	O
of	O
these	O
and	O
other	O
scale-space	B-Algorithm
interest	O
point	O
detectors	O
are	O
analyzed	O
in	O
detail	O
in	O
(	O
Lindeberg	O
2013	O
,	O
2015	O
)	O
.	O
</s>
<s>
Inspired	O
by	O
the	O
structurally	O
similar	O
properties	O
of	O
the	O
Hessian	O
matrix	O
of	O
a	O
function	O
and	O
the	O
second-moment	B-Algorithm
matrix	I-Algorithm
(	O
structure	B-Algorithm
tensor	I-Algorithm
)	O
,	O
as	O
can	O
e.g.	O
</s>
<s>
Lindeberg	O
(	O
2013	O
,	O
2015	O
)	O
proposed	O
to	O
define	O
four	O
feature	O
strength	O
measures	O
from	O
the	O
Hessian	O
matrix	O
in	O
related	O
ways	O
as	O
the	O
Harris	O
and	O
Shi-and-Tomasi	O
operators	O
are	O
defined	O
from	O
the	O
structure	B-Algorithm
tensor	I-Algorithm
(	O
second-moment	B-Algorithm
matrix	I-Algorithm
)	O
.	O
</s>
<s>
denote	O
the	O
trace	O
and	O
the	O
determinant	O
of	O
the	O
Hessian	O
matrix	O
of	O
the	O
scale-space	B-Algorithm
representation	I-Algorithm
at	O
any	O
scale	O
,	O
</s>
<s>
By	O
experiments	O
on	O
image	B-Algorithm
matching	I-Algorithm
under	O
scaling	O
transformations	O
on	O
a	O
poster	O
dataset	O
with	O
12	O
posters	O
with	O
multi-view	O
matching	O
over	O
scaling	O
transformations	O
up	O
to	O
a	O
scaling	O
factor	O
of	O
6	O
and	O
viewing	O
direction	O
variations	O
up	O
to	O
a	O
slant	O
angle	O
of	O
45	O
degrees	O
with	O
local	O
image	O
descriptors	O
defined	O
from	O
reformulations	O
of	O
the	O
pure	O
image	O
descriptors	O
in	O
the	O
SIFT	B-Algorithm
and	O
SURF	B-Algorithm
operators	O
to	O
image	O
measurements	O
in	O
terms	O
of	O
Gaussian	O
derivative	O
operators	O
(	O
Gauss-SIFT	O
and	O
Gauss-SURF	O
)	O
instead	O
of	O
original	O
SIFT	B-Algorithm
as	O
defined	O
from	O
an	O
image	O
pyramid	O
or	O
original	O
SURF	B-Algorithm
as	O
defined	O
from	O
Haar	O
wavelets	O
,	O
it	O
was	O
shown	O
that	O
scale-space	B-Algorithm
interest	O
point	O
detection	O
based	O
on	O
the	O
unsigned	O
Hessian	O
feature	O
strength	O
measure	O
allowed	O
for	O
the	O
best	O
performance	O
and	O
better	O
performance	O
than	O
scale-space	B-Algorithm
interest	O
points	O
obtained	O
from	O
the	O
determinant	O
of	O
the	O
Hessian	O
.	O
</s>
<s>
Furthermore	O
,	O
it	O
was	O
shown	O
that	O
all	O
these	O
differential	O
scale-space	B-Algorithm
interest	O
point	O
detectors	O
defined	O
from	O
the	O
Hessian	O
matrix	O
allow	O
for	O
the	O
detection	O
of	O
a	O
larger	O
number	O
of	O
interest	O
points	O
and	O
better	O
matching	O
performance	O
compared	O
to	O
the	O
Harris	O
and	O
Shi-and-Tomasi	O
operators	O
defined	O
from	O
the	O
structure	B-Algorithm
tensor	I-Algorithm
(	O
second-moment	B-Algorithm
matrix	I-Algorithm
)	O
.	O
</s>
<s>
A	O
theoretical	O
analysis	O
of	O
the	O
scale	O
selection	O
properties	O
of	O
these	O
four	O
Hessian	O
feature	O
strength	O
measures	O
and	O
other	O
differential	O
entities	O
for	O
detecting	O
scale-space	B-Algorithm
interest	O
points	O
,	O
including	O
the	O
Laplacian	O
of	O
the	O
Gaussian	O
and	O
the	O
determinant	O
of	O
the	O
Hessian	O
,	O
is	O
given	O
in	O
Lindeberg	O
(	O
2013	O
)	O
and	O
an	O
analysis	O
of	O
their	O
affine	O
transformation	O
properties	O
as	O
well	O
as	O
experimental	O
properties	O
in	O
Lindeberg	O
(	O
2015	O
)	O
.	O
</s>
<s>
The	O
images	O
that	O
constitute	O
the	O
input	O
to	O
a	O
computer	B-Application
vision	I-Application
system	O
are	O
,	O
however	O
,	O
also	O
subject	O
to	O
perspective	O
distortions	O
.	O
</s>
<s>
In	O
practice	O
,	O
affine	O
invariant	O
interest	O
points	O
can	O
be	O
obtained	O
by	O
applying	O
affine	B-Algorithm
shape	I-Algorithm
adaptation	I-Algorithm
where	O
the	O
shape	O
of	O
the	O
smoothing	O
kernel	O
is	O
iteratively	O
warped	O
to	O
match	O
the	O
local	O
image	O
structure	O
around	O
the	O
interest	O
point	O
or	O
equivalently	O
a	O
local	O
image	O
patch	O
is	O
iteratively	O
warped	O
while	O
the	O
shape	O
of	O
the	O
smoothing	O
kernel	O
remains	O
rotationally	O
symmetric	O
(	O
Lindeberg	O
1993	O
,	O
2008	O
;	O
Lindeberg	O
and	O
Garding	O
1997	O
;	O
Mikolajzcyk	O
and	O
Schmid	O
2004	O
)	O
.	O
</s>
<s>
Hence	O
,	O
besides	O
the	O
commonly	O
used	O
multi-scale	O
Harris	O
operator	O
,	O
affine	B-Algorithm
shape	I-Algorithm
adaptation	I-Algorithm
can	O
be	O
applied	O
to	O
other	O
corner	B-Algorithm
detectors	I-Algorithm
as	O
listed	O
in	O
this	O
article	O
as	O
well	O
as	O
to	O
differential	B-Algorithm
blob	I-Algorithm
detectors	I-Algorithm
such	O
as	O
the	O
Laplacian/difference	O
of	O
Gaussian	O
operator	O
,	O
the	O
determinant	O
of	O
the	O
Hessian	O
and	O
the	O
Hessian	O
–	O
Laplace	O
operator	O
.	O
</s>
<s>
The	O
Wang	O
and	O
Brady	O
detector	O
considers	O
the	O
image	O
to	O
be	O
a	O
surface	O
,	O
and	O
looks	O
for	O
places	O
where	O
there	O
is	O
large	O
curvature	O
along	O
an	O
image	B-Algorithm
edge	I-Algorithm
.	O
</s>
<s>
If	O
is	O
large	O
enough	O
,	O
then	O
this	O
becomes	O
an	O
edge	B-Algorithm
detector	I-Algorithm
.	O
</s>
<s>
For	O
corner	B-Algorithm
detection	I-Algorithm
,	O
two	O
further	O
steps	O
are	O
used	O
.	O
</s>
<s>
is	O
a	O
discretised	O
circle	O
(	O
a	O
Bresenham	B-Algorithm
circle	I-Algorithm
)	O
,	O
so	O
interpolation	B-Algorithm
is	O
used	O
for	O
intermediate	O
diameters	O
to	O
give	O
a	O
more	O
isotropic	O
response	O
.	O
</s>
<s>
Instead	O
of	O
evaluating	O
the	O
circular	O
disc	O
,	O
only	O
the	O
pixels	O
in	O
a	O
Bresenham	B-Algorithm
circle	I-Algorithm
of	O
radius	O
around	O
the	O
candidate	O
point	O
are	O
considered	O
.	O
</s>
<s>
The	O
first	O
corner	B-Algorithm
detection	I-Algorithm
algorithm	O
based	O
on	O
the	O
AST	O
is	O
FAST	O
(	O
features	B-Algorithm
from	I-Algorithm
accelerated	I-Algorithm
segment	I-Algorithm
test	I-Algorithm
)	O
.	O
</s>
<s>
The	O
order	O
in	O
which	O
pixels	O
are	O
tested	O
is	O
determined	O
by	O
the	O
ID3	B-Algorithm
algorithm	I-Algorithm
from	O
a	O
training	O
set	O
of	O
images	O
.	O
</s>
<s>
Trujillo	O
and	O
Olague	O
introduced	O
a	O
method	O
by	O
which	O
genetic	B-Algorithm
programming	I-Algorithm
is	O
used	O
to	O
automatically	O
synthesize	O
image	O
operators	O
that	O
can	O
detect	O
interest	O
points	O
.	O
</s>
<s>
In	O
Lindeberg	O
,	O
it	O
was	O
shown	O
that	O
and	O
implies	O
better	O
scale	O
selection	O
properties	O
in	O
the	O
sense	O
that	O
the	O
selected	O
scale	O
levels	O
obtained	O
from	O
a	O
spatio-temporal	O
Gaussian	O
blob	O
with	O
spatial	O
extent	O
and	O
temporal	O
extent	O
will	O
perfectly	O
match	O
the	O
spatial	O
extent	O
and	O
the	O
temporal	O
duration	O
of	O
the	O
blob	O
,	O
with	O
scale	O
selection	O
performed	O
by	O
detecting	O
spatio-temporal	O
scale-space	B-Algorithm
extrema	O
of	O
the	O
differential	O
expression	O
.	O
</s>
